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--git a/09E5T4oBgHgl3EQfqw9o/content/tmp_files/2301.05710v1.pdf.txt b/09E5T4oBgHgl3EQfqw9o/content/tmp_files/2301.05710v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33c92a3d6ffadb3e84cd84c193d6d28d43a31d97 --- /dev/null +++ b/09E5T4oBgHgl3EQfqw9o/content/tmp_files/2301.05710v1.pdf.txt @@ -0,0 +1,2148 @@ +arXiv:2301.05710v1 [gr-qc] 13 Jan 2023 +Perusing Buchbinder–Lyakhovich canonical formalism for Higher-Order +Theories of Gravity. +Dalia Saha†, Abhik Kumar Sanyal‡ +January 18, 2023 +† Dept. of Physics, University of Kalyani, West Bengal, India - 741235. +†,‡ Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213. +Abstract +Ostrogradsky’s, Dirac’s and Horowitz’s techniques of higher order theories of gravity produce identical +phase-space structures. The problem is manifested in the case of Gauss-Bonnet-dilatonic coupled action in the +presence of higher-order term, in which case, classical correspondence can’t be established. Here, we explore yet +another technique developed by Buchbinder and his collaborators (BL) long back and show that it also suffers +from the same disease. However, expressing the action in terms of the three-space curvature, and removing “the +total derivative terms”, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies +disappear. Here we show that the same is true for BL formalism, which appears to be the simplest of all the +techniques, to handle. +Keywords: Higher Order theory; Canonical Formulation. +1 +Introduction +Canonical formulation of higher-order theories was developed by Ostrogradsky almost two centuries back [1, 2]. +However, it did not draw much attention, since other than toy mechanical models, practically no such physical +theories were persuaded at that time. Exactly a century elapsed, when it was applied to a physically motivated +problem, such as fourth order harmonic oscillator [3]. The real physical problem in this context appeared for +the first time, while a renormalized quantum theory of gravity was attempted to formulate [4]. Higher-derivative +theory of gravity is usually considered as a model of quantum gravity. The reason being, Einstein-Hilbert ac- +tion is supplemented by curvature squared terms (R2, RµνRµν ) to ensure renormalizability [4] and asymptotic +freedom [5–7]. Unfortunately, curvature-squared gravity theories have been found to suffer from the unresolved +problem of physical unitarity in perturbative analysis, which is usual for higher-derivative theories. However, +possibilities to overcome this difficulty were also discussed in some literatures [6, 8] and references therein. It is +also ascertained that curvature squared gravity would arise as a low-energy effective theory derived from super- +string theory in D = 10 dimensions [9–11]. Over the last couple of decades, higher order theories of gravity e.g., +F(R), F(G), F(R, T ) etc, theories, (R, G, T being the Ricci scalar, the Gauss-Bonnet term, and the torsion +term respectively) have drawn much attention in search of alternatives to dark energy. Nonetheless, it is always +suggestive to test viability of such modified theories of gravity in different contexts. In the context of the very +early universe, a canonical formulation is required as a precursor, particularly to study quantum cosmology. +Since Ostrogradsky’s technique does not apply in the degenerate case of singular Lagrangian, for which the +Hessian determinant vanishes, Dirac’s constraint analysis [12] may be applied for the purpose. Nonetheless, a host +of theories have been formulated over decades to bypass the constraint analysis. One of these in this direction +was originally proposed by Boulware [13], and later reformulated by Horowitz’ [14], in particular in the context of +higher-order theory of gravity. Since the canonical formulation of higher order theories requires an extra degree +of freedom, in Horowitz’s formalism apart from the scale factor (‘a′ in the Robertson-Walker minisuperspace) an +auxiliary variable is introduced by taking derivative of the action (say A) with respect to the highest derivative of +1Electronic address: +† daliasahamandal1983@gmail.com +‡ sanyal ak@yahoo.com +1 + +the field variable present (Q = ∂A +∂¨a ). In the end, the auxiliary variable is replaced by the basic variable (extrinsic +curvature tensor) through a canonical transformation. The important finding in this regard is as follows: all the +three formalisms, viz, Ostrogradsky’s (once degeneracy has been removed), Dirac’s and Horowitz’s formalisms, +produce an identical phase-space structure [15]. Meanwhile, certain pathologies with Horowitz’ formalism have +been identified. For example, it was noticed that Horowitz’s formalism can even be applied in the case of linear +gravity theory (Einstein-Hilbert action) leading to wrong quantum dynamics [16–18], as well as some superfluous +total derivative terms are eliminated [18, 19], which neither may be obtained from the variational principle, nor +having any connection with Gibbons-Hawking–York term [20,21], nor any of its modified versions, associated with +higher-order gravity. Further, the coupling parameter, in the case of the “non-minimally coupled scalar tensor the- +ory of gravity associated with higher order term”, has not been found to play any particular role, since its derivative +does not appear in the Hamiltonian [22]. The same is true for the “Dilatonic coupled Gauss-Bonnet-theory in +the presence of higher order term”, where additionally, the classical correspondence with quantum counterpart, +could not be established [22]. In view of such an uncanny situation, yet another technique was developed, called +the “modified Horowitz’s formalism” (MHF), which was successfully applied to different modified higher-order +theories of gravity, to explore the evolution of the very early universe [15,17–19,22–32]. In the MHF, the action +is expressed in terms of the three-space curvature (instead of the scale factor), “the total derivative terms” are +removed by integrating the action by parts, and Horowitz’s formalism (the introduction of the auxiliary variable +etc.) was followed, thereafter. +To be very specific, let us consider the following isotropic and homogeneous Robertson–Walker (RW) metric: +ds2 = −N 2(t) dt2 + a2(t) +� +dr2 +1 − kr2 + r2(dθ2 + sin2θdφ2) +� +, +(1) +for which the degeneracy in the Lagrangian disappears if the gauge (N ) is fixed a priori, in which case, Ostrograd- +sky’s technique applies as well. Once such degeneracy is removed, it is observed that Ostrogradsky’s technique +produces the same Hamiltonian, obtained following Horowitz’s as well as Dirac’s formalism [15]. Therefore, it +certainly follows that both the Ostrogradsky’s and Dirac’s formalism implicitly suffer from the same problem, +in disguise, as was noticed in Horowitz’s technique, as discussed above. Therefore in the MHF, instead of the +scale factor, the action is expressed in terms of the basic variable hij — the three space metric from the very +beginning—so that redundant total derivative terms do not appear [18, 19]. Thereafter, all the total derivative +terms are integrated out by parts, which become cancelled by the supplementary boundary (Gibbons–Hawking— +York and modified Gibbons–Hawking—York) terms. Subsequently, the auxiliary variable is introduced following +Horowitz’s proposal. In the end, the auxiliary variable is replaced by the other basic variable Kij — the extrinsic +curvature tensor. In this process, the unwanted problems that appeared following Horowitz’s formalism disappear, +while it produces a different Hamiltonian altogether. We mention that although both Hamiltonians (obtained +following the MHF and Ostrogradky’s, Dirac’s and Horowitz’s formalisms) are related through the canonical +transformation, they indeed produce different dynamics in the quantum domain. It is also important to mention +that it is not possible to carry over the classical canonical transformations to the quantum domain for higher-order +theories, due to the non-linearity. The MHF leads to an effective Hermitian Hamiltonian, a standard quantum +mechanical probabilistic interpretation, and a viable semiclassical treatment, which exhibit oscillation of the wave +function about the classical de-Sitter solution. As a result, the classical correspondence is established. In this +regard, the MHF may be considered as the most-viable technique to handle the higher-order theories. It has later +been established that, if the action is expressed in terms of the three-space metric (hij ) from the very beginning +and the total derivative terms are addressed, Dirac’s constraint analysis [12] also produces the Hamiltonian iden- +tical to that of the MHF [22,28–30]. +Amongst other techniques, Hawking-Luttrell technique [33] has limited application, since conformal trans- +formation is not possible in general [19], Schmidt’s technique [34] is identical to the Horowitz’s formalism in +disguise [17]. However, there is yet another technique developed in the 80’s by Buchbinder and his collabora- +tors [35–39], which did not receive much attention. Querella [40] only noticed that although at a first glance, the +general formalism developed by Buchbinder and his collaborators (BL) appears to be satisfactory, nevertheless it +has pitfalls. BL formalism is our current concern. Here, we test this abstract theoretical settings of BL formalism +in simple minisuperspace model to explore the pitfall, if any. The underlying essence of this formalism is to bypass +Dirac’s constrained analysis, very much like Horowitz’s technique, but instead of introducing auxiliary variable, +here the program is initiated with the basic variables {hij, Kij}, the three-space curvature and the extrinsic +curvature tensors respectively from the very beginning. In our present attempt to explore the outcome of this +2 + +technique, we discover that the formalism leads to identical phase space structure as was found in the case of +Ostrogradsky’s/Dirac’s/Horowitz’s formalism. +This paper is organized as follows. In the following section, we study scalar tensor theory of gravity (both the +minimal and non-minimal cases), and Gauss-Bonnet-Dilatonic coupled action being supplemented by the scalar +curvature squared (R2 ) term, following BL formalism. In Section ??, we explore the fact that once total derivative +terms are taken care of, the Hamiltonian does not differ from MHF. Section ?? discusses its physical application, +in connection with some earlier work. Section ??, concludes our work. +2 +BL Formalism in Three Different Higher Order Theories +In view of the very importance of higher-order curvature invariant terms required to construct a renormalizable +quantum theory of gravity when the curvature is extremely strong, a unique canonical formulation of the Einstein– +Hilbert action being supplemented by higher-order curvature invariant terms, is therefore necessary. Here, we shall +consider three different cases, minimally and non-minimally coupled scalar-tensor theory of gravity supplemented +by R2 term, and the scalar-tensor theory of gravity being supplemented by R2 and Gauss-Bonnet terms. In the +Robertson-Walker minisuperspace (1) under consideration, the Ricci scalar and the Gauss-Bonnet terms are +R = +6 +N 2 +� +¨a +a + ˙a2 +a2 + N 2 k +a2 − +˙N ˙a +Na +� +. +(2) +G = R2 − 4RµνRµν + RαβµνRαβµν = +24 +N 3a3 +� +N¨a − ˙N ˙a +� � ˙a2 +N 2 + k +� +. +(3) +respectively. For the sake of comparison with earlier results, we express actions in terms of the three space metric, +instead of the scale factor, as its importance has been mentioned already, and will be explicitly shown at the +beginning of Section ??. Since construction of higher-order theory to its canonical form requires an additional +degree of freedom, hence, in addition to the three-space metric hij , the extrinsic curvature tensor Kij is treated +as basic variable, as already stated. We therefore choose the basic variables hij = zδij = a2δij , so that Kij = +− +˙hij +2N = − a ˙a +N δij = − +˙z +2N δij . In terms of z = a2, the Ricci scalar and the Gauss-Bonnet terms take the following +forms, +R = +6 +N 2 +� +¨z +2z + N 2 k +z − 1 +2 +˙N ˙z +Nz +� +, +(4) +G = 12 +N 2 +� +¨z +z − ˙z2 +2z2 − +˙N ˙z +Nz +� � +˙z2 +4N 2z2 + k +z +� +, +(5) +respectively. It is noteworthy that since, +RµνRµν = 12 +N 4 +� +¨a2 +a2 + ˙a2¨a +a3 + ˙a4 +a4 − 2 +˙N ˙a¨a +Na2 − +˙N ˙a3 +Na3 + +˙N 2 ˙a2 +N 2a2 + k N 2¨a +a3 ++ 2k N 2 ˙a2 +a4 +− k N ˙N ˙a +a3 ++ k2 N 4 +a4 +� +, +(6) +therefore, +RµνRµν − 1 +3R2 = − +� 12 +Na3 +� d +dt +�1 +3 +˙a3 +N 3 + k ˙a +N +� +, +(7) +and as a result, +� � +RµνRµν − 1 +3R2 +� √−gd4x = −12C +� � d +dt +�1 +3 +˙a3 +N 3 + k ˙a +N +�� +dt +(8) +3 + +is a total derivative term. Thus, RµνRµν term is redundant in RW metric, once R2 term is taken (the constant C +appears due to the integration of the three space). Hence, to scrutinize the BL formalism presented by Buchbinder +and his collaborators in RW minisuperspace model (1), we consider scalar-tensor theories of gravity and also +Gauss-Bonnet-Dilatonic coupled gravity theory, being associated with scalar curvature squared term R2. +2.1 +Minimal coupling: +Let us start with the following minimally coupled case, +A1 = +� √−g +� +αR + βR2 − 1 +2φ,µφ,ν − V (φ) +� +d4x + αΣR + βΣR2. +(9) +In the above, α = +1 +16πG , β is a constant coupling parameter, αΣR = 2α +� +∂V K +√ +hd3x is the Gibbons- +Hawking–York boundary term [21] associated with Einstein–Hilbert sector of the above action, and βΣR2 = +4β +� +∂V RK +√ +hd3x is its modified version corresponding to R2 term, while, K is the trace of the extrinsic curva- +ture tensor Kij . Note that, both the counter terms are required under the condition δR = 0, at the boundary. +Instead, if the condition δKij = 0 is chosen at the boundary, the counter terms are not required, as in the case +of Horowitz’s formalism [14], since both the boundary terms appearing under metric variation vanish. However, +in the case of Ostrogradsky’s technique [1] and Dirac constrained analysis [12], boundary terms are not taken +care of. This is true for BL formalism too, as we shall see shortly. Nevertheless, the modified Horowitz’s for- +malism [17–19, 23–25] fixes δhij = 0 = δR at the boundary, and hence requires supplementary boundary terms. +We have demonstrated earlier that proper attention to all the boundary terms is paid in modified Horowitz’s +formalism (MHF). As a result, it presents a different phase space structure of the Hamiltonian for a particular +action being supplemented by higher-order terms. Nonetheless, it is related to the others under a suitable set +of canonical transformation [15]. Although, as mentioned, such transformations cannot be carried over in the +quantum domain, due to non-linearity. So, it’s indeed required to check if BL formalism also produces the same. +The action (9) in the RW minisuperspace model (1) may be written in terms of the basic variable hij = zδij , as +A1 = +� � +3α√z +� ¨z +N − +˙N ˙z +N 2 + 2kN +� ++ 9β +√z +� ¨z2 +N 3 − 2 ˙N ˙z¨z +N 4 ++ +˙N 2 ˙z2 +N 5 +− 4k ˙N ˙z +N 2 ++ 4k¨z +N + 4k2N +� ++ z +3 +2 +� ˙φ2 +2N − V N +�� +dt + αΣR + βΣR2. +(10) +The (0 +0) component of the field equation in terms of the scale factor ‘a′ takes the following form +6α +a2 +� ˙a2 +N 2 + k +� ++ 36β +a2N 4 +� +2˙a...a − 2˙a2 ¨N +N − ¨a2 − 4˙a¨a +˙N +N + 2˙a2 ¨a +a + 5˙a2 ˙N 2 +N 2 − 2 ˙a3 ˙N +aN +− 3 ˙a4 +a2 − 2kN 2 ˙a2 +a2 + k2N 4 +a2 +� +− +� ˙φ2 +2N 2 + V +� += 0, +(11) +which contains term upto third derivative. This is the energy constraint equation (E = 0), and when expressed +in terms of the phase space variables, becomes the Hamiltonian constraint equation, (due to diffeomorphic invari- +ance) of the theory under consideration. This we aim at, following the formalism presented by Buchbinder and +his collaborators (BL). +The action (9) has already been expressed in terms of the basic variable {hij}, instead of the scale factor. +Canonical formulation of higher order theories requires additional degree of freedom, and the only choice is the +extrinsic curvature tensor {Kij}. +In contrast to Horowitz’s formalism, where apart from {hij} an auxiliary +variable is introduced and at the end the Hamiltonian is expressed in terms of the basic variables {hij, Kij}, +in BL formalism, these basic variables are associated from the very beginning. In Robertson-Walker metric, the +extrinsic curvature tensor is expressed as, +Kij = − +˙hij +2N = −2a˙a +2N δij = − ˙z +2N δij = −qij say. +(12) +4 + +Since there is only one independent component, so instead of qij , the new generalized coordinate is chosen to be +its trace viz, +q = 3 ˙z +2N , +i.e. qij = q +3δij. +(13) +To express the action in terms of velocities, we choose, +v ≡ ˙q, vφ ≡ ˙φ. +(14) +The scalar curvature (4) therefore takes the following form, +R = 2 ˙q +Nz + 6k +z ≡ Rq = +2 +Nz (v + 3Nk), +(15) +and action (10) can be expressed as, +A1q = +� � +2α√z(v + 3kN) + +4β +N√z (v + 3kN)2 + z +3 +2 +� +v2 +φ +2N − NV +�� +dt, +(16) +while the Lagrangian density is, +L1q = 2α√z(v + 3kN) + +4β +N√z (v + 3kN)2 + z +3 +2 +� +v2 +φ +2N − NV +� +. +(17) +Note that the boundary terms remain intact in the action as well as in the point Lagrangian. Canonical momenta +are +pq = ∂Lq +∂v = 2α√z + +8β +N√z (v + 3kN), pN = ∂L1q +∂vN += 0, pz = ∂Lq +∂vz += 0 and pφ = ∂Lq +∂vφ += z +3 +2 vφ +N +. +(18) +Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0. Therefore, Dirac constraint analysis +appears to be essential. However, here is a wonderful twist in the BL formalism. For example, one can express +the modified Lagrangian density as, +L∗ +1 = L1q + pq ( ˙q − v) + pN +� +˙N − vN +� ++ pz +� +˙z − 2Nq +3 +� ++ pφ +� +˙φ − vφ +� +, +(19) +and equivalently, the Hamiltonian density as, +H∗ +1 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗ +1 = pqv + pNvN + pφvφ + pz +2Nq +3 +− L1q. +(20) +As a consequence, one can immediately find that the primary constraint D ≡ pz ≈ 0 disappears. Further, since +N is a non-dynamical Lagrange multiplier, hence the constraint C vanishes strongly. Therefore, one arrives at, +H∗ +1 = pqv + CvN + pφvφ + pz +2qN +3 +− L1q = CvN + pqv + pφvφ + pz +2qN +3 +− L1q = CvN + NHm +BL, +(21) +where, +NHm +BL = pqv + pφvφ + 2N +3 qpz − L1q += pqv + pφvφ + 2N +3 qpz − 2α√z(v + 3kN) − +4β +N√z (v + 3kN)2 − z +3 +2 +� +v2 +φ +2N + NV +� +. +(22) +5 + +In the above, m in the superscript stands for minimally coupled theory. Now upon substituting v and vφ from +the definition of momentum (18), we obtain, +NHm +BL = N +�2q +3 pz + +√z +16β p2 +q − +�αz +4β + 3k +� +pq + α2 +4β z +3 +2 + +1 +2z +3 +2 p2 +φ + V z +3 +2 +� +, +(23) +so that the canonical Hamiltonian finally reads as, +Hm +BL = 2q +3 pz + +√z +16β p2 +q − +�αz +4β + 3k +� +pq + α2 +4β z +3 +2 + +1 +2z +3 +2 p2 +φ + V z +3 +2 . +(24) +The action (10) may also be cast in the canonical form, +A1q = +� � +˙zpz + ˙qpq + ˙φpφ − NHBL +� +dt d3x += +� � +˙hijπij + ˙KijΠij + ˙φpφ − NHBL +� +dt d3x, +(25) +where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. For the sake of comparison, +let us make the following canonical transformation: +q → 3 +2x; +pq → 2 +3px, +(26) +to express the above Hamiltonian (24) as: +Hm +BL = xpz + +√z +36β p2 +x − +�αz +6β + 2k +� +px + α2z +3 +2 +4β ++ +p2 +φ +2z +3 +2 + V z +3 +2 . +(27) +It is revealed that the above Hamiltonian (27) is exactly the one obtained earlier, following Ostrogradsky, Dirac +as well as Horowitz’s formalisms [15]. Note that, very much like the Ostrogradsky’s and Dirac’s formalisms here +also, once the formalism is initiated, i.e. (R is expressed in terms of {hij, Kij} (15) as well as the action (16) +and the point Lagrangian (17)), there remains no option to integrate the action by parts. As a result, even the +Gibbons-Hawking–York term [20,21] which is physically meaningful, being associated with the entropy of the black +hole, along with its higher-order counterpart, also remain obscure. On the contrary, following Modified Horowitz’s +Formalism (MHF), where boundary terms are taken care of, we earlier obtained [15] +Hm +MHF = xpz + +√z +36β p2 +x + 3αx2 +2√z − 18βkx2 +z +3 +2 +− 36βk2 +√z +− 6kα√z + +p2 +φ +2z +3 +2 + V z +3 +2 . +(28) +Although the two (27) and (28) exactly match under the following set of canonical transformations, +pz → pz − 18kβx +z +3 +2 ++ 3αx +2√z , +z → z, +px → px + 36kβ +√z − 3α√z, +x → x, +pφ → pφ, +φ → φ. +and apparently there is no contradiction between the two, note the essential difference: linear term in the mo- +mentum (px ), which is very much present in (27), remains absent from the Hamiltonian (28). As a result, the +two Hamiltonians (27) and (28) induce completely different quantum dynamics, since in the quantum domain, as +mentioned, canonical transformation cannot be carried over due to non-linearity. +6 + +2.2 +Non-minimally coupled case +We find that the two different Hamiltonians (27) and (28) render two different quantum descriptions of the +same classical model. Although, some of the essential features (Gibbons-Hawking-York term and its higher order +counterpart) are absent from the Hamiltonian (27), it is not clear, which one gives correct quantum description +of the theory. Further, there may exist a unitary transformation (we have not found it though) relating the two +Hamiltonian operators. Therefore, to inspect the situation more deeply, we consider the non-minimally coupled +case next, whose action +A2 = +� √−g d4x +� +f(φ)R + βR2 − 1 +2φ,µφ,ν − V (φ) +� ++ f(φ)ΣR + BΣR2, +(29) +may be expressed in the RW metric (1) as +A2 = +� � +3f(φ)√z +� ¨z +N − +˙N ˙z +N 2 + 2kN +� ++ 9β +√z +� ¨z2 +N 3 − 2 ˙N ˙z¨z +N 4 ++ +˙N 2 ˙z2 +N 5 +− 4k ˙N ˙z +N 2 ++ 4k¨z +N + 4k2N +� ++ z +3 +2 +� ˙φ2 +2N − V N +�� +dt + f(φ)ΣR + BΣR2, +(30) +where, as already mentioned, the supplementary boundary terms are required when MHF is taken into account. In +the above, we consider an arbitrary functional coupling parameter f(φ). Pursuing the same procedure as above, +one finally arrives at the following Hamiltonian: +HnmBL = xpz + +√z +36β p2 +x − +� +f(φ) z +6β + 2k +� +px + f 2(φ)z +3 +2 +4β + +p2 +φ +2z +3 +2 + V z +3 +2 , +(31) +which is again identical to the one found following Dirac formalism and may be found following Ostrogradsky’s +and Horowitz’s techniques as well [22]. In the superscript nm stands for non-minimal coupling. The action (30) +may also be cast in the canonical form as in (25). On the contrary, following MHF, one finds [22] +HnmMHF = xpz + +√z +36β p2 +x + 3f(φ) +� x2 +2√z − 2k√z +� +− 18kβ +√z +�x2 +z + 2k +� ++ +p2 +φ +2z +3 +2 ++ 3xf ′(φ)pφ +z ++ 9f ′(φ)2x2 +2√z ++ V z +3 +2 . +(32) +However, under the following set of canonical transformations, +pz → pz − 18βkx +z +3 +2 ++ 3f(φ)x +2√z , +z → z, +px → px + 36β k +√z + 3f(φ)√z, +x → x, +pφ → pφ + 3f ′(φ)x√z, +φ → φ, +the two Hamiltonians (31) and (32) match again [22]. Nevertheless, here the difference is predominant and explicit. +Note that f ′(φ) term does not appear in (31), while it is coupled to pφ in (32). This coupled (f ′(φ)pφ ) term +requires operator ordering in the quantum domain, which is different for different form of f(φ). Hence even if the +two Hamiltonians are related through unitary transformation, such transformation would be different for different +form of f(φ). +7 + +2.3 +Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher order term +Although, it is clear that two different quantum descriptions follow from the same classical action using different +techniques, it is still abstruse to select the correct description. Therefore, we next consider Einstein-Gauss-Bonnet- +Dilatonic coupled action in the presence of higher order curvature invariant term. Gauss-Bonnet (GB) term arises +quite naturally as the leading order of the α′ expansion of heterotic superstring theory, where α′ is the inverse +string tension [41–46]. Several interesting features of GB term have been explored in the past and appear in +the literature [47–67]. However, Gauss–Bonnet term is topological invariant in 4-dimensions, and so to get its +contribution in the field equations, a dynamic dilatonic scalar coupling is required. It is worth mentioning that, +in string induced gravity near initial singularity, GB coupling with scalar field plays a very crucial role for the +occurrence of nonsingular cosmology [68,69]. The particular hallmark of GB term is the fact that, despite being +formed from a combination of higher order curvature invariant terms (G = R2 − 4RµνRµν + RαβµνRαβµν) (3), +it ends up only with second order field equations, avoiding Ostrogradsky’s instability, and equivalently, ghost +degrees of freedom. Nonetheless, such a wonderful feature ultimately leads to a serious pathology of ‘Branched +Hamiltonian’, which has no unique resolution till date [70–72]. +Nevertheless, it has been revealed that, the +pathology may be bypassed upon supplementing the action with higher order curvature invariant term [24, 25]. +We therefore consider the following action, +A3 = +� √−g d4x +� +αR + βR2 + γ(φ)G − 1 +2φ,µφ,ν − V (φ) +� ++ αΣR + βΣR2 + γ(φ)ΣG. +(33) +In the above, Gauss-Bonnet term G , is coupled with γ(φ), while V (φ) is the dilatonic potential. Further, the +symbol K stands for K = K3 − 3KKijKij + 2KijKikKk +j , where, K is the trace of the extrinsic curvature +tensor Kij , and γ(φ)ΣG = 4γ(φ) +� +∂V +� +2GijKij + K +3 +� √ +hd3x is the supplementary boundary term associated with +Gauss-Bonnet sector. The (0 +0) component of the Einstein’s field equation in terms of the scale factor here reads +as, +6α +a2 +� ˙a2 +N 2 + k +� ++ 36β +a2N 4 +� +2˙a...a − 2˙a2 ¨N +N − ¨a2 − 4˙a¨a +˙N +N + 2˙a2 ¨a +a + 5˙a2 ˙N 2 +N 2 − 2 ˙a3 ˙N +aN +− 3 ˙a4 +a2 − 2kN 2 ˙a2 +a2 + k2N 4 +a2 +� ++ 24γ′ ˙a ˙φ +N 2a3 +� ˙a2 +N 2 + k +� +− +� ˙φ2 +2N 2 + V +� += 0. +(34) +The action (33) in terms of the basic variable (hij = a2δij = zδij ) may be expressed as, +A3 = +� � +3α√z +� +¨z +N − +˙N ˙z +N 2 + 2kN +� ++ 9β +√z +� +¨z2 +N 3 − 2 ˙N ˙z¨z +N 4 ++ +˙N 2 ˙z2 +N 5 +− 4k ˙N ˙z +N 2 ++ 4k¨z +N + 4k2N +� ++ 3γ(φ) +N√z +� +˙z2¨z +N 2z + 4k¨z − +˙z4 +2N 2z2 − +˙N ˙z3 +N 3z − 2k ˙z2 +z +− 4k ˙N ˙z +N +� ++ z +3 +2 +� 1 +2N +˙φ2 − V N +� � +dt ++ αΣR + βΣR2 + γ(φ)ΣG, +(35) +where the additional supplementary boundary term γ(φ)ΣG = −γ(φ) +˙z +N√z +� +˙z2 +N 2z + 12k +� +, is required in the case of +MHF. Inserting the other basic variable (Kij = − q +3δij ) and considering ˙q = v (13), the action (35) finally may +be expressed as, +A3q = +� � +2α√z(v + 3kN) + +4β +N√z (v + 3kN)2 + 8γ(φ) +√z +�vq2 +9z − q4N +27z2 + kv − kNq2 +3z +� ++ z +3 +2 +� +v2 +φ +2N − NV +� � +dt + αΣR + βΣR2 + Λ(φ)ΣG. +(36) +Thus, the Lagrangian density takes the following form, +L3q = 2α√z(v + 3kN) + +4β +N√z (v + 3kN)2 + 8γ(φ) +√z +�vq2 +9z − q4N +27z2 + kv − kNq2 +3z +� ++ z +3 +2 +� 1 +2N v2 +φ − V N +� +, +(37) +8 + +where, boundary terms are not taken care of. The canonical momenta are +pq = ∂Lq +∂v = 2α√z + +8β +N√z (v + 3kN) + 8γ(φ) +√z +� q2 +9z + k +� +, +pN = ∂L3q +∂vN += 0, +pφ = ∂Lq +∂vφ += z +3 +2 vφ +N +, +and, +pz = ∂L3q +∂vz += 0. +(38) +Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0, which are usually handled by Dirac +constraint analysis. However, as mentioned, such analysis is not at all required in the BL formalism. For example, +one can express the modified Lagrangian density as, +L∗ +3 = L3q + pq ( ˙q − v) + pN +� +˙N − vN +� ++ pz +� +˙z − 2Nq +3 +� ++ pφ +� +˙φ − vφ +� +, +(39) +so that the corresponding Hamiltonian density takes the following form, +H∗ +3 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗ +3 = pqv + pNvN + pφvφ + 2Nq +3 +pz − L3q. +(40) +As a result, the primary constraint D ≡ pz ≈ 0 disappears and one obtains, +H∗ +3 = pqv + CvN + pφvφ + pz +2qN +3 +− L3q = CvN + +� +pqv + pφvφ + 2qN +3 +pz − L3q +� += CvN + NHGB +BL. (41) +In the superscript GB stands for Hamiltonian in connection with Einstein-Gauss-Bonnet-Dilatonic coupling. +Note that the constraint C ≡ pN strongly vanishes, since the lapse function N is simply a Lagrange multiplier. +Therefore, +NHGBBL = pqv + pφvφ + 2qN +3 +pz − L3q += pqv + pφvφ + 2qN +3 +pz − 2α√z(v + 3kN) − +4β +N√z (v + 3kN)2 +− 8γ(φ) +√z +�vq2 +9z − q4N +27z2 + kv − kNq2 +3z +� +− z +3 +2 +� 1 +2N v2 +φ − V N +� +. +(42) +Now upon substituting v from the definition of momentum (38), one obtains, +NHGBBL =N +�2qpz +3 ++ +√zp2 +q +16β − pq +�αz +4β + 3k +� ++ α2z +3 +2 +4β +− pq +� γq2 +9βz + γk +β +� ++ 2αγ +β +� q2 +9√z + k√z +� ++ 4γq4 +27z +3 +2 +� γ +3β + 2 +� ++ 8γkq2 +3z +3 +2 +� γ +3β + 2 +� ++ 12γk2 +√z +� γ +3β + 2 +� ++ +p2 +φ +2z +3 +2 + V z +3 +2 +� +. +(43) +The canonical Hamiltonian therefore finally reads as, +HGB +BL =2qpz +3 ++ +√zp2 +q +16β − pq +�αz +4β + 3k +� ++ α2z +3 +2 +4β +− pq +� γq2 +9βz + γk +β +� ++ 2αγ +β +� q2 +9√z + k√z +� ++ 4γq4 +27z +3 +2 +� γ +3β + 2 +� ++ 8γkq2 +3z +3 +2 +� γ +3β + 2 +� ++ 12γk2 +√z +� γ +3β + 2 +� ++ +p2 +φ +2z +3 +2 + V z +3 +2 . +(44) +Again, for the sake of comparison, let us make the canonical transformation q → 3 +2x; pq → 2 +3px (26), to express +the above Hamiltonian (44) in the following form, +HGBBL = xpz + +√zp2 +x +36β + α2z +3 +2 +4β +− +�αz +6β + γx2 +6βz + 2kγ +3β + 2k +� +px + +p2 +φ +2z +3 +2 + +� γ2 +4βz +5 +2 + 3γ +2z +5 +2 +� +x4 ++ +� αγ +2β√z + 12kγ +z +3 +2 ++ 2kγ2 +βz +3 +2 +� +x2 + 2αkγ√z +β ++ 24k2γ +√z ++ 4k2γ2 +β√z + V z +3 +2 , +(45) +9 + +and notice that, it is similar to the one already found, following Dirac formalism and may be found following +Ostrogradsky’s and Horowitz’s techniques as well [22]. The action (35) may also be cast in the canonical form +with respect to the basic variables as, +A3q = +� � +˙zpz + ˙qpq + ˙φvφ − NHBL +� +dt d3x = +� � +˙hijπij + ˙KijΠij + ˙φvφ − NHMHF +� +dt d3x, +(46) +where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. Hence, everything appears +to be consistent. On the contrary, although following MHF, we found [22] +HGB +MHF = xpz + +√zp2 +x +36β + 3α +� x2 +2√z − 2k√z +� +− 18kβ +√z +�x2 +z + 2k +� ++ +� x6 +2z +9 +2 + 12kx4 +z +7 +2 ++ 72k2x2 +z +5 +2 +� +γ′2 ++ +�x3 +z3 + 12kx +z2 +� +γ′pφ + +p2 +φ +2Z +3 +2 + V z +3 +2 , +(47) +nonetheless, under the following set of canonical transformations, +pz → pz − 18βkx +z +3 +2 ++ 3αx +2√z − 6kγ(φ)x +z +3 +2 +− 3γ(φ)x3 +2z +5 +2 +, +z → z, +px → px + 36β k +√z + 3α√z + 3γ(φ)x2 +z +3 +2 ++ 12kΛ +√z , +x → x, +pφ → pφ − γ′(φ)x3 +z +3 +2 +− 12kγ′(φ)x +√z +, +φ → φ, +(48) +the two Hamiltonians (45) and (47) match again [22]. +Apparently therefore, there is absolutely no problem. +Nevertheless note that, the Hamiltonian (47) contains a term (γ′(φ)pφ ), which is absent from (45). Now, during +canonical quantization the presence of this term requires operator ordering, which is different for different form +of γ(φ). As a result, even if the two may be related through unitary transformation, such transformation would +be different for different form of γ(φ). Thus, there does not exist a unique unitary transformation. In a nutshell, +we repeat that the two Hamiltonians (45) and (47) induce two different descriptions in the quantum domain, and +apparently, there is no way to choose one to be the correct. +3 +The role of divergent terms: +The very first important point to mention is, in all the formalisms the scale factor is treated as the basic variable, +while we initiate our program treating three three-space curvature, instead. To explain the reason behind this +choice, let us consider curvature squared action, A = +� +βR2d4x, as an example. Under variation, it gives a total +derivative term σ = −4β +� +RK +√ +h d3x, as mentioned earlier, where K is the trace of the extrinsic curvature +tensor Kij . A counter term (−σ), known by the name modified Gibbons-Hawking–York term [20, 21], must be +added to the action in case, instead of δ ˙q, δR is kept fixed at the boundary, as in MHF. In the RW (1) metric +under consideration, the action reads as, +A = 36β +� � +a¨a2 + 2˙a2¨a + 2k¨a + ˙a4 +a + 2k ˙a2 +a ++ k2 +a +� +dt +� +d3x. +(49) +Under integration by parts, we end up with, +A = C +� � +a¨a2 + ˙a4 +a + 2k ˙a2 +a ++ k2 +a +� +dt + C +�2 +3 ˙a3 + 2k ˙a +� +. +(50) +where, C = 36β +� +d3x. Now following Horowitz’s program, we introduce an auxiliary variable Q = ∂A +∂¨a = 2Ca¨a, +judiciously into the action in the following manner, such that it may be cast in canonical form, +A = +� � +Q¨a − Q2 +4Ca + C +� ˙a4 +a + 2k ˙a2 +a ++ k2 +a +�� +dt + C +�2 +3 ˙a3 + 2k ˙a +� +. +(51) +10 + +Integrating the action again by parts we find +A = +� +− ˙Q˙a − Q2 +4Ca + C +� ˙a4 +a + 2k ˙a2 +a ++ k2 +a +�� ++ C +�Q˙a +C + 2 +3 ˙a3 + 2k ˙a +� +. +(52) +The action is canonical, since the Hessian determinant is non-zero. It is trivial to check that the above action +gives correct field equations, but the left out total derivative term may be expressed as, +σ′ = −4β +� +RK +√ +h d3x + 16β +� +K +√ +h +� ˙a2 +a2 +� +d3x, +(53) +and as a result σ ̸= σ′ . Thus some redundant total derivative terms are pulled out in the process, which has severe +consequence in the quantum domain. On the contrary, if we start with, z = a2, the action reads as, +A = C +� � ¨z2 +4√z + k¨z +√z + k2 +√z +� +dt = C +� � ¨z2 +4√z + +2 +√z +� ++ C k ˙z +√z , +(54) +where the last expression is found under integration by parts. Now following Horowitz’s program, we find the +auxiliary variable as Q = ∂A +∂¨z = C +¨z +2√z , which is again judiciously introduced in the action as, +A = +� � +Q¨z − +√zQ2 +C ++ C +� k¨z +√z + k2 +√z +�� +dt + C k ˙z +√z . +(55) +Finally, performing integration by parts again, one obtains, +A = +� � +− ˙Q ˙z − +√zQ2 +C ++ C +� k¨z +√z + k2 +√z +�� +dt + C +�Q ˙z +C + k ˙z +√z +� +, +(56) +The action is again canonical, the Euler-Lagrange equations here again lead to the appropriate field equations, +while one can express the total derivative term as σ. In a nut-shell, although total derivative terms do not affect +the classical field equations, for non-linear theories such as gravity, such terms tell upon the quantum dynam- +ics. Therefore, to establish consistency in every respect, hij should be treated as the basic variable, instead of +the scale factor. This is essentially the so-called MHF, which finally requires to replace the auxiliary variable +by the the second basic variable, viz., the extrinsic curvature tensor Kij = −a˙a = − ˙z = x(say), in the Hamiltonian. +Next, we observe that the phase-space structures obtained following BL formalism although are identical to +the Ostrogradsky/Dirac/Horowitz’s formalism, they all differ from the MHF upto a canonical transformation. We +quote from [22] the general argument in connection with the total derivative terms, which runs as; “it is just +the change of the variables in the wave function and the phase transformation, plus the change of the integra- +tion measure, and the transformation of the momenta respecting the change of the measure, and so a unitary +transformation relates the two”. It’s possible (we have not found though) that each pair of quantum equations +cast from {(27) and (28)}; {(31) and (32)}; {(45) and (47)}, are related by unitary transformation. However, +it was also mentioned [22] that different forms of coupling parameter yield different quantum dynamics in the +case of MHF, due to the presence of a coupling term (f ′(φ)pφ ) for non-minimal coupled case, and (γ′(φ)pφ ) for +the Gauss-Bonnet-Dilaton coupled case, in the Hamiltonian. Thus, different unitary transformations (if exist) +are required to relate the last two pairs. Such coupling as well as the derivative of coupling parameter remain +absent in other formalisms. In a nutshell, unitary transformation relating each pair is not unique. Further, the +semiclassical wave functions found for all the three cases studied here, exhibit different pre-factors and exponents +for each pair [22]. This generates different probability amplitude and the evolution of the wave function while +entering the classical domain. +Finally, it is important to note that, if the coupling parameter f(φ) is treated as constant in Subsection ??, +the Hamiltonian (32) merely reduces to (28), while the Hamiltonian (31) reduces to (27). Hence the question is: +which of the two should be treated as the correct quantum description of the models under consideration? In +this connection we mention that a serious problem arises with Ostrogradsky/Dirac/Horowitz as well as with BL +11 + +formalisms when considering Gauss-Bonnet-Dilaton induced action. To be specific, in Subsection ?? if γ(φ) is +treated as a constant, then the contribution of Gauss-Bonnet term disappears from the Hamiltonian (47), and +it reduces to (28). +Indeed, it should since as mentioned, Gauss-Bonnet term is topologically invariant in 4- +dimensions, and so without functional coupling, it does not contribute to the field equations and the Hamiltonian +as well. On the contrary, a constant γ does not affect the form of the Hamiltonian (45), and it does not reduce +to (27). This means, if we had started with a constant γ from the very beginning, all the terms appearing with +γ in (45) would have been absent, and the end result would be (27). While, after constructing the Hamiltonian +with arbitrary γ = γ(φ), if we set it equal to a constant, then its contribution remains present, and we obtain +a different Hamiltonian, altogether. Clearly this is wrong. Hence, we realize that boundary terms indeed play +a crucial role while constructing the phase-space structure of non-linear theories. In fact, if boundary terms are +taken into account from the very beginning, treating hij as the basic variable, then Horowitz’s formalism reduces +to the MHF, as already demonstrated. It was also noticed that if Dirac algorithm is applied after integrating the +action by parts, then it also yields Hamiltonian identical to MHF [22]. It is therefore suggestive to test the same +for BL formalism too. In this section we shall first integrate actions by parts to get rid of the total derivative +terms and follow the BL formalism thereafter, to explore the outcome. +3.1 +Scalar-tensor theory: minimal coupling; +Upon integrating the action (30) by parts, we obtain +A1 = +�  +− 3α ˙z2 +2N√z + 6αkN√z + +9β +N√z + + + +� +¨z +N − +˙N ˙z +N 2 +�2 ++ 2k ˙z2 +z ++ 4k2N 2 + + + + z +3 +2 +� ˙φ2 +2N − NV +� + dt. +(57) +Replacing ˙z by 2N +3 q in view of (13), the above action may be cast as, +A1q = +� � +−2 +3αN q2 +√z + 6αkN√z + +9β +N√z +�4 +9 ˙q2 + 8kN 2q2 +9z ++ 4k2N 2 +� ++ z +3 +2 +� ˙φ2 +2N − NV +�� +dt. +(58) +Note that the action (58) cannot be expressed only in terms of velocities, due to the explicit presence of q unlike +(16). However, similar situation arrived at, in the case of Gauss-Bonnet-Dilaton case, and so it doesn’t matter. +The canonical momenta are the following: +pq = +8β +N√z ˙q; +pφ = z +3 +2 +N +˙φ; +pz = 0 = pN. +(59) +Dirac constraint analysis appears to be inevitable, since the action is singular. However as mentioned, the lapse +function N being the Lagrange multiplier, the constraint strongly vanishes, so that one can ignore it without +loss of generality. Still, another primary constraint pz = 0 is apparent. Nonetheless, as already noticed, in BL +formalism, Dirac analysis may be bypassed despite the presence of the constraint pz = 0 in the following manner. +The Lagrangian density is: +L1q = −2 +3αN q2 +√z + 6αkN√z + +9β +N√z +�4 +9 ˙q2 + 8kN 2q2 +9z ++ 4k2N 2 +� ++ z +3 +2 +� ˙φ2 +2N − NV +� +, +(60) +and hence the Hamiltonian reads as, +NHm +MBL = pq ˙q + pz ˙z + pφ ˙φ − L1q += N√z +16β p2 +q + 2 +3Nqpz + N +2z +3 +2 p2 +φ + 2αNq2 +3√z +− 6αkN√z − 8βkNq2 +z +3 +2 +− 36βk2N +√z ++ NV z +3 +2 , +(61) +where we have used (59) and replaced ˙z by 2N +3 q, in view of (13), and the suffix {MBL} now stands for ‘Modified +Buchbinder-Lyakhovich’ formalism. Finally as before, for the sake of comparison, if we perform the canonical +12 + +transformation q → 3 +2x, +and +pq → 2 +3px, then the above Hamiltonian (61) may be expressed in the following +form, +Hm +MBL = xpz + +√z +36β p2 +x + +p2 +φ +2z +3 +2 + 3α +2√z (x2 − 4kz) − 18βk +z +3 +2 +(x2 + 2kz) + V z +3 +2 , +(62) +which is identical to Hm +MHF presented in (28). +3.2 +Scalar-tensor theory: non-minimal coupling; +Here again, upon integrating the action (30) by parts we obtain, +A2 = +� � +− 3f ˙z2 +2N√z − 3f ′ ˙φ ˙z√z +N ++ 6fkN√z + +9β +N√z +�� ¨z +N − +˙N ˙z +N 2 +�2 ++ 2k ˙z2 +z ++ 4k2N 2� ++ z +3 +2 +� ˙φ2 +2N − NV +�� +dt. (63) +Now, replacing ˙z by 2N +3 q in view of (13), the above action (63) may be cast as, +A2 = +� � +−2 +3fN q2 +√z − 2f ′√zq ˙φ + 6fkN√z + +9β +N√z +�4 +9 ˙q2 + 8kN 2q2 +9z ++ 4k2N 2 +� ++ z +3 +2 +� ˙φ2 +2N − NV +�� +dt. (64) +Canonical momenta may therefore be found as, +pq = +8β +N√z ˙q, +pφ = −2f ′q√z + z +3 +2 +N +˙φ, +pN = 0 = pz. +(65) +As before, leaving out the constraint associate with the lapse function, and replacing ˙z = 2N +3 q in view of (13), +the Hamiltonian may be cast as, +NHnmMBL = pq ˙q + pz ˙z + pφ ˙φ − L += N +� √z +16β p2 +q + 2 +3qpz + +p2 +φ +2z +3 +2 + 2f ′ +z qpφ + 2fq2 +3√z + 2f ′2q2 +√z +− 6kf√z − 8βkq2 +z +3 +2 +− 36βk2 +√z ++ V z +3 +2 +� +. +(66) +Finally, applying the canonical transformation relations q → 3 +2x, and pq → 2 +3px, we obtain +HnmMBL = xpz + +√z +36β p2 +x + +p2 +φ +2z +3 +2 + 3x +z f ′pφ + 3f +2√z (x2 − 4kz) − 18βk +z +3 +2 +(x2 + 2kz) + 9x2f ′2 +2√z ++ V z +3 +2 . +(67) +Clearly, HnmMBL ∼= HnmMHF presented in (32). +3.3 +Einstein-Gauss-Bonnet-Dilatonic action +Eventually, in order to construct the correct Hamiltonian in connection with the Einstein-Gauss-Bonnet-Dilatonic +action (35), let us integrate it by parts to obtain, +A3 = +� � +α +� +− +3 ˙z2 +2N√z + 6kN√z +� ++ +9β +N√z +�� ¨z +N − +˙N ˙z +N 2 +�2 ++ 2k ˙z2 +z ++ 4k2N 2� +− γ′(φ) ˙z ˙φ +N√z +� ˙z2 +N 2z + 12k +� ++ z +3 +2 +� ˙φ2 +2N − NV +�� +dt. +(68) +13 + +As before, replacing ˙z by 2N +3 q in view of (13), the above action may be cast as, +A3q = +� � +− 2 +3αN q2 +√z + 6αkN√z + +9β +N√z +�4 +9 ˙q2 + 8kN 2q2 +9z ++ 4k2N 2 +� +− 2qγ′(φ) ˙φ +3√z +�4q2 +9z + 12k +� ++ z +3 +2 +� ˙φ2 +2N − NV +� � +dt. +(69) +Canonical momenta are now found as, +pq = +8β +N√z ˙q, +pφ = −2qγ′(φ) +3√z +�4q2 +9z + 12k +� ++ z +3 +2 +N +˙φ, +pN = 0 = pz. +(70) +As always, leaving out the constraint associated with the lapse function, and replacing ˙z = 2N +3 q in view of (13), +the Hamiltonian may be cast as, +NHGB +MBL =pq ˙q + pz ˙z + pφ ˙φ − L += N +� √z +16β p2 +q + 2 +3qpz + +p2 +φ +2z +3 +2 + 2αq2 +3√z − 6kα√z − 8βkq2 +z +3 +2 +− 36βk2 +√z ++ 2qγ′(φ)pφ +3z2 +�4q2 +9z + 12k +� ++ 2q2γ′2(φ) +9z +5 +2 +�4q2 +9z + 12k +�2 ++ V z +3 +2 +� +, +(71) +Finally, the set of canonical transformations q → 3 +2x, and pq → 2 +3px, allows one to express the Hamiltonian (71) +as, +HGBMBL =xpz + +√z +36β p2 +x + +p2 +φ +2z +3 +2 + 3α +2√z (x2 − 4kz) − 18βk +z +3 +2 +(x2 + 2kz) ++ xγ′(φ) +z2 +�x2 +z + 12k +� +pφ + γ′2(φ)x2 +2z +5 +2 +�x2 +z + 12k +�2 ++ V z +3 +2 . +(72) +As a result one finds, HGBMBL ∼= HGBMHF presented in (47). It is important to note that in the process of +constructing the Hamiltonian starting from a divergent free action, the pathology discussed in regard of canonical +formulation of Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher-order term is also removed. +4 +Application +It is mentioned in the introduction that canonical formulation is a precursor to canonical quantization. In the +absence of a viable quantum theory of gravity, it is suggestive to canonically quantize the cosmological equa- +tion and study quantum cosmology to extract some ethos of pre-Planck era. For example, one can explore the +Euclidean wormhole solution. Nonetheless, ‘cosmological inflationary scenario’ has been developed since 1980, +to solve horizon, flatness (fine tuning), structure formation and monopole problems, singlehandedly. Short-lived +(10−36 −10−26)s. inflation, occurred just after Planck’s era and falls within the periphery of ‘quantum field theory +in curved space-time’. To be more specific, ‘inflation is a quantum theory of perturbations on the top of the +classical background’, so that the energy scale of the background remains much below Planck’s scale. Nonetheless +in this context, Hartle [73] prescribed that, most of the important physics may still be extracted from the classical +action provided, the semiclassical wave-function is strongly peaked. The reason being, in that case correlation +between the geometrical and matter degrees of freedom is established, and hence the emergence of classical trajec- +tories (i.e. the universe) is expected. Hence, quantization and an appropriate semiclassical approximation must +be treated as a forerunner to study inflation. +Canonical quantization and the semiclassical wave-function in connection with the Hamiltonian (67) for non- +minimally coupled higher order theory had been presented in [26], which reduces to the minimally coupled case +14 + +when the coupling parameter becomes constant [19]. The Hamiltonian operator was found to be hermitian, stan- +dard probabilistic interpretation holds, and the semiclassical wave-functions was found to be oscillatory about the +classical inflationary solution. Inflation has been studied and the parameters are found with excellent agreement +with the observational constraints [74,75]. Gravitational perturbation has also been studied. +In [29] again, the quantum counterpart of the Hamiltonian (72) in connection with Einstein-Gauss-Bonnet- +Dilatonic coupled action has been presented. +Hermiticity of the Hamiltonian operator has been established, +probabilistic interpretation is explored, and the semiclassical wave-function is found to be oscillatory about a +classical inflationary solution. Finally, we have studied inflation and found that the inflationary parameters more- +or-less satisfy observational constraints [74,75]. In a nut-shell, the results obtained in [29] are the following. +iℏ∂Ψ +∂σ = +� +− ℏ2φ +54β0x +� ∂2 +∂x2 + n +x +∂ +∂x +� +− +ℏ2 +3xσ +4 +3 +∂2 +∂φ2 + 2iℏα0 +σ +� 1 +φ2 +∂ +∂φ − 1 +φ3 +� +− 2iℏγ0x2 +3σ +7 +3 +� +2φ ∂ +∂φ + 1 +� ++ Ve +� +Ψ += �HeΨ, +(73) +where, the proper volume, σ = z +3 +2 = a3 plays the role of internal time parameter, and n is the operator ordering +index. In the above equation, �He is the effective hermitian Hamiltonian operator, while the the effective potential +Ve is given by, +Ve = 3α2 +0x +σ +2 +3 φ4 − 4α0γ0x3 +σ2φ ++ 4γ2 +0x5φ2 +3σ +10 +3 ++ α0x +σ +2 +3 φ ++ λ2σ +2 +3 φ2 +3x ++ 2σ +2 +3 ΛM 2 +P +x +. +(74) +The effective Hamiltonian operator is found to be hermitian for n = −1, which selects the operator ordering +parameter from physical consideration. Standard quantum mechanical probability interpretation also holds. Under +a suitable (WKB) semiclassical approximation, the wave-function has been found to be, +Ψ = Ψ0e +i +ℏ +� +− 6α0λz2 +a0φ0 +16γ0a2 +0φ2 +0λ3√z +� +, +(75) +which exhibits oscillatory behaviour about the classical inflationary solution a = a0eλt , where, α0, φ0, γ0 are +constants. We have also presented several sets of inflationary parameters in [29], which depict that the spectral +index of scalar perturbation and the scalar to tensor ratio lie within the range 0.967 ≤ ns ≤ 0.979 and 0.056 ≤ +r ≤ 0.089 respectively, showing reasonably good agreement with the recently released data [74,75]. The number +of e-folding also remains within the acceptable range 46 < N < 73, which is sufficient to solve the horizon and +flatness problems. +5 +Concluding remarks +Although initiated two centuries back, canonical formulation of higher-order theory of gravity is particularly +non-trivial. In fact, only after probing Dilatonic coupled Gauss-Bonnet action, it is learnt that divergent terms +play a vital role to formulate correct quantum dynamics of non-linear gravity theory. The scheme is therefore +first, to express the action in terms of the basic variable hij , otherwise if expressed in terms of the scale factor, +as commonly done, some unwanted divergent terms are removed in the process of integration by parts, which are +unaccredited by the variational principle. Next, unless divergent terms are taken care of, the Hamiltonian is found +to be different, which is related through canonical transformation though, such transformation cannot be carried +over in the quantum domain due to non-linearity. It is shown that in the case of Einstein-Gauss-Bonnet-Dilatonic +coupled action in 4-dimension that, unless the action is divergent free, an erroneous Hamiltonian is constructed, +since it does not reflect the topological invariance of the theory. This proves the importance of divergent terms in +higher order theories. In this respect the difference of BL formalism with MHF is apparent. In fact BL formalism +produces identical Hamiltonian as obtained earlier following Ostrogradsky’s, Dirac’s or Horowitz’s formalisms. +However, MHF is essentially the Horowitz formalism, after expressing the action in terms of the three space +curvature and taking care of the total derivative terms under integration by parts. It was shown that following the +same route if Dirac’s algorithm is applied, the Hamiltonian becomes identical to the one found following MHF, +15 + +and one obtains unique quantum description. Here, we reveal that the same is true with BL formalism. In fact, +BL formalism not only bypasses constraint analysis, as in the case of Horowitz’s formalism, it also does not require +auxiliary variable to cast the action in canonical form, which is a bit intricate. 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Astro- +phys. 641 A10. +19 + diff --git a/09E5T4oBgHgl3EQfqw9o/content/tmp_files/load_file.txt b/09E5T4oBgHgl3EQfqw9o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f3934e174028187145f84ee143b91a2737adb72 --- /dev/null +++ b/09E5T4oBgHgl3EQfqw9o/content/tmp_files/load_file.txt @@ -0,0 +1,941 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf,len=940 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='05710v1 [gr-qc] 13 Jan 2023 Perusing Buchbinder–Lyakhovich canonical formalism for Higher-Order Theories of Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Dalia Saha†, Abhik Kumar Sanyal‡ January 18, 2023 † Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' of Physics, University of Kalyani, West Bengal, India - 741235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' †,‡ Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Abstract Ostrogradsky’s, Dirac’s and Horowitz’s techniques of higher order theories of gravity produce identical phase-space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The problem is manifested in the case of Gauss-Bonnet-dilatonic coupled action in the presence of higher-order term, in which case, classical correspondence can’t be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here, we explore yet another technique developed by Buchbinder and his collaborators (BL) long back and show that it also suffers from the same disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, expressing the action in terms of the three-space curvature, and removing “the total derivative terms”, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here we show that the same is true for BL formalism, which appears to be the simplest of all the techniques, to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Keywords: Higher Order theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Canonical Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 1 Introduction Canonical formulation of higher-order theories was developed by Ostrogradsky almost two centuries back [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, it did not draw much attention, since other than toy mechanical models, practically no such physical theories were persuaded at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Exactly a century elapsed, when it was applied to a physically motivated problem, such as fourth order harmonic oscillator [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The real physical problem in this context appeared for the first time, while a renormalized quantum theory of gravity was attempted to formulate [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Higher-derivative theory of gravity is usually considered as a model of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The reason being, Einstein-Hilbert ac- tion is supplemented by curvature squared terms (R2, RµνRµν ) to ensure renormalizability [4] and asymptotic freedom [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Unfortunately, curvature-squared gravity theories have been found to suffer from the unresolved problem of physical unitarity in perturbative analysis, which is usual for higher-derivative theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, possibilities to overcome this difficulty were also discussed in some literatures [6, 8] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is also ascertained that curvature squared gravity would arise as a low-energy effective theory derived from super- string theory in D = 10 dimensions [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Over the last couple of decades, higher order theories of gravity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=', F(R), F(G), F(R, T ) etc, theories, (R, G, T being the Ricci scalar, the Gauss-Bonnet term, and the torsion term respectively) have drawn much attention in search of alternatives to dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, it is always suggestive to test viability of such modified theories of gravity in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the context of the very early universe, a canonical formulation is required as a precursor, particularly to study quantum cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Since Ostrogradsky’s technique does not apply in the degenerate case of singular Lagrangian, for which the Hessian determinant vanishes, Dirac’s constraint analysis [12] may be applied for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, a host of theories have been formulated over decades to bypass the constraint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' One of these in this direction was originally proposed by Boulware [13], and later reformulated by Horowitz’ [14], in particular in the context of higher-order theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Since the canonical formulation of higher order theories requires an extra degree of freedom, in Horowitz’s formalism apart from the scale factor (‘a′ in the Robertson-Walker minisuperspace) an auxiliary variable is introduced by taking derivative of the action (say A) with respect to the highest derivative of 1Electronic address: † daliasahamandal1983@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='com ‡ sanyal ak@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='com 1 the field variable present (Q = ∂A ∂¨a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the end, the auxiliary variable is replaced by the basic variable (extrinsic curvature tensor) through a canonical transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The important finding in this regard is as follows: all the three formalisms, viz, Ostrogradsky’s (once degeneracy has been removed), Dirac’s and Horowitz’s formalisms, produce an identical phase-space structure [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Meanwhile, certain pathologies with Horowitz’ formalism have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For example, it was noticed that Horowitz’s formalism can even be applied in the case of linear gravity theory (Einstein-Hilbert action) leading to wrong quantum dynamics [16–18], as well as some superfluous total derivative terms are eliminated [18, 19], which neither may be obtained from the variational principle, nor having any connection with Gibbons-Hawking–York term [20,21], nor any of its modified versions, associated with higher-order gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Further, the coupling parameter, in the case of the “non-minimally coupled scalar tensor the- ory of gravity associated with higher order term”, has not been found to play any particular role, since its derivative does not appear in the Hamiltonian [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The same is true for the “Dilatonic coupled Gauss-Bonnet-theory in the presence of higher order term”, where additionally, the classical correspondence with quantum counterpart, could not be established [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In view of such an uncanny situation, yet another technique was developed, called the “modified Horowitz’s formalism” (MHF), which was successfully applied to different modified higher-order theories of gravity, to explore the evolution of the very early universe [15,17–19,22–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the MHF, the action is expressed in terms of the three-space curvature (instead of the scale factor), “the total derivative terms” are removed by integrating the action by parts, and Horowitz’s formalism (the introduction of the auxiliary variable etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=') was followed, thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' To be very specific, let us consider the following isotropic and homogeneous Robertson–Walker (RW) metric: ds2 = −N 2(t) dt2 + a2(t) � dr2 1 − kr2 + r2(dθ2 + sin2θdφ2) � , (1) for which the degeneracy in the Lagrangian disappears if the gauge (N ) is fixed a priori, in which case, Ostrograd- sky’s technique applies as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Once such degeneracy is removed, it is observed that Ostrogradsky’s technique produces the same Hamiltonian, obtained following Horowitz’s as well as Dirac’s formalism [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, it certainly follows that both the Ostrogradsky’s and Dirac’s formalism implicitly suffer from the same problem, in disguise, as was noticed in Horowitz’s technique, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore in the MHF, instead of the scale factor, the action is expressed in terms of the basic variable hij — the three space metric from the very beginning—so that redundant total derivative terms do not appear [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Thereafter, all the total derivative terms are integrated out by parts, which become cancelled by the supplementary boundary (Gibbons–Hawking— York and modified Gibbons–Hawking—York) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Subsequently, the auxiliary variable is introduced following Horowitz’s proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the end, the auxiliary variable is replaced by the other basic variable Kij — the extrinsic curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In this process, the unwanted problems that appeared following Horowitz’s formalism disappear, while it produces a different Hamiltonian altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We mention that although both Hamiltonians (obtained following the MHF and Ostrogradky’s, Dirac’s and Horowitz’s formalisms) are related through the canonical transformation, they indeed produce different dynamics in the quantum domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is also important to mention that it is not possible to carry over the classical canonical transformations to the quantum domain for higher-order theories, due to the non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The MHF leads to an effective Hermitian Hamiltonian, a standard quantum mechanical probabilistic interpretation, and a viable semiclassical treatment, which exhibit oscillation of the wave function about the classical de-Sitter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' As a result, the classical correspondence is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In this regard, the MHF may be considered as the most-viable technique to handle the higher-order theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It has later been established that, if the action is expressed in terms of the three-space metric (hij ) from the very beginning and the total derivative terms are addressed, Dirac’s constraint analysis [12] also produces the Hamiltonian iden- tical to that of the MHF [22,28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Amongst other techniques, Hawking-Luttrell technique [33] has limited application, since conformal trans- formation is not possible in general [19], Schmidt’s technique [34] is identical to the Horowitz’s formalism in disguise [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, there is yet another technique developed in the 80’s by Buchbinder and his collabora- tors [35–39], which did not receive much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Querella [40] only noticed that although at a first glance, the general formalism developed by Buchbinder and his collaborators (BL) appears to be satisfactory, nevertheless it has pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' BL formalism is our current concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here, we test this abstract theoretical settings of BL formalism in simple minisuperspace model to explore the pitfall, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The underlying essence of this formalism is to bypass Dirac’s constrained analysis, very much like Horowitz’s technique, but instead of introducing auxiliary variable, here the program is initiated with the basic variables {hij, Kij}, the three-space curvature and the extrinsic curvature tensors respectively from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In our present attempt to explore the outcome of this 2 technique, we discover that the formalism leads to identical phase space structure as was found in the case of Ostrogradsky’s/Dirac’s/Horowitz’s formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the following section, we study scalar tensor theory of gravity (both the minimal and non-minimal cases), and Gauss-Bonnet-Dilatonic coupled action being supplemented by the scalar curvature squared (R2 ) term, following BL formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=', we explore the fact that once total derivative terms are taken care of, the Hamiltonian does not differ from MHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' discusses its physical application, in connection with some earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=', concludes our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 2 BL Formalism in Three Different Higher Order Theories In view of the very importance of higher-order curvature invariant terms required to construct a renormalizable quantum theory of gravity when the curvature is extremely strong, a unique canonical formulation of the Einstein– Hilbert action being supplemented by higher-order curvature invariant terms, is therefore necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here, we shall consider three different cases, minimally and non-minimally coupled scalar-tensor theory of gravity supplemented by R2 term, and the scalar-tensor theory of gravity being supplemented by R2 and Gauss-Bonnet terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the Robertson-Walker minisuperspace (1) under consideration, the Ricci scalar and the Gauss-Bonnet terms are R = 6 N 2 � ¨a a + ˙a2 a2 + N 2 k a2 − ˙N ˙a Na � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (2) G = R2 − 4RµνRµν + RαβµνRαβµν = 24 N 3a3 � N¨a − ˙N ˙a � � ˙a2 N 2 + k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For the sake of comparison with earlier results, we express actions in terms of the three space metric, instead of the scale factor, as its importance has been mentioned already, and will be explicitly shown at the beginning of Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='. Since construction of higher-order theory to its canonical form requires an additional degree of freedom, hence, in addition to the three-space metric hij , the extrinsic curvature tensor Kij is treated as basic variable, as already stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We therefore choose the basic variables hij = zδij = a2δij , so that Kij = − ˙hij 2N = − a ˙a N δij = − ˙z 2N δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In terms of z = a2, the Ricci scalar and the Gauss-Bonnet terms take the following forms, R = 6 N 2 � ¨z 2z + N 2 k z − 1 2 ˙N ˙z Nz � , (4) G = 12 N 2 � ¨z z − ˙z2 2z2 − ˙N ˙z Nz � � ˙z2 4N 2z2 + k z � , (5) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is noteworthy that since, RµνRµν = 12 N 4 � ¨a2 a2 + ˙a2¨a a3 + ˙a4 a4 − 2 ˙N ˙a¨a Na2 − ˙N ˙a3 Na3 + ˙N 2 ˙a2 N 2a2 + k N 2¨a a3 + 2k N 2 ˙a2 a4 − k N ˙N ˙a a3 + k2 N 4 a4 � , (6) therefore, RµνRµν − 1 3R2 = − � 12 Na3 � d dt �1 3 ˙a3 N 3 + k ˙a N � , (7) and as a result, � � RµνRµν − 1 3R2 � √−gd4x = −12C � � d dt �1 3 ˙a3 N 3 + k ˙a N �� dt (8) 3 is a total derivative term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Thus, RµνRµν term is redundant in RW metric, once R2 term is taken (the constant C appears due to the integration of the three space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence, to scrutinize the BL formalism presented by Buchbinder and his collaborators in RW minisuperspace model (1), we consider scalar-tensor theories of gravity and also Gauss-Bonnet-Dilatonic coupled gravity theory, being associated with scalar curvature squared term R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='1 Minimal coupling: Let us start with the following minimally coupled case, A1 = � √−g � αR + βR2 − 1 2φ,µφ,ν − V (φ) � d4x + αΣR + βΣR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (9) In the above, α = 1 16πG , β is a constant coupling parameter, αΣR = 2α � ∂V K √ hd3x is the Gibbons- Hawking–York boundary term [21] associated with Einstein–Hilbert sector of the above action, and βΣR2 = 4β � ∂V RK √ hd3x is its modified version corresponding to R2 term, while, K is the trace of the extrinsic curva- ture tensor Kij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Note that, both the counter terms are required under the condition δR = 0, at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Instead, if the condition δKij = 0 is chosen at the boundary, the counter terms are not required, as in the case of Horowitz’s formalism [14], since both the boundary terms appearing under metric variation vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, in the case of Ostrogradsky’s technique [1] and Dirac constrained analysis [12], boundary terms are not taken care of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This is true for BL formalism too, as we shall see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nevertheless, the modified Horowitz’s for- malism [17–19, 23–25] fixes δhij = 0 = δR at the boundary, and hence requires supplementary boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We have demonstrated earlier that proper attention to all the boundary terms is paid in modified Horowitz’s formalism (MHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' As a result, it presents a different phase space structure of the Hamiltonian for a particular action being supplemented by higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, it is related to the others under a suitable set of canonical transformation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Although, as mentioned, such transformations cannot be carried over in the quantum domain, due to non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' So, it’s indeed required to check if BL formalism also produces the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The action (9) in the RW minisuperspace model (1) may be written in terms of the basic variable hij = zδij , as A1 = � � 3α√z � ¨z N − ˙N ˙z N 2 + 2kN � + 9β √z � ¨z2 N 3 − 2 ˙N ˙z¨z N 4 + ˙N 2 ˙z2 N 5 − 4k ˙N ˙z N 2 + 4k¨z N + 4k2N � + z 3 2 � ˙φ2 2N − V N �� dt + αΣR + βΣR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (10) The (0 0) component of the field equation in terms of the scale factor ‘a′ takes the following form 6α a2 � ˙a2 N 2 + k � + 36β a2N 4 � 2˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='a − 2˙a2 ¨N N − ¨a2 − 4˙a¨a ˙N N + 2˙a2 ¨a a + 5˙a2 ˙N 2 N 2 − 2 ˙a3 ˙N aN − 3 ˙a4 a2 − 2kN 2 ˙a2 a2 + k2N 4 a2 � − � ˙φ2 2N 2 + V � = 0, (11) which contains term upto third derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This is the energy constraint equation (E = 0), and when expressed in terms of the phase space variables, becomes the Hamiltonian constraint equation, (due to diffeomorphic invari- ance) of the theory under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This we aim at, following the formalism presented by Buchbinder and his collaborators (BL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The action (9) has already been expressed in terms of the basic variable {hij}, instead of the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Canonical formulation of higher order theories requires additional degree of freedom, and the only choice is the extrinsic curvature tensor {Kij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In contrast to Horowitz’s formalism, where apart from {hij} an auxiliary variable is introduced and at the end the Hamiltonian is expressed in terms of the basic variables {hij, Kij}, in BL formalism, these basic variables are associated from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In Robertson-Walker metric, the extrinsic curvature tensor is expressed as, Kij = − ˙hij 2N = −2a˙a 2N δij = − ˙z 2N δij = −qij say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (12) 4 Since there is only one independent component, so instead of qij , the new generalized coordinate is chosen to be its trace viz, q = 3 ˙z 2N , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' qij = q 3δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (13) To express the action in terms of velocities, we choose, v ≡ ˙q, vφ ≡ ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (14) The scalar curvature (4) therefore takes the following form, R = 2 ˙q Nz + 6k z ≡ Rq = 2 Nz (v + 3Nk), (15) and action (10) can be expressed as, A1q = � � 2α√z(v + 3kN) + 4β N√z (v + 3kN)2 + z 3 2 � v2 φ 2N − NV �� dt, (16) while the Lagrangian density is, L1q = 2α√z(v + 3kN) + 4β N√z (v + 3kN)2 + z 3 2 � v2 φ 2N − NV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (17) Note that the boundary terms remain intact in the action as well as in the point Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Canonical momenta are pq = ∂Lq ∂v = 2α√z + 8β N√z (v + 3kN), pN = ∂L1q ∂vN = 0, pz = ∂Lq ∂vz = 0 and pφ = ∂Lq ∂vφ = z 3 2 vφ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (18) Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, Dirac constraint analysis appears to be essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, here is a wonderful twist in the BL formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For example, one can express the modified Lagrangian density as, L∗ 1 = L1q + pq ( ˙q − v) + pN � ˙N − vN � + pz � ˙z − 2Nq 3 � + pφ � ˙φ − vφ � , (19) and equivalently, the Hamiltonian density as, H∗ 1 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗ 1 = pqv + pNvN + pφvφ + pz 2Nq 3 − L1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (20) As a consequence, one can immediately find that the primary constraint D ≡ pz ≈ 0 disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Further, since N is a non-dynamical Lagrange multiplier, hence the constraint C vanishes strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, one arrives at, H∗ 1 = pqv + CvN + pφvφ + pz 2qN 3 − L1q = CvN + pqv + pφvφ + pz 2qN 3 − L1q = CvN + NHm BL, (21) where, NHm BL = pqv + pφvφ + 2N 3 qpz − L1q = pqv + pφvφ + 2N 3 qpz − 2α√z(v + 3kN) − 4β N√z (v + 3kN)2 − z 3 2 � v2 φ 2N + NV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (22) 5 In the above, m in the superscript stands for minimally coupled theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Now upon substituting v and vφ from the definition of momentum (18), we obtain, NHm BL = N �2q 3 pz + √z 16β p2 q − �αz 4β + 3k � pq + α2 4β z 3 2 + 1 2z 3 2 p2 φ + V z 3 2 � , (23) so that the canonical Hamiltonian finally reads as, Hm BL = 2q 3 pz + √z 16β p2 q − �αz 4β + 3k � pq + α2 4β z 3 2 + 1 2z 3 2 p2 φ + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (24) The action (10) may also be cast in the canonical form, A1q = � � ˙zpz + ˙qpq + ˙φpφ − NHBL � dt d3x = � � ˙hijπij + ˙KijΠij + ˙φpφ − NHBL � dt d3x, (25) where, πij and Πij are momenta canonically conjugate to hij and Kij respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For the sake of comparison, let us make the following canonical transformation: q → 3 2x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pq → 2 3px, (26) to express the above Hamiltonian (24) as: Hm BL = xpz + √z 36β p2 x − �αz 6β + 2k � px + α2z 3 2 4β + p2 φ 2z 3 2 + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (27) It is revealed that the above Hamiltonian (27) is exactly the one obtained earlier, following Ostrogradsky, Dirac as well as Horowitz’s formalisms [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Note that, very much like the Ostrogradsky’s and Dirac’s formalisms here also, once the formalism is initiated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (R is expressed in terms of {hij, Kij} (15) as well as the action (16) and the point Lagrangian (17)), there remains no option to integrate the action by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' As a result, even the Gibbons-Hawking–York term [20,21] which is physically meaningful, being associated with the entropy of the black hole, along with its higher-order counterpart, also remain obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' On the contrary, following Modified Horowitz’s Formalism (MHF), where boundary terms are taken care of, we earlier obtained [15] Hm MHF = xpz + √z 36β p2 x + 3αx2 2√z − 18βkx2 z 3 2 − 36βk2 √z − 6kα√z + p2 φ 2z 3 2 + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (28) Although the two (27) and (28) exactly match under the following set of canonical transformations, pz → pz − 18kβx z 3 2 + 3αx 2√z , z → z, px → px + 36kβ √z − 3α√z, x → x, pφ → pφ, φ → φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' and apparently there is no contradiction between the two, note the essential difference: linear term in the mo- mentum (px ), which is very much present in (27), remains absent from the Hamiltonian (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' As a result, the two Hamiltonians (27) and (28) induce completely different quantum dynamics, since in the quantum domain, as mentioned, canonical transformation cannot be carried over due to non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='2 Non-minimally coupled case We find that the two different Hamiltonians (27) and (28) render two different quantum descriptions of the same classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Although, some of the essential features (Gibbons-Hawking-York term and its higher order counterpart) are absent from the Hamiltonian (27), it is not clear, which one gives correct quantum description of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Further, there may exist a unitary transformation (we have not found it though) relating the two Hamiltonian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' to inspect the situation more deeply,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' we consider the non-minimally coupled case next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' whose action A2 = � √−g d4x � f(φ)R + βR2 − 1 2φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='µφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='ν − V (φ) � + f(φ)ΣR + BΣR2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (29) may be expressed in the RW metric (1) as A2 = � � 3f(φ)√z � ¨z N − ˙N ˙z N 2 + 2kN � + 9β √z � ¨z2 N 3 − 2 ˙N ˙z¨z N 4 + ˙N 2 ˙z2 N 5 − 4k ˙N ˙z N 2 + 4k¨z N + 4k2N � + z 3 2 � ˙φ2 2N − V N �� dt + f(φ)ΣR + BΣR2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (30) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' as already mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' the supplementary boundary terms are required when MHF is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the above, we consider an arbitrary functional coupling parameter f(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Pursuing the same procedure as above, one finally arrives at the following Hamiltonian: HnmBL = xpz + √z 36β p2 x − � f(φ) z 6β + 2k � px + f 2(φ)z 3 2 4β + p2 φ 2z 3 2 + V z 3 2 , (31) which is again identical to the one found following Dirac formalism and may be found following Ostrogradsky’s and Horowitz’s techniques as well [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the superscript nm stands for non-minimal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The action (30) may also be cast in the canonical form as in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' On the contrary, following MHF, one finds [22] HnmMHF = xpz + √z 36β p2 x + 3f(φ) � x2 2√z − 2k√z � − 18kβ √z �x2 z + 2k � + p2 φ 2z 3 2 + 3xf ′(φ)pφ z + 9f ′(φ)2x2 2√z + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (32) However, under the following set of canonical transformations, pz → pz − 18βkx z 3 2 + 3f(φ)x 2√z , z → z, px → px + 36β k √z + 3f(φ)√z, x → x, pφ → pφ + 3f ′(φ)x√z, φ → φ, the two Hamiltonians (31) and (32) match again [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nevertheless, here the difference is predominant and explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Note that f ′(φ) term does not appear in (31), while it is coupled to pφ in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This coupled (f ′(φ)pφ ) term requires operator ordering in the quantum domain, which is different for different form of f(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence even if the two Hamiltonians are related through unitary transformation, such transformation would be different for different form of f(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='3 Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher order term Although, it is clear that two different quantum descriptions follow from the same classical action using different techniques, it is still abstruse to select the correct description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, we next consider Einstein-Gauss-Bonnet- Dilatonic coupled action in the presence of higher order curvature invariant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Gauss-Bonnet (GB) term arises quite naturally as the leading order of the α′ expansion of heterotic superstring theory, where α′ is the inverse string tension [41–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Several interesting features of GB term have been explored in the past and appear in the literature [47–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, Gauss–Bonnet term is topological invariant in 4-dimensions, and so to get its contribution in the field equations, a dynamic dilatonic scalar coupling is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is worth mentioning that, in string induced gravity near initial singularity, GB coupling with scalar field plays a very crucial role for the occurrence of nonsingular cosmology [68,69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The particular hallmark of GB term is the fact that, despite being formed from a combination of higher order curvature invariant terms (G = R2 − 4RµνRµν + RαβµνRαβµν) (3), it ends up only with second order field equations, avoiding Ostrogradsky’s instability, and equivalently, ghost degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, such a wonderful feature ultimately leads to a serious pathology of ‘Branched Hamiltonian’, which has no unique resolution till date [70–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nevertheless, it has been revealed that, the pathology may be bypassed upon supplementing the action with higher order curvature invariant term [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We therefore consider the following action, A3 = � √−g d4x � αR + βR2 + γ(φ)G − 1 2φ,µφ,ν − V (φ) � + αΣR + βΣR2 + γ(φ)ΣG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (33) In the above, Gauss-Bonnet term G , is coupled with γ(φ), while V (φ) is the dilatonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Further, the symbol K stands for K = K3 − 3KKijKij + 2KijKikKk j , where, K is the trace of the extrinsic curvature tensor Kij , and γ(φ)ΣG = 4γ(φ) � ∂V � 2GijKij + K 3 � √ hd3x is the supplementary boundary term associated with Gauss-Bonnet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The (0 0) component of the Einstein’s field equation in terms of the scale factor here reads as, 6α a2 � ˙a2 N 2 + k � + 36β a2N 4 � 2˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='a − 2˙a2 ¨N N − ¨a2 − 4˙a¨a ˙N N + 2˙a2 ¨a a + 5˙a2 ˙N 2 N 2 − 2 ˙a3 ˙N aN − 3 ˙a4 a2 − 2kN 2 ˙a2 a2 + k2N 4 a2 � + 24γ′ ˙a ˙φ N 2a3 � ˙a2 N 2 + k � − � ˙φ2 2N 2 + V � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (34) The action (33) in terms of the basic variable (hij = a2δij = zδij ) may be expressed as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' A3 = � � 3α√z � ¨z N − ˙N ˙z N 2 + 2kN � + 9β √z � ¨z2 N 3 − 2 ˙N ˙z¨z N 4 + ˙N 2 ˙z2 N 5 − 4k ˙N ˙z N 2 + 4k¨z N + 4k2N � + 3γ(φ) N√z � ˙z2¨z N 2z + 4k¨z − ˙z4 2N 2z2 − ˙N ˙z3 N 3z − 2k ˙z2 z − 4k ˙N ˙z N � + z 3 2 � 1 2N ˙φ2 − V N � � dt + αΣR + βΣR2 + γ(φ)ΣG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (35) where the additional supplementary boundary term γ(φ)ΣG = −γ(φ) ˙z N√z � ˙z2 N 2z + 12k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' is required in the case of MHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Inserting the other basic variable (Kij = − q 3δij ) and considering ˙q = v (13), the action (35) finally may be expressed as, A3q = � � 2α√z(v + 3kN) + 4β N√z (v + 3kN)2 + 8γ(φ) √z �vq2 9z − q4N 27z2 + kv − kNq2 3z � + z 3 2 � v2 φ 2N − NV � � dt + αΣR + βΣR2 + Λ(φ)ΣG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (36) Thus, the Lagrangian density takes the following form, L3q = 2α√z(v + 3kN) + 4β N√z (v + 3kN)2 + 8γ(φ) √z �vq2 9z − q4N 27z2 + kv − kNq2 3z � + z 3 2 � 1 2N v2 φ − V N � , (37) 8 where, boundary terms are not taken care of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The canonical momenta are pq = ∂Lq ∂v = 2α√z + 8β N√z (v + 3kN) + 8γ(φ) √z � q2 9z + k � , pN = ∂L3q ∂vN = 0, pφ = ∂Lq ∂vφ = z 3 2 vφ N , and, pz = ∂L3q ∂vz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (38) Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0, which are usually handled by Dirac constraint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, as mentioned, such analysis is not at all required in the BL formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For example, one can express the modified Lagrangian density as, L∗ 3 = L3q + pq ( ˙q − v) + pN � ˙N − vN � + pz � ˙z − 2Nq 3 � + pφ � ˙φ − vφ � , (39) so that the corresponding Hamiltonian density takes the following form, H∗ 3 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗ 3 = pqv + pNvN + pφvφ + 2Nq 3 pz − L3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (40) As a result, the primary constraint D ≡ pz ≈ 0 disappears and one obtains, H∗ 3 = pqv + CvN + pφvφ + pz 2qN 3 − L3q = CvN + � pqv + pφvφ + 2qN 3 pz − L3q � = CvN + NHGB BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (41) In the superscript GB stands for Hamiltonian in connection with Einstein-Gauss-Bonnet-Dilatonic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Note that the constraint C ≡ pN strongly vanishes, since the lapse function N is simply a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, NHGBBL = pqv + pφvφ + 2qN 3 pz − L3q = pqv + pφvφ + 2qN 3 pz − 2α√z(v + 3kN) − 4β N√z (v + 3kN)2 − 8γ(φ) √z �vq2 9z − q4N 27z2 + kv − kNq2 3z � − z 3 2 � 1 2N v2 φ − V N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (42) Now upon substituting v from the definition of momentum (38), one obtains, NHGBBL =N �2qpz 3 + √zp2 q 16β − pq �αz 4β + 3k � + α2z 3 2 4β − pq � γq2 9βz + γk β � + 2αγ β � q2 9√z + k√z � + 4γq4 27z 3 2 � γ 3β + 2 � + 8γkq2 3z 3 2 � γ 3β + 2 � + 12γk2 √z � γ 3β + 2 � + p2 φ 2z 3 2 + V z 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (43) The canonical Hamiltonian therefore finally reads as, HGB BL =2qpz 3 + √zp2 q 16β − pq �αz 4β + 3k � + α2z 3 2 4β − pq � γq2 9βz + γk β � + 2αγ β � q2 9√z + k√z � + 4γq4 27z 3 2 � γ 3β + 2 � + 8γkq2 3z 3 2 � γ 3β + 2 � + 12γk2 √z � γ 3β + 2 � + p2 φ 2z 3 2 + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (44) Again, for the sake of comparison, let us make the canonical transformation q → 3 2x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pq → 2 3px (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' to express the above Hamiltonian (44) in the following form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' HGBBL = xpz + √zp2 x 36β + α2z 3 2 4β − �αz 6β + γx2 6βz + 2kγ 3β + 2k � px + p2 φ 2z 3 2 + � γ2 4βz 5 2 + 3γ 2z 5 2 � x4 + � αγ 2β√z + 12kγ z 3 2 + 2kγ2 βz 3 2 � x2 + 2αkγ√z β + 24k2γ √z + 4k2γ2 β√z + V z 3 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (45) 9 and notice that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' it is similar to the one already found,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' following Dirac formalism and may be found following Ostrogradsky’s and Horowitz’s techniques as well [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The action (35) may also be cast in the canonical form with respect to the basic variables as, A3q = � � ˙zpz + ˙qpq + ˙φvφ − NHBL � dt d3x = � � ˙hijπij + ˙KijΠij + ˙φvφ − NHMHF � dt d3x, (46) where, πij and Πij are momenta canonically conjugate to hij and Kij respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence, everything appears to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' On the contrary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' although following MHF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' we found [22] HGB MHF = xpz + √zp2 x 36β + 3α � x2 2√z − 2k√z � − 18kβ √z �x2 z + 2k � + � x6 2z 9 2 + 12kx4 z 7 2 + 72k2x2 z 5 2 � γ′2 + �x3 z3 + 12kx z2 � γ′pφ + p2 φ 2Z 3 2 + V z 3 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (47) nonetheless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' under the following set of canonical transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pz → pz − 18βkx z 3 2 + 3αx 2√z − 6kγ(φ)x z 3 2 − 3γ(φ)x3 2z 5 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' z → z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' px → px + 36β k √z + 3α√z + 3γ(φ)x2 z 3 2 + 12kΛ √z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' x → x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pφ → pφ − γ′(φ)x3 z 3 2 − 12kγ′(φ)x √z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' φ → φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (48) the two Hamiltonians (45) and (47) match again [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Apparently therefore, there is absolutely no problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nevertheless note that, the Hamiltonian (47) contains a term (γ′(φ)pφ ), which is absent from (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Now, during canonical quantization the presence of this term requires operator ordering, which is different for different form of γ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' As a result, even if the two may be related through unitary transformation, such transformation would be different for different form of γ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Thus, there does not exist a unique unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In a nutshell, we repeat that the two Hamiltonians (45) and (47) induce two different descriptions in the quantum domain, and apparently, there is no way to choose one to be the correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 3 The role of divergent terms: The very first important point to mention is, in all the formalisms the scale factor is treated as the basic variable, while we initiate our program treating three three-space curvature, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' To explain the reason behind this choice, let us consider curvature squared action, A = � βR2d4x, as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Under variation, it gives a total derivative term σ = −4β � RK √ h d3x, as mentioned earlier, where K is the trace of the extrinsic curvature tensor Kij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' A counter term (−σ), known by the name modified Gibbons-Hawking–York term [20, 21], must be added to the action in case, instead of δ ˙q, δR is kept fixed at the boundary, as in MHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the RW (1) metric under consideration, the action reads as, A = 36β � � a¨a2 + 2˙a2¨a + 2k¨a + ˙a4 a + 2k ˙a2 a + k2 a � dt � d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (49) Under integration by parts, we end up with, A = C � � a¨a2 + ˙a4 a + 2k ˙a2 a + k2 a � dt + C �2 3 ˙a3 + 2k ˙a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (50) where, C = 36β � d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Now following Horowitz’s program, we introduce an auxiliary variable Q = ∂A ∂¨a = 2Ca¨a, judiciously into the action in the following manner, such that it may be cast in canonical form, A = � � Q¨a − Q2 4Ca + C � ˙a4 a + 2k ˙a2 a + k2 a �� dt + C �2 3 ˙a3 + 2k ˙a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (51) 10 Integrating the action again by parts we find A = � − ˙Q˙a − Q2 4Ca + C � ˙a4 a + 2k ˙a2 a + k2 a �� + C �Q˙a C + 2 3 ˙a3 + 2k ˙a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (52) The action is canonical, since the Hessian determinant is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is trivial to check that the above action gives correct field equations, but the left out total derivative term may be expressed as, σ′ = −4β � RK √ h d3x + 16β � K √ h � ˙a2 a2 � d3x, (53) and as a result σ ̸= σ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Thus some redundant total derivative terms are pulled out in the process, which has severe consequence in the quantum domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' On the contrary, if we start with, z = a2, the action reads as, A = C � � ¨z2 4√z + k¨z √z + k2 √z � dt = C � � ¨z2 4√z + 2 √z � + C k ˙z √z , (54) where the last expression is found under integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Now following Horowitz’s program, we find the auxiliary variable as Q = ∂A ∂¨z = C ¨z 2√z , which is again judiciously introduced in the action as, A = � � Q¨z − √zQ2 C + C � k¨z √z + k2 √z �� dt + C k ˙z √z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (55) Finally, performing integration by parts again, one obtains, A = � � − ˙Q ˙z − √zQ2 C + C � k¨z √z + k2 √z �� dt + C �Q ˙z C + k ˙z √z � , (56) The action is again canonical, the Euler-Lagrange equations here again lead to the appropriate field equations, while one can express the total derivative term as σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In a nut-shell, although total derivative terms do not affect the classical field equations, for non-linear theories such as gravity, such terms tell upon the quantum dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Therefore, to establish consistency in every respect, hij should be treated as the basic variable, instead of the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This is essentially the so-called MHF, which finally requires to replace the auxiliary variable by the the second basic variable, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=', the extrinsic curvature tensor Kij = −a˙a = − ˙z = x(say), in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Next, we observe that the phase-space structures obtained following BL formalism although are identical to the Ostrogradsky/Dirac/Horowitz’s formalism, they all differ from the MHF upto a canonical transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We quote from [22] the general argument in connection with the total derivative terms, which runs as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' “it is just the change of the variables in the wave function and the phase transformation, plus the change of the integra- tion measure, and the transformation of the momenta respecting the change of the measure, and so a unitary transformation relates the two”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It’s possible (we have not found though) that each pair of quantum equations cast from {(27) and (28)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' {(31) and (32)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' {(45) and (47)}, are related by unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, it was also mentioned [22] that different forms of coupling parameter yield different quantum dynamics in the case of MHF, due to the presence of a coupling term (f ′(φ)pφ ) for non-minimal coupled case, and (γ′(φ)pφ ) for the Gauss-Bonnet-Dilaton coupled case, in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Thus, different unitary transformations (if exist) are required to relate the last two pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Such coupling as well as the derivative of coupling parameter remain absent in other formalisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In a nutshell, unitary transformation relating each pair is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Further, the semiclassical wave functions found for all the three cases studied here, exhibit different pre-factors and exponents for each pair [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This generates different probability amplitude and the evolution of the wave function while entering the classical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Finally, it is important to note that, if the coupling parameter f(φ) is treated as constant in Subsection ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=', the Hamiltonian (32) merely reduces to (28), while the Hamiltonian (31) reduces to (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence the question is: which of the two should be treated as the correct quantum description of the models under consideration?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In this connection we mention that a serious problem arises with Ostrogradsky/Dirac/Horowitz as well as with BL 11 formalisms when considering Gauss-Bonnet-Dilaton induced action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' To be specific, in Subsection ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' if γ(φ) is treated as a constant, then the contribution of Gauss-Bonnet term disappears from the Hamiltonian (47), and it reduces to (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Indeed, it should since as mentioned, Gauss-Bonnet term is topologically invariant in 4- dimensions, and so without functional coupling, it does not contribute to the field equations and the Hamiltonian as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' On the contrary, a constant γ does not affect the form of the Hamiltonian (45), and it does not reduce to (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This means, if we had started with a constant γ from the very beginning, all the terms appearing with γ in (45) would have been absent, and the end result would be (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' While, after constructing the Hamiltonian with arbitrary γ = γ(φ), if we set it equal to a constant, then its contribution remains present, and we obtain a different Hamiltonian, altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Clearly this is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence, we realize that boundary terms indeed play a crucial role while constructing the phase-space structure of non-linear theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In fact, if boundary terms are taken into account from the very beginning, treating hij as the basic variable, then Horowitz’s formalism reduces to the MHF, as already demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It was also noticed that if Dirac algorithm is applied after integrating the action by parts, then it also yields Hamiltonian identical to MHF [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is therefore suggestive to test the same for BL formalism too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In this section we shall first integrate actions by parts to get rid of the total derivative terms and follow the BL formalism thereafter, to explore the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='1 Scalar-tensor theory: minimal coupling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Upon integrating the action (30) by parts, we obtain A1 = � \uf8ee \uf8f0− 3α ˙z2 2N√z + 6αkN√z + 9β N√z \uf8f1 \uf8f2 \uf8f3 � ¨z N − ˙N ˙z N 2 �2 + 2k ˙z2 z + 4k2N 2 \uf8fc \uf8fd \uf8fe + z 3 2 � ˙φ2 2N − NV �\uf8f9 \uf8fb dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (57) Replacing ˙z by 2N 3 q in view of (13), the above action may be cast as, A1q = � � −2 3αN q2 √z + 6αkN√z + 9β N√z �4 9 ˙q2 + 8kN 2q2 9z + 4k2N 2 � + z 3 2 � ˙φ2 2N − NV �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (58) Note that the action (58) cannot be expressed only in terms of velocities, due to the explicit presence of q unlike (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, similar situation arrived at, in the case of Gauss-Bonnet-Dilaton case, and so it doesn’t matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The canonical momenta are the following: pq = 8β N√z ˙q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pφ = z 3 2 N ˙φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' pz = 0 = pN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (59) Dirac constraint analysis appears to be inevitable, since the action is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However as mentioned, the lapse function N being the Lagrange multiplier, the constraint strongly vanishes, so that one can ignore it without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Still, another primary constraint pz = 0 is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, as already noticed, in BL formalism, Dirac analysis may be bypassed despite the presence of the constraint pz = 0 in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The Lagrangian density is: L1q = −2 3αN q2 √z + 6αkN√z + 9β N√z �4 9 ˙q2 + 8kN 2q2 9z + 4k2N 2 � + z 3 2 � ˙φ2 2N − NV � , (60) and hence the Hamiltonian reads as, NHm MBL = pq ˙q + pz ˙z + pφ ˙φ − L1q = N√z 16β p2 q + 2 3Nqpz + N 2z 3 2 p2 φ + 2αNq2 3√z − 6αkN√z − 8βkNq2 z 3 2 − 36βk2N √z + NV z 3 2 , (61) where we have used (59) and replaced ˙z by 2N 3 q, in view of (13), and the suffix {MBL} now stands for ‘Modified Buchbinder-Lyakhovich’ formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Finally as before, for the sake of comparison, if we perform the canonical 12 transformation q → 3 2x, and pq → 2 3px, then the above Hamiltonian (61) may be expressed in the following form, Hm MBL = xpz + √z 36β p2 x + p2 φ 2z 3 2 + 3α 2√z (x2 − 4kz) − 18βk z 3 2 (x2 + 2kz) + V z 3 2 , (62) which is identical to Hm MHF presented in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='2 Scalar-tensor theory: non-minimal coupling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here again, upon integrating the action (30) by parts we obtain, A2 = � � − 3f ˙z2 2N√z − 3f ′ ˙φ ˙z√z N + 6fkN√z + 9β N√z �� ¨z N − ˙N ˙z N 2 �2 + 2k ˙z2 z + 4k2N 2� + z 3 2 � ˙φ2 2N − NV �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (63) Now, replacing ˙z by 2N 3 q in view of (13), the above action (63) may be cast as, A2 = � � −2 3fN q2 √z − 2f ′√zq ˙φ + 6fkN√z + 9β N√z �4 9 ˙q2 + 8kN 2q2 9z + 4k2N 2 � + z 3 2 � ˙φ2 2N − NV �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (64) Canonical momenta may therefore be found as, pq = 8β N√z ˙q, pφ = −2f ′q√z + z 3 2 N ˙φ, pN = 0 = pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (65) As before, leaving out the constraint associate with the lapse function, and replacing ˙z = 2N 3 q in view of (13), the Hamiltonian may be cast as, NHnmMBL = pq ˙q + pz ˙z + pφ ˙φ − L = N � √z 16β p2 q + 2 3qpz + p2 φ 2z 3 2 + 2f ′ z qpφ + 2fq2 3√z + 2f ′2q2 √z − 6kf√z − 8βkq2 z 3 2 − 36βk2 √z + V z 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (66) Finally, applying the canonical transformation relations q → 3 2x, and pq → 2 3px, we obtain HnmMBL = xpz + √z 36β p2 x + p2 φ 2z 3 2 + 3x z f ′pφ + 3f 2√z (x2 − 4kz) − 18βk z 3 2 (x2 + 2kz) + 9x2f ′2 2√z + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (67) Clearly, HnmMBL ∼= HnmMHF presented in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='3 Einstein-Gauss-Bonnet-Dilatonic action Eventually, in order to construct the correct Hamiltonian in connection with the Einstein-Gauss-Bonnet-Dilatonic action (35), let us integrate it by parts to obtain, A3 = � � α � − 3 ˙z2 2N√z + 6kN√z � + 9β N√z �� ¨z N − ˙N ˙z N 2 �2 + 2k ˙z2 z + 4k2N 2� − γ′(φ) ˙z ˙φ N√z � ˙z2 N 2z + 12k � + z 3 2 � ˙φ2 2N − NV �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (68) 13 As before, replacing ˙z by 2N 3 q in view of (13), the above action may be cast as, A3q = � � − 2 3αN q2 √z + 6αkN√z + 9β N√z �4 9 ˙q2 + 8kN 2q2 9z + 4k2N 2 � − 2qγ′(φ) ˙φ 3√z �4q2 9z + 12k � + z 3 2 � ˙φ2 2N − NV � � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (69) Canonical momenta are now found as, pq = 8β N√z ˙q, pφ = −2qγ′(φ) 3√z �4q2 9z + 12k � + z 3 2 N ˙φ, pN = 0 = pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (70) As always,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' leaving out the constraint associated with the lapse function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' and replacing ˙z = 2N 3 q in view of (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' the Hamiltonian may be cast as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' NHGB MBL =pq ˙q + pz ˙z + pφ ˙φ − L = N � √z 16β p2 q + 2 3qpz + p2 φ 2z 3 2 + 2αq2 3√z − 6kα√z − 8βkq2 z 3 2 − 36βk2 √z + 2qγ′(φ)pφ 3z2 �4q2 9z + 12k � + 2q2γ′2(φ) 9z 5 2 �4q2 9z + 12k �2 + V z 3 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (71) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' the set of canonical transformations q → 3 2x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' and pq → 2 3px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' allows one to express the Hamiltonian (71) as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' HGBMBL =xpz + √z 36β p2 x + p2 φ 2z 3 2 + 3α 2√z (x2 − 4kz) − 18βk z 3 2 (x2 + 2kz) + xγ′(φ) z2 �x2 z + 12k � pφ + γ′2(φ)x2 2z 5 2 �x2 z + 12k �2 + V z 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (72) As a result one finds, HGBMBL ∼= HGBMHF presented in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is important to note that in the process of constructing the Hamiltonian starting from a divergent free action, the pathology discussed in regard of canonical formulation of Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher-order term is also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 4 Application It is mentioned in the introduction that canonical formulation is a precursor to canonical quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the absence of a viable quantum theory of gravity, it is suggestive to canonically quantize the cosmological equa- tion and study quantum cosmology to extract some ethos of pre-Planck era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' For example, one can explore the Euclidean wormhole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless, ‘cosmological inflationary scenario’ has been developed since 1980, to solve horizon, flatness (fine tuning), structure formation and monopole problems, singlehandedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Short-lived (10−36 −10−26)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' inflation, occurred just after Planck’s era and falls within the periphery of ‘quantum field theory in curved space-time’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' To be more specific, ‘inflation is a quantum theory of perturbations on the top of the classical background’, so that the energy scale of the background remains much below Planck’s scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Nonetheless in this context, Hartle [73] prescribed that, most of the important physics may still be extracted from the classical action provided, the semiclassical wave-function is strongly peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The reason being, in that case correlation between the geometrical and matter degrees of freedom is established, and hence the emergence of classical trajec- tories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' the universe) is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hence, quantization and an appropriate semiclassical approximation must be treated as a forerunner to study inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Canonical quantization and the semiclassical wave-function in connection with the Hamiltonian (67) for non- minimally coupled higher order theory had been presented in [26], which reduces to the minimally coupled case 14 when the coupling parameter becomes constant [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The Hamiltonian operator was found to be hermitian, stan- dard probabilistic interpretation holds, and the semiclassical wave-functions was found to be oscillatory about the classical inflationary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Inflation has been studied and the parameters are found with excellent agreement with the observational constraints [74,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Gravitational perturbation has also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In [29] again, the quantum counterpart of the Hamiltonian (72) in connection with Einstein-Gauss-Bonnet- Dilatonic coupled action has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Hermiticity of the Hamiltonian operator has been established, probabilistic interpretation is explored, and the semiclassical wave-function is found to be oscillatory about a classical inflationary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Finally, we have studied inflation and found that the inflationary parameters more- or-less satisfy observational constraints [74,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In a nut-shell, the results obtained in [29] are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' iℏ∂Ψ ∂σ = � − ℏ2φ 54β0x � ∂2 ∂x2 + n x ∂ ∂x � − ℏ2 3xσ 4 3 ∂2 ∂φ2 + 2iℏα0 σ � 1 φ2 ∂ ∂φ − 1 φ3 � − 2iℏγ0x2 3σ 7 3 � 2φ ∂ ∂φ + 1 � + Ve � Ψ = �HeΨ, (73) where, the proper volume, σ = z 3 2 = a3 plays the role of internal time parameter, and n is the operator ordering index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In the above equation, �He is the effective hermitian Hamiltonian operator, while the the effective potential Ve is given by, Ve = 3α2 0x σ 2 3 φ4 − 4α0γ0x3 σ2φ + 4γ2 0x5φ2 3σ 10 3 + α0x σ 2 3 φ + λ2σ 2 3 φ2 3x + 2σ 2 3 ΛM 2 P x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' (74) The effective Hamiltonian operator is found to be hermitian for n = −1, which selects the operator ordering parameter from physical consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Standard quantum mechanical probability interpretation also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Under a suitable (WKB) semiclassical approximation, the wave-function has been found to be, Ψ = Ψ0e i ℏ � − 6α0λz2 a0φ0 +16γ0a2 0φ2 0λ3√z � , (75) which exhibits oscillatory behaviour about the classical inflationary solution a = a0eλt , where, α0, φ0, γ0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' We have also presented several sets of inflationary parameters in [29], which depict that the spectral index of scalar perturbation and the scalar to tensor ratio lie within the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='967 ≤ ns ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='979 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='056 ≤ r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content='089 respectively, showing reasonably good agreement with the recently released data [74,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The number of e-folding also remains within the acceptable range 46 < N < 73, which is sufficient to solve the horizon and flatness problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' 5 Concluding remarks Although initiated two centuries back, canonical formulation of higher-order theory of gravity is particularly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In fact, only after probing Dilatonic coupled Gauss-Bonnet action, it is learnt that divergent terms play a vital role to formulate correct quantum dynamics of non-linear gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' The scheme is therefore first, to express the action in terms of the basic variable hij , otherwise if expressed in terms of the scale factor, as commonly done, some unwanted divergent terms are removed in the process of integration by parts, which are unaccredited by the variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Next, unless divergent terms are taken care of, the Hamiltonian is found to be different, which is related through canonical transformation though, such transformation cannot be carried over in the quantum domain due to non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It is shown that in the case of Einstein-Gauss-Bonnet-Dilatonic coupled action in 4-dimension that, unless the action is divergent free, an erroneous Hamiltonian is constructed, since it does not reflect the topological invariance of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' This proves the importance of divergent terms in higher order theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In this respect the difference of BL formalism with MHF is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In fact BL formalism produces identical Hamiltonian as obtained earlier following Ostrogradsky’s, Dirac’s or Horowitz’s formalisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' However, MHF is essentially the Horowitz formalism, after expressing the action in terms of the three space curvature and taking care of the total derivative terms under integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' It was shown that following the same route if Dirac’s algorithm is applied, the Hamiltonian becomes identical to the one found following MHF, 15 and one obtains unique quantum description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' Here, we reveal that the same is true with BL formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In fact, BL formalism not only bypasses constraint analysis, as in the case of Horowitz’s formalism, it also does not require auxiliary variable to cast the action in canonical form, which is a bit intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfqw9o/content/2301.05710v1.pdf'} +page_content=' In a straightforward manner, it establishes diffeomorphic invariance, and therefore is the easiest technique to handle higher-order theories.' metadata={'source': 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a/4tE3T4oBgHgl3EQfowri/content/tmp_files/2301.04637v1.pdf.txt b/4tE3T4oBgHgl3EQfowri/content/tmp_files/2301.04637v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..651417767b62508a3509d0dc240cb8fce962ad91 --- /dev/null +++ b/4tE3T4oBgHgl3EQfowri/content/tmp_files/2301.04637v1.pdf.txt @@ -0,0 +1,3551 @@ +DRAFT VERSION JANUARY 12, 2023 +Typeset using LATEX twocolumn style in AASTeX62 +A Systematic Study of Ia-CSM Supernovae from the ZTF Bright Transient Survey +YASHVI SHARMA,1 JESPER SOLLERMAN,2 CHRISTOFFER FREMLING,1 SHRINIVAS R. KULKARNI,1 KISHALAY DE,3 IDO IRANI,4 +STEVE SCHULZE,5 NORA LINN STROTJOHANN,4 AVISHAY GAL-YAM,4 KATE MAGUIRE,6 DANIEL A. PERLEY,7 ERIC C. BELLM,8 +ERIK C. KOOL,2 THOMAS BRINK,9 RACHEL BRUCH,4 MAXIME DECKERS,6 RICHARD DEKANY,10 ALISON DUGAS,11 +SAMANTHA GOLDWASSER,4 MATTHEW J. GRAHAM,1 MELISSA L. GRAHAM,8 STEVEN L. GROOM,12 MATT HANKINS,13 +JACOB JENCSON,14 JOEL P. JOHANSSON,5 VIRAJ KARAMBELKAR,1 MANSI M. KASLIWAL,1 FRANK J. MASCI,12 +MICHAEL S. MEDFORD,15, 16 JAMES D. NEILL,1 GUY NIR,9 REED L. RIDDLE,10 MICKAEL RIGAULT,17 TASSILO SCHWEYER,2 +JACCO H. TERWEL,6, 18 LIN YAN,1 YI YANG (杨轶) ,9 AND YUHAN YAO1 +1Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA +2Department of Astronomy, The Oskar Klein Center, Stockholm University, AlbaNova, 10691 Stockholm, Sweden +3MIT-Kavli Institute for Astrophysics and Space Research, Cambridge, MA 02139, USA +4Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 234 Herzl St, 76100 Rehovot, Israel +5Department of Physics, The Oskar Klein Center, Stockholm University, AlbaNova, 10691 Stockholm, Sweden +6School of Physics, Trinity College Dublin, the University of Dublin, College Green, Dublin, Ireland +7Astrophysics Research Institute, Liverpool John Moores University, Liverpool Science Park, 146 Brownlow Hill, Liverpool L35RF, UK +8DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA +9Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA +10Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA +11Institute for Astronomy, University of Hawai’i, Honolulu, HI 96822, USA +12IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA +13Arkansas Tech University, Russellville, AR 72801, USA +14Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA +15Department of Astronomy, University of California, Berkeley, Berkeley, CA 94720 +16Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley, CA 94720 +17Universit´e Clermont Auvergne, CNRS/IN2P3, Laboratoire de Physique de Clermont, 63000 Clermont-Ferrand, France +18Isaac Newton Group (ING), Apt. de correos 321, E-38700, Santa Cruz de La Palma, Canary Islands, Spain +ABSTRACT +Among the supernovae (SNe) that show strong interaction with the circumstellar medium, there is a rare +subclass of Type Ia supernovae, SNe Ia-CSM, that show strong narrow hydrogen emission lines much like SNe +IIn but on top of a diluted over-luminous Type Ia spectrum. In the only previous systematic study of this class +(Silverman et al. 2013), 16 objects were identified, 8 historic and 8 from the Palomar Transient Factory (PTF). +Now using the successor survey to PTF, the Zwicky Transient Facility (ZTF), we have classified 12 additional +objects of this type through the systematic Bright Transient Survey (BTS). In this study, we present and analyze +the optical and mid-IR light curves, optical spectra and host galaxy properties of this sample. Consistent with +previous studies, we find the objects to have slowly evolving light curves compared to normal SNe Ia with peak +absolute magnitudes between −19.1 and −21, spectra having weak Hβ, large Balmer decrements of ∼ 7 and +strong Ca NIR emission. Out of 10 SNe from our sample observed by NEOWISE, 9 have 3σ detections, along +with some showing a clear reduction in red-wing of Hα, indicative of newly formed dust. We do not find our +SN Ia-CSM sample to have significantly different distribution of equivalent width of He I λ5876 than SNe IIn +as observed in Silverman et al. (2013). The hosts tend to be late-type galaxies with recent star formation. We +also derive a rate estimate of 29+27 +−21 Gpc−3 yr−1 for SNe Ia-CSM which is ∼0.02–0.2% of the SN Ia rate. This +work nearly doubles the sample of well studied Ia-CSM objects in Silverman et al. (2013), increasing the total +number to 28. +Corresponding author: Yashvi Sharma +yssharma@astro.caltech.edu +arXiv:2301.04637v1 [astro-ph.HE] 11 Jan 2023 + +2 +Keywords: circumstellar matter – supernovae: general – supernovae: individual (SN 1997cy, SN 2002ic, SN +2005gj, SN 2005ip, SN 2006jc, SN 2008J, SN 2009ip, SN 2010jl, PTF11kx, SN 2012ca, SN 2013dn, +SN 2018crl, SN 2018gkx, SN 2018evt, SN 2019agi, SN 2019ibk, SN 2019rvb, SN 2020onv, SN +2020qxz, SN 2020uem, SN 2020xtg, SN 2020abfe, SN 2020aekp) +1. INTRODUCTION +When it comes to supernovae (SNe) interacting with cir- +cumstellar material (CSM), a number of sub-types of core- +collapse SNe (CCSNe) show signs of strong interaction, like +SNe IIn (Schlegel 1990; Filippenko 1997), SNe Ibn (Pas- +torello et al. 2008; Foley et al. 2007; Chugai 2009; Hos- +seinzadeh et al. 2017) and most recently SNe Icn (Gal-Yam +et al. 2021, 2022; Perley et al. 2022). SN IIn progenitors are +generally thought to be massive stars (like Luminous Blue +Variables, LBVs) that lose their hydrogen envelopes to wind- +driven mass loss and outbursts (Gal-Yam et al. 2007; Gal- +Yam & Leonard 2009; Kiewe et al. 2012; Taddia et al. 2013; +Smith 2014). Helium-rich but hydrogen-deficient CSM in +the case of SNe Ibn (Pastorello et al. 2008; Foley et al. 2007; +Chugai 2009) and both hydrogen and helium deficient CSM +in SNe Icn (Gal-Yam et al. 2022; Perley et al. 2022; Pel- +legrino et al. 2022) are thought to arise from high-velocity +wind mass loss or stripping of the envelope in binary con- +figurations of massive Wolf-Rayet (WR) like stars. For SNe +IIn in most cases, the mass-loss rate derived from the CSM +velocity is consistent with estimates from LBV-like eruptive +mass loss. +However, there exists a rare sub-type of thermonuclear su- +pernovae (SNe Ia) which also interacts strongly with CSM +i.e. SNe Ia-CSM. This class poses a challenge to the progen- +itor debate of SNe Ia. There is some consensus on there being +at least two major progenitor channels for SNe Ia; the double- +degenerate (DD) channel (Webbink 1984; Iben & Tutukov +1984) which is the merging of two C/O white dwarfs and +the single-degenerate (SD) channel (Whelan & Iben 1973) +where the white dwarf accretes enough material from a non- +degenerate companion to explode. Although there are more +arguments for the DD scenario from observations of nearby +SNe Ia (Nugent et al. 2011; Li et al. 2011; Brown et al. 2012; +Bloom et al. 2011), the strongest observational evidence for +the SD scenario are SNe Ia with CSM. +Indications of CSM around SNe Ia ranges from detec- +tion of time varying narrow Na ID absorption lines (Patat +et al. 2007; Blondin et al. 2009; Simon et al. 2009) in high- +resolution spectra (found in at least 20% of SNe Ia in spiral +hosts, Sternberg et al. 2011; Maguire et al. 2013; Clark et al. +2021), to strong intermediate and narrow Balmer emission +features in the spectra and large deviations of the light curves +from the standard shape. The latter phenomena have been +named SNe Ia-CSM (Silverman et al. 2013), but were ear- +lier referred to as “SNe IIna” or “SNe Ian” due to the strong +similarity between their spectra and those of SNe IIn. The +first two examples of this class studied in detail were SNe +2002ic (Hamuy et al. 2003; Deng et al. 2004; Wang et al. +2004; Wood-Vasey et al. 2004; Kotak & Meikle 2005; Chugai +et al. 2004) and 2005gj (Aldering et al. 2006; Prieto et al. +2007), but for a long time there was ambiguity regarding their +thermonuclear nature (Benetti et al. 2006). These SNe were +dominated by interaction from the first spectrum and were +quite over-luminous compared to normal SNe Ia. The first +clear example of a thermonuclear SN Ia-CSM was PTF11kx +(Dilday et al. 2012; Silverman et al. 2013). It looked like a +luminous SN Ia (99aa-like) at early phases but started show- +ing interaction at ∼ 60 days from explosion and thereafter +strongly resembled SNe 2002ic and 2005gj at late times. +Higher resolution spectra taken at early times indicated mul- +tiple shells of CSM with some evacuated regions in between. +Dilday et al. (2012) suggested a symbiotic nova progenitor +involving a WD and a red giant (similar to RS Ophiuchi) +could produce such CSM distribution, however later studies +argued that the massive CSM of PTF11kx was inconsistent +with the mass-loss rates from symbiotic nova systems (Sil- +verman et al. 2013; Soker et al. 2013). +Ever since, a handful of SNe of this class have been stud- +ied in detail to investigate their progenitors and to distinguish +them from their spectroscopic cousins, the Type IIn SNe. +Both SN Ia-CSM and SN IIn spectra share a blue quasi- +continuum, a strong Hα feature with an intermediate and a +narrow component, and often a broad Ca NIR triplet fea- +ture, but they differ with regards to the line strength of Hβ, +strength/presence of helium and presence of emission lines +from intermediate mass elements often found in CCSNe. +There are some individual SNe with unclear type often re- +ferred to as SN Ia-CSM/IIn, like SN 2012ca for which some +papers argue for core-collapse (Inserra et al. 2014, 2016) and +others for a thermonuclear origin (Fox et al. 2015). This am- +biguity becomes more dominant as the underlying SN flux +gets smaller compared to the interaction power (Leloudas +et al. 2015). Silverman et al. (2013, hereafter S13) is the +only study to analyze a sample of SNe Ia-CSM, 16 objects +in total including 6 previously known, 3 re-discovered (re- +classified SNe IIn) and 7 new from the Palomar Transient +Factory (PTF). Their paper presents the common properties +of optical light curves, spectra and host galaxies and contrast +them against SN IIn properties. In this paper, we present +12 new SNe Ia-CSM discovered as part of the Zwicky Tran- +sient Facility’s (ZTF; Bellm et al. 2019; Graham et al. 2019; + +3 +Dekany et al. 2020) Bright Transient Survey (BTS; Fremling +et al. 2020; Perley et al. 2020) and analyze their optical light +curves, spectra, hosts and rates. Throughout this paper, we +have compared the results derived from our sample to the +ones in S13. +This paper is organised as follows; we first discuss the sam- +ple selection criteria, the photometric and spectroscopic data +collection in §2, then the analysis of light- and color-curves +and the bolometric luminosities is done in §3.1. The analysis +of early and late-time spectra and emission line identification +is presented in §3.2, and analysis of the host galaxies is pro- +vided in §3.3. The rates are estimated from the BTS survey +in §3.4. We end with a discussion about the nature of SN +Ia-CSM progenitors and a summary in §4 and §5. +2. OBSERVATIONS AND DATA REDUCTION +In this section, we outline our selection criteria, and +present the optical photometry and spectroscopic observa- +tions of the 12 SNe Ia-CSM in our sample. +2.1. Selection Criteria +To carefully curate our sample of SNe Ia-CSM, we used +the BTS sample and its publicly available BTS Sample Ex- +plorer1 website to obtain the list of all classified Type Ia sub- +types during the period 2018-05-01 to 2021-05-01. We then +filter out oddly behaving Type Ia SNe based on their light- +curve properties. We used two criteria; the primary being +rest-frame duration considering flux above 20% of peak flux, +and the second being change in magnitude after 30 days from +peak (∆m30). We calculated these two properties from either +g or r-band light curves (whichever had maximum number of +detections) grouped into 3-day bins and used Gaussian Pro- +cess Regression2 to interpolate the light curves where cov- +erage was missing. For the first filtering, we calculated the +mean (µ ≈ 35 days) and standard deviation (σ ≈ 16 days) +of the duration distribution and selected everything that had +a duration greater than µ + 3σ. Given the large sample size +(N = 3486), the standard error on the mean is ∼ 0.5 days, +hence our duration cut of 3σ is suitable. This filtering se- +lected 41 out of 3486 BTS SNe Ia. +Then from these 41 +SNe, we calculated the mean and standard deviation of the +∆m30 distribution and removed SNe that were more than 1σ +away from the mean on the higher side to reject the relatively +steeply declining long SNe, which resulted in 35 SNe being +kept. Again, the mean and standard deviation of ∆m30 dis- +tribution of these 41 long duration SNe are 0.48 mag and 0.27 +mag respectively and the standard error on mean is ∼ 0.04, +making our 1σ cut suitable. Finally, we manually inspected +1 https://sites.astro.caltech.edu/ztf/bts/explorer.php +2 Pedregosa et al. (2011) https://scikit-learn.org/stable/modules/gaussian +process.html +the 35 selected SNe Ia to confirm their classification. 20 out +of the 35 SNe that passed the above filtering criteria were +just normal SNe Ia either caught late or missing some post- +peak coverage in ZTF or had spurious detections that resulted +in long duration estimates, 2 had incorrect duration estimate +due to an interpolation error and were recalculated and 1 +(AT2020caa; Soraisam et al. 2021) had some detections be- +fore the SN explosion which could be connected to a different +SN (i.e. a sibling; Graham et al. 2022). +The remaining 12 long-duration SNe Ia all turned out to be +spectroscopically classified SNe Ia-CSM in BTS, and none +of the classified BTS SNe Ia-CSM were missed in this fil- +tering. No other SNe apart from these stood out in particu- +lar, indicating the classification reliability of the BTS sample. +During the same period, 9 SNe Ia-CSM were reported to the +Transient Name Server (TNS), out of which 7 are already in +our sample, 1 was detected by ZTF but did not meet the BTS +criteria, and 1 was not detected in ZTF as the transient lo- +cation fell too close to the field edges and was masked by +the automated image subtraction pipeline. Yao et al. (2019) +presented early photometric observations of one SN Ia-CSM +in our sample, SN 2018crl. Table 1 summarizes the coor- +dinates, redshifts, peak absolute magnitudes, durations, host +galaxy information and Milky Way extinction for the 12 SNe +Ia-CSM in our sample. +Furthermore, we re-checked the classifications of 142 SNe +IIn classified in BTS during the same period as above, in case +any SN Ia-CSM was masquerading among them and found 6 +to have ambiguous classifications. These are discussed fur- +ther in Appendix A. +2.2. Discovery +All SNe Ia-CSM were detected by the ZTF (Bellm et al. +2019; Graham et al. 2019; Dekany et al. 2020) and passed +the criteria for the BTS (Fremling et al. 2020; Perley et al. +2020) automatic filtering, i.e. extra-galactic real transients +with peak magnitudes brighter than 19 mag. +These were +saved and classified as part of BTS which aims to classify all +transients brighter than 18.5 magnitude, and reported to the +Transient Name Server3 (TNS) during the period 2018-05- +01 to 2021-05-01. Out of the 12 SNe, 6 were first reported to +TNS (i.e. discovered) by ZTF (AMPEL, Nordin et al. 2019; +Soumagnac & Ofek 2018 and BTS), 3 were first reported by +GaiaAlerts (Hodgkin et al. 2021), 2 by ATLAS (Smith et al. +2020) and 1 by ASAS-SN (Shappee et al. 2014). For classi- +fication, 9 were classified by the ZTF group, 1 by ePESSTO +(Smartt et al. 2015; Stein et al. 2018a), 1 by SCAT (Tucker +et al. 2018; Payne et al. 2019) and 1 by the Trinity College +Dublin (TCD) group (Prentice et al. 2020). The follow-up +spectral series for these SNe were obtained as part of the +3 https://www.wis-tns.org/ + +4 +Table 1. Properties of the 12 BTS SNe Ia-CSM +ZTF Name +IAU Name +z +M peak +r +Duration1 +Host Name +Host Mag2 +(mag) +(days) +(mr) +ZTF18aaykjei +SN 2018crl +0.097 +-19.66 +130 +SDSS J161938.90+491104.5 +18.89 +ZTF18abuatfp +SN 2018gkx +0.1366 +-20.07 +322 +SDSS J135219.22+553830.2 +18.23 +ZTF18actuhrs +SN 2018evt +0.02378 +-19.10 +447 +MCG-01-35-011 +14.07 +ZTF19aaeoqst +SN 2019agi +0.0594 +<-18.76 +>303 +SDSS J162244.06+240113.4 +17.82 +ZTF19abidbqp +SN 2019ibk +0.04016 +<-17.55 +>576 +SDSS J014611.93-161701.1 +15.55 +ZTF19acbjddp +SN 2019rvb +0.1835 +-20.74 +172 +WISEA J163809.90+682746.3 +20.44 +ZTF20abmlxrx +SN 2020onv +0.095 +<-20.36 +>154 +WISEA J231646.31-231839.9 +17.95 +ZTF20abqkbfx +SN 2020qxz +0.0964 +-20.00 +166 +WISEA J180400.99+740050.0 +17.65 +ZTF20accmutv +SN 2020uem +0.041 +<-20.17 +>279 +WISEA J082423.32-032918.6 +15.88 +ZTF20aciwcuz +SN 2020xtg +0.0612 +<-19.60 +>336 +SDSS J153317.64+450022.8 +15.42 +ZTF20acqikeh +SN 2020abfe +0.093 +-20.24 +171 +SDSS J200003.30+100904.2 +20.18 +ZTF21aaabwzx +SN 2020aekp +0.046 +-19.62 +458 +SDSS J154311.45+174843.7 +18.41 +1 Rest frame duration above 20% of r-band peak flux, uncertainty of ±2 − 3 days from ZTF cadence. +2 Corrected for Galactic extinction. +BTS classification campaign as many were difficult to clas- +sify with the ultra-low resolution spectrograph P60/SEDM +(Blagorodnova et al. 2018) and hence were followed up with +intermediate resolution spectrographs. The SEDM spectra +were helpful in determining an initial redshift but the tem- +plate matches were unclear (matched to SN IIn as well as +SN Ia-CSM and SN Ia-pec templates, some matched poorly +to SN Ia/Ic at early times). SNe 2019agi (classification and +spectrum taken from TNS), 2019rvb, 2020onv, 2020qxz and +2020uem were classified as Ia-CSM ∼ 1 − 2 month after +discovery using spectra at phases of 42, 26, 38, 45 and 51 +days respectively. SNe 2018crl, 2018gkx and 2019ibk were +classified ∼ 2 − 3 months after discovery using spectra at +phases of 92, 75 and 103 days respectively. SNe 2018evt, +2020abfe and 2020aekp were classified ∼ 4 − 5 months af- +ter discovery using the spectra at phases of 144, 146 and 132 +days respectively. SN 2020xtg immediately went behind the +sun after its first detection in ZTF therefore its first spectrum +(using SEDM) was taken at 91 days since explosion which +was dominated by strong Hα emission, and thus SN 2020xtg +was initially classified as a Type II. As this SN was exhibiting +a long lasting light curve, an intermediate resolution spec- +trum was taken at 340 days which matched very well to SN +Ia-CSM and therefore its classification was updated. SNe +2020uem and 2020aekp showed peculiar features and were +followed up for more optical spectroscopy for single object +studies (to be presented in future papers). +2.3. Optical photometry +To assemble our sample light curves, we obtained forced +PSF photometry via the ZTF forced-photometry service +(Masci et al. 2019; IRSA 2022) in g, r and i bands and +also added data from ATLAS (Tonry et al. 2018; Smith et al. +2020) forced-photometry service in c and o bands. The high +cadence ZTF partnership survey in i band contributed some +photometry to SNe 2018crl, 2018gkx, 2019agi, 2019ibk and +2019rvb. The ZTF and ATLAS data were supplemented with +data from the Rainbow camera (RC, Ben-Ami et al. 2012) +on the robotic Palomar 60-inch telescope (P60, Cenko et al. +2006) and the Optical wide field camera (IO:O) on the Liver- +pool telescope (LT, Steele et al. 2004). The P60 data was pro- +cessed with the automatic image subtraction pipeline FPipe +(Fremling et al. 2016) using reference images from SDSS +when available, and otherwise from Pan-STARRS1. +The +IO:O data was initially reduced with their standard pipeline4 +then image subtraction was carried out using the method +outlined in Taggart (2020). +For SN 2018evt, some early +time data available from ASAS-SN (Shappee et al. 2014; +Kochanek et al. 2017) in the V band was obtained through +their Sky Patrol5 interface. +We corrected all photometry for Milky Way extinction +with the Python package extinction (Barbary 2016) us- +ing the dust extinction function from Fitzpatrick (1999), the +Schlafly & Finkbeiner (2011) dust map, and an RV of 3.1. +Then we converted all measurements into flux units for anal- +ysis and considered anything less than a 3σ detection an up- +per limit. There is moderate to good coverage in g, r, c and +o bands for all SNe in our sample. Figure 1 shows a multi- +paneled figure of the light curves of the objects in our sample. +2.4. Mid-IR photometry +4 https://telescope.livjm.ac.uk/TelInst/Pipelines/ +5 https://asas-sn.osu.edu/ + +5 +0 +80 +160 +240 +320 +400 +-15.5 +-16.9 +-18.2 +-19.6 +-21.0 +Co decay +SN 2018crl +2.11 mag/100 d +0.36 mag/100 d +0 +100 +200 +300 +400 +500 +-15.0 +-16.5 +-18.0 +-19.5 +-21.0 +Co decay +SN 2018gkx +1.18 mag/100 d +0.42 mag/100 d +0 +150 +300 +450 +600 +-14.0 +-15.6 +-17.2 +-18.9 +-20.5 +Co decay +SN 2018evt +0.51 mag/100 d +1.36 mag/100 d +0 +80 +160 +240 +320 +400 +-15.5 +-16.9 +-18.2 +-19.6 +-21.0 +Co decay +SN 2019rvb +0.93 mag/100 d +0.5 mag/100 d +0 +100 +200 +300 +400 +500 +-15.0 +-16.5 +-18.0 +-19.5 +-21.0 +Co decay +SN 2019agi +0.96 mag/100 d +0.45 mag/100 d +0 +150 +300 +450 +600 +-14.0 +-15.6 +-17.2 +-18.9 +-20.5 +Co decay +SN 2019ibk +0.61 mag/100 d +0.18 mag/100 d +0 +80 +160 +240 +320 +400 +-15.5 +-16.9 +-18.2 +-19.6 +-21.0 +Co decay +SN 2020qxz +1.75 mag/100 d +0.29 mag/100 d +0 +100 +200 +300 +400 +500 +-15.0 +-16.5 +-18.0 +-19.5 +-21.0 +D +Co decay +SN 2020onv +1.26 mag/100 d +1.06 mag/100 d +0 +150 +300 +450 +600 +-14.0 +-15.6 +-17.2 +-18.9 +-20.5 +Co decay +SN 2020uem* +0.6 mag/100 d +1.27 mag/100 d +0 +80 +160 +240 +320 +400 +D +-15.5 +-16.9 +-18.2 +-19.6 +-21.0 +Co decay +SN 2020abfe +1.51 mag/100 d +1.51 mag/100 d +0 +100 +200 +300 +400 +500 +-15.0 +-16.5 +-18.0 +-19.5 +-21.0 +Co decay +SN 2020xtg +0.62 mag/100 d +1.21 mag/100 d +0 +150 +300 +450 +600 +-14.0 +-15.6 +-17.2 +-18.9 +-20.5 +Co decay +SN 2020aekp +1.52 mag/100 d +0.07 mag/100 d +0.66 mag/100 d +Rest-frame days since explosion +Absolute magnitude +P48:ZTF: g +P48:ZTF: r +P48:ZTF: i +ATLAS: c +ATLAS: o +P60:RC: g +P60:RC: r +P60:RC: i +LT:IOO: g +LT:IOO: r +LT:IOO: i +LT:IOO: z +LCO: sdssg +LCO: sdssr +LCO: sdssi +ASASSN: V +Figure 1. Optical light curves of the ZTF BTS SN Ia-CSM sample. The SNe Ia-CSM have longer duration than the average SN Ia, with some +variety like bumpy light curves or long plateaus. The one SN marked with an asterisk (SN 2020uem) has an unconstrained explosion time +estimate (∼ ±50 d). The decline rate from Cobalt decay is marked with black dashed line, the light curve decline rates measured from r-band +data are shown in the subplot legends. + +6 +The transients were observed during the ongoing NEO- +WISE all-sky mid-IR survey in the W1 (3.4 µm) and W2 +(4.5 µm) bands (Wright et al. 2010a; Mainzer et al. 2014). +We retrieved time-resolved coadded images of the field cre- +ated as part of the unWISE project (Lang 2014a; Meisner +et al. 2018). To remove contamination from the host galax- +ies, we used a custom code (De et al. 2020) based on the +ZOGY algorithm (Zackay et al. 2016) to perform image sub- +traction on the NEOWISE images using the full-depth coadds +of the WISE and NEOWISE mission (obtained during 2010- +2014) as reference images. Photometric measurements were +obtained by performing forced PSF photometry at the tran- +sient position on the subtracted WISE images until the epoch +of the last NEOWISE data release (data acquired until De- +cember 2021). Further analysis of the mid-IR photometry is +presented in §3.1.4 +2.5. Optical spectroscopy +The main instruments used for taking spectra and the soft- +ware used to reduce the data are summarized in Table 2. Ad- +ditionally, the spectrum Reguitti (2020) obtained using the +Asiago Faint Object Spectrograph and Camera (AFOSC) on +the 1.8 m telescope at Cima Ekar, and the spectrum Stein +et al. (2018b) obtained using the ESO Faint Object Spectro- +graph and Camera version 2 (EFOSC2) on ESO New Tech- +nology Telescope (NTT) were taken from TNS. +The details for all optical spectra (61 for the sample in to- +tal) presented in this paper are provided in Table 3. Further- +more, all spectra were corrected for Milky Way extinction +using extinction and the same procedure as for the pho- +tometry. The SN redshifts were derived using narrow host +lines for the objects which did not already have a host red- +shift available in the NASA/IPAC Extragalactic Database6 +(NED). Photometric calibration was done for all spectra i.e. +they were scaled such that the synthetic photometry from the +spectrum matched the contemporaneous host-subtracted ZTF +r-band data. For SN 2018crl, a host galaxy spectrum taken +using P200/DBSP was available, which was subtracted from +the P200/DBSP SN spectrum taken at +92 days. For SN +2020aekp, more spectra beyond ∼ 350 days were obtained +but will be presented in a future study of the object (34 addi- +tional spectra up to ∼600 day). +These processed spectra were used for the rest of the anal- +ysis as detailed in §3.2 and will be available on WISeREP7 +(Yaron & Gal-Yam 2012). +3. ANALYSIS +3.1. Photometry +6 https://ned.ipac.caltech.edu/ +7 https://www.wiserep.org/ +Table 2. Description of spectrographs used for follow-up and the +corresponding data reduction pipelines +Inst. +Telescope +Reduction Software +SEDM1 +Palomar 60-inch (P60) +pySEDM2 +ALFOSC3 +Nordic Optical Telescope +IRAF4, PyNOT14, pypeit +DBSP5 +Palomar 200-inch (P200) +IRAF6, DBSP DRP7 +KAST8 +Shane 3-m +IRAF +LRIS9 +Keck-I +LPipe10 +SPRAT11 +Liverpool Telescope +Barnsley et al. (2012) +DIS12 +APO13 +IRAF +1 Spectral Energy Distribution Machine (Blagorodnova et al. 2018) +2 Rigault et al. (2019) +3 Andalucia Faint Object Spectrograph and Camera +4 Tody (1986, 1993) +5 Double Beam Spectrograph (Oke & Gunn 1982) +6 Standard pipeline by Bellm & Sesar (2016) used prior to Fall +2020 +7 pypeit (Prochaska et al. 2020) based pipeline (https://github. +com/finagle29/dbsp drp) used since Fall 2020 +8 Kast Double Spectrograph (Miller & Stone 1987) +9 Low Resolution Imaging Spectrometer (Oke et al. 1995) +10 IDL based automatic reduction pipelinea (Perley 2019) +11 Spectrograph for the Rapid Acquisition of Transients (Piascik +et al. 2014) +12 Dual Imaging Spectrograph +13 Astrophysics Research Consortium telescope at the Apache +Point Observatory +14 https://github.com/jkrogager/PyNOT +ahttps://sites.astro.caltech.edu/∼dperley/programs/lpipe.html +3.1.1. Explosion epoch estimates +For the purpose of this paper, the ‘explosion time’ simply +refers to the time when optical flux rises above the zero-point +baseline (i.e. first light). We used pre-peak g, r, i-band ZTF +photometry and c, o-band ATLAS photometry (binned in 1- +day bins), when available, for our analysis. For each SN, the +light curve was interpolated using Gaussian process regres- +sion to obtain the peak flux epoch, then a power-law (PL) +model was fit using epochs from baseline to 60% of peak +brightness in each band following Miller et al. (2020). The +PL fits converged in at least one band for 6 out of 12 BTS +SNe Ia-CSM. For the rest, we simply took the middle point +between the first 5σ detection and the last upper limit before +this detection as the explosion epoch with half of the separa- +tion between these two points as the uncertainty. +The explosion time estimates, light curve bands used for +the PL fits and the 1σ uncertainties on explosion times are +listed in Table 4. The unfilled ‘PL fit filters’ column in the +table are the SNe for which the PL fit did not converge and +averages were used. For the PL fits this typically constrains +the time of explosion to within a fraction of a day. Given the +high cadence of the ZTF survey, even in the cases where we + +7 +Table 3. Summary of optical spectra +SN +JD +Epoch +Telescope/Instrument +Int +SN +JD +Epoch +Tel./Instr. +Int +(−2450000) +(days) +(sec) +(−2450000) +(days) +(sec) +SN 2018crl +8282 +9 +APO/DIS +2400 +SN 2020uem +9128 +11 +P60/SEDM +1800 +8288 +15 +P60/SEDM +2700 +9136 +18 +P60/SEDM +1800 +8295 +21 +P60/SEDM +2700 +9170 +51 +Ekar/AFOSC +1200 +8306 +31 +P60/SEDM +2700 +9222 +101 +Lick-3m/KAST +3600 +8373 +92 +P200/DBSP +600 +9252 +130 +Lick-3m/KAST +2700 +(Host) +8627 +324 +P200/DBSP +900 +9263 +140 +Lick-3m/KAST +2400 +SN 2018gkx +8457 +75 +Keck1/LRIS +300 +9291 +167 +NOT/ALFOSC +900 +SN 2018evt +8343 +9 +NTT/EFOSC2 +300 +9481 +349 +P60/SEDM +2160 +8465 +127 +P60/SEDM +1200 +9492 +360 +Keck1/LRIS +600 +8481 +143 +P60/SEDM +1200 +9583 +448 +P60/SEDM +2160 +8481 +144 +LT/SPRAT +1000 +9586 +451 +P60/SEDM +2160 +8534 +195 +P60/SEDM +1200 +SN 2020xtg +9226 +91 +P60/SEDM +2160 +SN 2019agi +8547 +42 +UH88/SNIFS +1820 +9491 +340 +Keck1/LRIS +600 +SN 2019ibk +8691 +35 +P60/SEDM +2250 +9606 +448 +Keck1/LRIS +1200 +8695 +39 +P60/SEDM +2250 +SN 2020abfe +9189 +27 +P60/SEDM +2700 +8697 +41 +P60/SEDM +2250 +9319 +146 +Keck1/LRIS +400 +8748 +90 +P60/SEDM +2250 +SN 2020aekp +9224 +19 +P60/SEDM +2160 +8761 +103 +P200/DBSP +600 +9342 +132 +P60/SEDM +2160 +SN 2019rvb +8766 +14 +P60/SEDM +2250 +9343 +132 +NOT/ALFOSC +1200 +8780 +26 +P200/DBSP +600 +9362 +151 +P60/SEDM +2700 +SN 2020onv +9058 +23 +P60/SEDM +1800 +9381 +169 +NOT/ALFOSC +2400 +9062 +27 +P60/SEDM +1800 +9404 +191 +P60/SEDM +2700 +9069 +33 +P60/SEDM +1800 +9425 +211 +NOT/ALFOSC +1800 +9070 +34 +LT/SPRAT +750 +9434 +220 +P60/SEDM +2700 +9073 +37 +P60/SEDM +1800 +9448 +233 +P60/SEDM +2700 +9074 +38 +NOT/ALFOSC +450 +9468 +252 +P60/SEDM +2700 +SN 2020qxz +9076 +13 +P60/SEDM +2250 +9569 +348 +P60/SEDM +2700 +9087 +22 +P60/SEDM +2250 +9092 +26 +NOT/ALFOSC +1800 +9098 +32 +P60/SEDM +2250 +9101 +34 +NOT/ALFOSC +1200 +9107 +40 +P200/DBSP +900 +9112 +45 +Keck1/LRIS +300 +9121 +53 +P60/SEDM +2250 +9141 +71 +Keck1/LRIS +399 +use only the last non-detection the uncertainty range is typ- +ically less than 3 days. Only for SN 2020uem is the date of +explosion virtually unconstrained (±57 days) as it was be- +hind the sun at the time of explosion. +Although for SN 2019ibk the explosion time is formally +constrained with a ±3 day uncertainty, this estimate was +derived using only ATLAS o-band data right after the SN +emerges from behind the sun. There is not a clear rise ob- +served over a few epochs but two non-detections before a +5σ detection. It is possible that the actual peak of this SN +occurred earlier while it was behind the sun and the rising +o-band points after it emerged are due to a second peak or +bump (similar to SN 2018evt, in that case the actual rise was +caught before the SN went behind the sun in ASAS-SN data). +If the former explosion epoch estimate from o-band is to be +believed then SN 2019ibk would be the most sub-luminous +among the SNe Ia-CSM, peaking at −17.5. +3.1.2. Duration and absolute magnitudes +Figure 2 shows the SNe Ia-CSM (colored squares) in our +sample in the duration-luminosity and duration-∆m30 phase +space. In the top panel, the x-axis is duration above half-max +and the y-axis is the peak absolute magnitude (see Table 1) +when we have photometric coverage both pre-peak and post- +peak. For SNe missing the pre-peak coverage, their discov- +ery magnitude is taken to be the upper limit to peak absolute +magnitude and the duration from discovery the lower limit + +8 +Table 4. Explosion time epoch estimates derived from pre-peak +multi-band light curves. For 6 out of 12 SNe Ia-CSM, we were able +to fit a power-law model to multi-band data following Miller et al. +(2020). For the remaining 6 SNe, the explosion epoch was estimated +by taking the mean of the first 5σ detection and last upper-limit +before the first detection. +IAU Name +PL fit filters +to +1σ interval +(MJD) +(days) +SN 2018crl +g, r, o +58271.83 +[−0.48,+0.38] +SN 2018gkx +r, o +58371.34 +[−0.64,+0.53] +SN 2018evt +- +58334.26 +[−2.00,+2.00] +SN 2019agi +- +58502.48 +[−1.51,+1.51] +SN 2019ibk +- +58654.61 +[−2.99,+2.99] +SN 2019rvb +g, r, i, o +58749.16 +[−0.79,+0.60] +SN 2020onv +o +59032.75 +[−2.49,+1.10] +SN 2020qxz +g, r, o +59063.05 +[−0.51,+0.45] +SN 2020uem +- +59117.03 +[−56.63,+56.63] +SN 2020xtg +- +59130.14 +[−0.04,+0.04] +SN 2020abfe +g, r, o +59159.36 +[−2.16,+2.23] +SN 2020aekp +- +59204.53 +[−5.50,+5.50] +to duration above half-max (marked by arrows in Figure 2). +The BTS SN Ia sample is shown in gray points, and we also +show the SNe Ia-CSM presented in S13 with empty trian- +gles for comparison in the top panel. In the bottom panel, +the x-axis is duration above 20% of peak flux (∆t20) and the +y-axis is ∆m30, the two parameters used in the selection cri- +teria. Most of the SNe Ia-CSM lie on the longer duration +and brighter luminosity side, and are even more distinctly +separated in the ∆t20-∆m30 phase space. This makes the +SN initial decline rate and duration useful tools for identify- +ing thermonuclear SNe potentially interacting with CSM, if +they have not revealed themselves already in their early time +spectra. The gray points lying in the same phase space as +SNe Ia-CSM are the false positive cases described in §2.1. +Also worth noting is that the duration calculated by taking +the flux above half of peak flux value does not capture the +true duration of the light curve when the plateau phase falls +below half-max as is the case for SN 2020aekp (> 500 days +light curve) but ∆t20 and ∆m30 do. +3.1.3. Light and color curves +We have good pre-peak coverage in ZTF data for 8 of +the 12 SNe in our sample8. +SN 2018evt was discovered +by ASAS-SN on JD 2458341.91 (Nicholls & Dong 2018) +and classified by ePESSTO the next day (Stein et al. 2018a), +around 115 days before the first detection in ZTF when the +SN came back from behind the sun. Hence we have only one +8 except for SNe 2018evt, 2019ibk, 2020onv and 2020uem. +0 +25 +50 +75 +100 +125 +150 +175 +Rest-Frame duration above half-max (days) +22 +21 +20 +19 +18 +17 +16 +15 +Peak absolute magnitude +Ia +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +Silverman 2013 +0 +100 +200 +300 +400 +500 +Rest-frame duration above 20% of max (days) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +mpeak + 30d +mpeak (r-band) +Figure 2. Top: Location of our 12 SNe Ia-CSM in the peak absolute +magnitude vs. rest-frame duration above half max phase space. The +colored points are the BTS SNe Ia-CSM and the gray points are the +rest of the BTS SNe Ia. Also shown with empty triangles are the +SNe Ia-CSM from S13. The vertical arrows mark the upper limits +to peak absolute magnitudes and horizontal arrows mark the lower +limits to durations of SNe not having pre-peak coverage. Bottom: +Change in magnitude 30 days after peak (∆m30) vs. rest-frame +duration above 20% of peak-flux for BTS SNe Ia and SNe Ia-CSM. +These criteria were used to filter out potential SNe Ia-CSM from all +SNe Ia and demonstrate that SNe Ia-CSM occupy a distinct portion +in this phase space. However some gray points (not SN Ia-CSM) +remain on the longer duration side and are the false positive cases +described in §2.1. +epoch of pre-peak photometry and one early spectrum for SN +2018evt. +Our mixed bag of SNe Ia-CSM show post-maximum de- +cline rates ranging from 0.5 to 2.0 mag 100d−1 in the r +band from peak to ∼ 100 days post peak. The median de- +cline rate is 1.07 mag 100d−1, which is much slower than +the decline rates of normal SNe Ia. +We see a variety of +changes in decline rates after around 100 days from peak. +Two SNe (2020onv and 2020abfe) show no change and have +a constant slow decline throughout. +Four SNe (2018gkx, +2019agi, 2019ibk and 2019rvb) evolve to a shallower slope +going from ∼ 0.6–1 mag 100d−1 to ∼ 0.2–0.5 mag 100d−1. +Three SNe (2018crl, 2020qxz and 2020aekp) show a ma- +jor change in decline rate with the light curves becoming + +9 +almost flat, and SN 2020aekp shifts back to a slow de- +cline from this plateau after ∼ 200 days. In three cases, +the decline rate actually becomes steeper, SN 2018evt goes +from 0.52 mag 100d−1 to 1.4 mag 100d−1, SN 2020uem goes +from 0.52 mag 100d−1 to 1.25 mag 100d−1 and SN 2020xtg +seems to go from 0.61 mag 100d−1 to 1.35 mag 100d−1 (even +though there is only one epoch at late times to measure +this change). +The 3 SNe with fastest initial decline rates +(≳ 1.5 mag 100d−1 in the r band) are similar to SN 2002ic +(initial decline of 1.66 mag 100d−1 in V ) and PTF11kx (ini- +tial decline of 3.3 mag 100d−1 in R) and coincidentally are +also the ones that evolve into a plateau. +The rest of the +sample have initial decline rates comparable to SN 1997cy +(0.75 mag 100d−1) and SN 2005gj (0.88 mag 100d−1) (In- +serra et al. 2016). From these observations, we can conclude +that SNe Ia-CSM exhibit a range of slow evolution indicat- +ing that there exists a continuum of phases at which strong +CSM interaction begins to dominate the powering of the light +curves for these SNe. It is, however, difficult to pinpoint +the exact phase when interaction starts from the light curve +without modeling. CSM interaction could be affecting the +peak brightness significantly even in cases where interaction +only appears to dominate after a few weeks (SNe 2018crl, +2020qxz 2020aekp). Considering the average peak phase to +be ∼ 20 days past explosion from the light curves and as- +suming an ejecta velocity of ∼ 20000 km s−1, the CSM is +located at ∼ 3.5 × 1015 cm. This estimate can be refined +by considering the phase of the earliest spectrum that shows +interaction signatures (see §3.2). At late times, all the de- +cline rates are slower than that expected from Cobalt decay +(0.98 mag 100d−1), confirming that the power from CSM in- +teraction dominates the light curve behaviour for a long time. +Figure 3 shows the g − r color evolution of our sam- +ple SNe as a function of phase (rest-frame days from r- +band maximum), comparing them with some famous SNe +Ia-CSM (SNe 2005gj, 1997cy, 1999E), and SNe 2012ca (Ia- +CSM/IIn), 2010jl (IIn) and 1991T (over-luminous Type Ia). +The color evolution of normal SNe Ia from ZTF (Dhawan +et al. 2022) is shown in grey lines. We use g − r colors +when available, otherwise we estimate the g − r color by +fitting Planck functions to estimate the black-body tempera- +tures from the V − R colors. Our SNe Ia-CSM show simi- +lar color evolution as the older Type Ia-CSM/IIn interacting +SNe, i.e. the g − r color increases gradually for about 100 +days and then settles onto a plateau or slowly declines, and +one object (SN 2019ibk) becomes redder at late times similar +to SN 2012ca. The interacting SNe are redder at late times +compared to the normal SNe Ia. +3.1.4. Mid-IR brightness comparison +Out of 12 SNe in our sample, only one observed (SN +2020abfe) did not have 3σ detections post explosion in the +unWISE difference photometry light curves and two (SNe +2019rvb and 2020qxz) did not have coverage post explosion. +The unWISE light curves for the rest of the SNe Ia-CSM +having > 3σ detections in W1 (3.3 µm) and W2 (4.6 µm) +bands are shown in Figure 4 (black and red stars) along with +Spitzer IRAC survey data of SN 2008cg (indigo and ma- +genta empty triangles), SN 2008J (indigo and magenta empty +squares) (both Ia-CSM) and some SNe IIn (blue and orange +crosses) taken from Fox et al. (2011). The most nearby SN +in our sample, SN 2018evt, is among the brightest (∼ 17 AB +mag) in MIR at least until ∼1000 days after explosion and +has a bumpy light curve. SNe 2019ibk and 2018crl however +are the most luminous with an absolute magnitude of −18.7 +mag in the W1 band. The brightness of the BTS SNe Ia-CSM +is comparable with other interacting SNe and span a similar +range (−16 to −19). However, SNe IIn have been detected +until even later epochs (up to 1600 days) than SNe Ia-CSM, +probably due to the larger number of SNe IIn at closer dis- +tances. SN 2020abfe has upper limits around ∼ −18 in W1 +band and ∼ −18.5 in W2 band up to ∼300 days post explo- +sion shown with upside down filled triangles. As the mid-IR +luminosity can be fainter than these limits for SNe Ia-CSM +(as can be seen for other nearby SNe in this sample) and SN +2020abfe is at a redshift of 0.093, it might just be out of reach +for WISE. +This brightness of SNe Ia-CSM in mid-IR can be indica- +tive of existing or newly formed dust. A clear signature of +new dust is reduced flux in the red wing of the Hα emission +line at late phases as the new dust formed in the cold dense +shell behind forward shock absorbs the far-side (redshifted) +intermediate and narrow line emission (see bottom panel of +Fig. 7). For our sample, this reduction in Hα red wing is the +most pronounced for SN 2018evt. +3.1.5. Bolometric luminosity +As the SN Ia-CSM luminosity is dominated by CSM inter- +action, their spectra comprise of a pseudo-continuum on the +blue side and strong Hα emission on the red side, hence a +blackbody fit to multi-band photometric data is not appropri- +ate to estimate the bolometric luminosity. Instead we calcu- +late a pseudo-bolometric luminosity from the available multi- +band optical data by linearly interpolating the flux between +the bands and integrating over the optical wavelength range +spanned by the ATLAS and ZTF bands. The individual band +light curves are first interpolated using Gaussian process re- +gression to fill in the missing epochs. This estimate places a +strict lower limit on the bolometric luminosity. +In Figure 5 we show the pseudo-bolometric luminosity +of our SN Ia-CSM sample in comparison with SN 1991T +(Type Ia), SNe 1997cy, 1999E, 2002ic, 2005gj, 2013dn and +PTF11kx (Ia-CSM). Multi-band photometric data were taken +from the Open Supernova Catalog (Guillochon et al. 2017) + +10 +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2018crl +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2018gkx +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2018evt +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2019agi +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2019ibk +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2019rvb +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +D +SN 2020onv +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2020qxz +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2020uem +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2020xtg +0 +150 +300 +450 +600 +D +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2020abfe +0 +150 +300 +450 +600 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +SN 2020aekp +Rest-frame days from r-band maximum +g +r color (mag) +SN2005gj +SN2012ca +SN1997cy +SN1999E +SN1991T +SN2010jl +SN Ia +Figure 3. Color evolution (g − r) of BTS SNe Ia-CSM from r-band maximum (plotted in black) compared with SNe 2005gj, 1997cy, 1999E +(Ia-CSM), SN 2012ca (IIn/Ia-CSM), SN 2010jl (IIn), SN 1991T (SN Ia) and ZTF SNe Ia (gray lines). As can be seen for up to ∼ 150 days, +our SNe Ia-CSM tend to be redder than SNe Ia and at late times develop a plateau similar to other interacting SNe (IIn/Ia-CSM). + +11 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +Days since explosion +19.5 +19.0 +18.5 +18.0 +17.5 +17.0 +16.5 +16.0 +15.5 +Absolute Magnitude (AB) +BTS SN Ia-CSM: W1 +BTS SN Ia-CSM: W2 +SN 2008J: 3.6 m +SN 2008J: 4.5 m +SN 2008cg: 3.6 m +SN 2008cg: 4.5 m +SN IIn: 3.6 m +SN IIn: 4.5 m +Figure 4. unWISE detections in the W1 and W2 bands of BTS +SNe Ia-CSM. The W1 and W2 points are marked with black and +red filled stars respectively. Spitzer IRAC photometry of SNe IIn +(blue and orange crosses) and two SNe Ia-CSM from Fox et al. +(2011) (SNe 2008cg and 2008J in empty triangle and square) are +also shown for comparison. 9 out of 12 BTS SNe Ia-CSM are as +bright in mid-IR as other interacting SNe (∼ −16 to ∼ −19). The +upper limits for SN 2020abfe are shown in black and red filled up- +side down triangles. +for SN 1991T (Filippenko et al. 1992; Ford et al. 1993; +Schmidt et al. 1994) to generate the bolometric luminos- +ity light curve through black body fitting. +The pseudo- +bolometric luminosity light curve for SN 1997cy was ob- +tained from Germany et al. (2000), for SN 2013dn from +Fox et al. (2015) and for SNe 2002ic, 2005gj, 1999E and +PTF11kx from Inserra et al. (2016). +All BTS SNe Ia-CSM show a slow evolution in bolomet- +ric luminosity, inconsistent with the decay of 56Co to 56Fe. +The sample’s overall luminosity decline rates are comparable +to those of SNe 1997cy and 2013dn, as shown in Figure 5. +Only SNe 2018crl and 2020aekp seem to show early decline +in their pseudo-bolometric light curves similar to SN 1991T +for about 40 days after peak like SN 2002ic and PTF11kx. +Another BTS interacting SN Ia, ZTF20aatxryt (Kool et al. +2022), was found to follow the PTF11kx light-curve evo- +lution very closely and as its light curve fell into a plateau +the SN started showing signs of interaction with a helium- +rich CSM and evolved into a helium-rich SN Ia-CSM. We +have excluded ZTF20aatxryt from the sample as we focus on +typical SNe Ia-CSM interacting with hydrogen-rich CSM in +this study. At late phases (∼ 300 days), the SNe Ia-CSM +are approximately 100 times brighter than normal SNe Ia +at the same epoch. Therefore, at these late phases, the lu- +minosity and spectral features of SNe Ia-CSM are entirely +dominated by CSM-interaction with little emergent SN flux. +From the pseudo-bolometric light curves, we place a lower +limit on the total radiated energy for SNe Ia-CSM to be 0.1– +1.5 ×1050erg. This is well below the thermonuclear budget +(Ekin ∼ 1051 erg), but as this is a lower limit and some SNe +in the sample have unconstrained peaks, the true total radia- +tive energy might come close to the thermonuclear budget, +requiring high conversion efficiency to achieve their lumi- +nosity. +0 +80 +160 +240 +320 +400 +Rest-frame days since explosion +41.0 +41.5 +42.0 +42.5 +43.0 +43.5 +44.0 +log(Lopt (erg/s)) +SN2013dn +SN1997cy +SN1991T +PTF11kx +SN2002ic +SN2005gj +SN1999E +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +Figure 5. Pseudo-bolometric luminosity light curves of BTS SNe +Ia-CSM compared with pseudo-bolometric light curves of SNe +1991T, 1997cy, 1999E, 2002ic, 2005gj, 2013dn, and PTF11kx from +literature. The light curves in each filter having more than 10 epochs +were interpolated using Gaussian process regression to fill in the +missing epochs, and at each epoch the fluxes between the bands +were linearly interpolated and integrated over the optical wave- +length range spanned by ZTF and ATLAS filters to get the pseudo- +bolometric luminosity. For BTS SNe, the phases are with respect +to the estimated explosion epochs, while for comparison SNe the +phases are with respect to discovery. +3.2. Spectroscopy +Figure 6 displays the spectral series obtained for the BTS +SNe Ia-CSM. Most of the early time spectra were taken +with the SEDM, the BTS workhorse instrument (R ∼100), +which is not able to resolve the narrow CSM lines. There- +fore, these SNe were followed up with higher resolution in- +struments to get more secure classifications. For each spec- +trum in Figure 6, the phase is provided with respect to the +explosion epoch estimate given in Table 4. We have spec- +tra ranging from a few to around 470 days from explo- +sion. Considering the well constrained explosion times of +SN 2018evt, presence of narrow Hα in its first spectrum at +8 days since explosion and assuming a typical ejecta veloc- +ity of ∼20000 km s−1, this implies that the CSM interaction +start as close as ∼1.4×1015 cm. +Figure 7 shows the early time (left) and late time (right) +spectral behaviour of the BTS SNe Ia-CSM together with a +few historical SNe for comparison, namely SNe Ia-CSM SN +2011jb (Silverman et al. 2013), SN 2005gj and PTF11kx, the +Type Ia SN 1991T and the well-observed Type IIn SN 2010jl. +Vertical gray regions mark typical SN Ia absorption features + +12 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +2 +0 +2 +4 +6 +8 +10 +12 +14 +9 d +15 d +21 d +31 d +92 d +H +H +H +He I +H +Ca II +SN2018crl +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +1 +0 +1 +2 +3 +4 +75 d +H +H +H +He I +H +Ca II +SN2018gkx +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +2 +4 +6 +8 +10 +9 d +127 d +143 d +144 d +195 d +H +H +H +He I +H +Ca II +SN2018evt +3000 +4000 +5000 +6000 +7000 +8000 +9000 +0 +2 +4 +6 +8 +10 +12 +14 +16 ++ Constant +35 d +39 d +41 d +90 d +103 d +H +H +H +He I +H +Ca II +SN2019ibk +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +1 +2 +3 +42 d +H +H +H +He I +H +Ca II +SN2019agi +3000 +4000 +5000 +6000 +7000 +8000 +9000 +0 +1 +2 +3 +4 +5 +14 d +26 d +H +H +H +He I +H +Ca II +SN2019rvb +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +2 +4 +6 +8 +10 +12 +Normalized Flux +23 d +27 d +33 d +34 d +37 d +38 d +H +H +H +He I +H +Ca II +SN2020onv +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +13 d +22 d +26 d +32 d +34 d +40 d +45 d +53 d +71 d +H +H +H +He I +H +Ca II +SN2020qxz +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +Rest Wavelength (Å) +0 +5 +10 +15 +20 +25 +19 d +132 d +132 d +151 d +169 d +191 d +211 d +220 d +233 d +252 d +348 d +H +H +H +He I +H +Ca II +SN2020aekp +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +2 +4 +6 +8 +10 +12 +27 d +146 d +H +H +H +He I +H +Ca II +SN2020abfe +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +5 +10 +15 +20 +25 +11 d +18 d +51 d +101 d +130 d +140 d +167 d +349 d +360 d +448 d +451 d +H +H +H +He I +H +Ca II +SN2020uem +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +0 +5 +10 +15 +20 +91 d +340 d +448 d +H +H +H +He I +H +Ca II +SN2020xtg +P60/SEDM +NOT/ALFOSC +P200/DBSP +Keck1/LRIS +LT/SPRAT +Lick-3m/KAST +Ekar/AFOSC +NTT/EFOSC2 +UH88/SNIFS +APO/DIS +Figure 6. Spectral series of all SNe Ia-CSM presented in this paper. The rest-frame phases are shown alongside the spectra in each subplot and +have been calculated using the explosion epoch estimate. The colors depict different instruments used to obtain this data. Major emission lines +are marked with vertical dashed lines. + +13 +and [Fe II/III] line regions, and vertical dashed lines mark the +Balmer emission lines. The sample spectra have been mul- +tiplied by a constant factor to magnify relevant spectral fea- +tures. In the following paragraphs, we compare the observa- +tions of some of the spectral features with previous analysis +of this class (Silverman et al. 2013; Fox et al. 2015; Inserra +et al. 2016). +A few of our early time SNe Ia-CSM show underlying SN +Ia absorption features like PTF11kx and SN 2002ic (most +are, however, quite diluted and also affected by the low res- +olution and signal-to-noise ratio (SNR) of the SEDM spec- +tra), the most notable being SNe 2018evt, 2020qxz and +2020aekp. SNe 2020qxz and 2020aekp also have among the +fastest initial post-peak decline rates in the sample, similar to +PTF11kx, while coverage around peak is not available for SN +2018evt. On the other hand, SNe with slower decline rates +similar to SN 1997cy and SN 2005gj have more SN IIn-like +early time spectra dominated by blue pseudo-continuum and +Balmer emission. The faster decline rate suggests we are still +seeing some of the emission from the ejecta at those phases. +To unveil the nature of the progenitor of interacting SNe, it is +therefore necessary to obtain some spectroscopic follow-up +before peak light. Spectroscopic data at the phase of tran- +sition to interaction-dominated luminosity would also help +in deducing the extent and density structure of the optically +thick CSM. +Late time spectra of SNe Ia-CSM look very similar to those +of SNe IIn, heavily dominated by Hα emission. The CSM in- +teraction masks the underlying SN signature and we instead +see late-time spectra riddled with photoionized CSM lines. +In some cases, the photosphere might lie in an optically thick +cold dense shell (CDS) formed between the forward and re- +verse shocks which obscures the ejecta completely (Smith +et al. 2008; Chugai et al. 2004). The continuum is also en- +shrouded under a blue quasi-continuum from a forest of iron- +group element lines (S13) as identified and analyzed for SNe +2012ca and 2013dn by Fox et al. (2015). +The blue quasi-continuum blend of iron lines ([Fe III] lines +around ∼4700 ˚A and [Fe II] around ∼5200 ˚A) in the spectra +of the BTS SN Ia-CSM sample (see Figure 7 top right panel) +is the dominant feature blue-ward of 5500 ˚A but the ratio +of [Fe III]/[Fe II] is much weaker compared to for SNe Ia +(like SN 1991T). This feature is more apparent in the SNe Ia- +CSM like PTF11kx and SN 2002ic that became interaction- +dominated later than for other SNe Ia-CSM such as SNe +1997cy, 1999E and SN 2012ca (SN Ia-CSM/IIn, for which +a clear type has not been established). Inserra et al. (2014) +argues for a core-collapse origin for SN 2012ca given this +low amount of [Fe III] along with the detection of blueshifted +Carbon and Oxygen lines (which however, were later argued +to be [Fe II] lines by Fox et al. 2015). S13 instead argues +in favor of a thermonuclear origin given the presence of this +blue quasi-continuum, despite [Fe III] being weaker. Fox +et al. (2015) points out that a similarly suppressed ratio of +[Fe III]/[Fe II] is observed in some SNe Ia, particularly the +super-Chandra candidate SN 2009dc, for which the expla- +nation was suggested to be a low ionization nebular phase +owing to high central ejecta density and low expansion ve- +locities (Taubenberger et al. 2013). Fox et al. (2015) argue +that in the case of SNe Ia-CSM, a lower ionization state could +arise owing to the deceleration of ejecta by the dense CSM +explaining the Fe line ratio suppression. Since Ca has lower +first and second ionization potentials than Fe, the detection +of [Ca II] λλ7291, 7324 would be consistent with this low +ionization, which Fox et al. (2015) confirms for SNe 2012ca +and 2013dn. Indeed, we find clear evidence of [Ca II] emis- +sion for 8 out of 12 SNe in our sample and moderate to weak +signal for the remaining 4. +Although this does favor the +argument for a thermonuclear origin, a similar blue quasi- +continuum is also observed in other interacting SN types like +SNe Ibn (SN 2006jc, Foley et al. 2007) and SNe IIn (SNe +2005ip and 2009ip), making Fe an incomplete indicator of +the progenitor nature (see detailed discussion in Fox et al. +2015). +We do not find strong evidence of O I λ7774 or [O I] +λλ6300, 6364 emission in our sample, although they might +be present at very weak levels in some SNe (e.g. SN +2020uem). SN 2020uem has strong emission lines at 6248, +7155 and 7720 ˚A which are consistent with being iron lines +and were also observed in SNe 2012ca, 2013dn and 2008J. +S13 note that the very broad emission around 7400 ˚A can be +due to a blend of [Ca II] λλ7291, 7324 and [O II] λλ7319, +7330, however we note that this broad emission is likely to +be from calcium as O II is harder to excite than O I which is +either very weak or absent in our spectra. The broad Ca NIR +triplet feature resulting from electron scattering is the next +strongest feature after the Balmer emission and is present in +all mid to late-time spectra of the SNe in our sample where +the wavelength coverage is available. We observe it increas- +ing in relative strength with phase, at least for a year, after +which we no longer have spectral coverage. +The bottom panel of Figure 7 shows the line profile of Hα, +with the blue side reflected over the red side at the maxi- +mum flux after continuum removal. We do see evidence of +diminished flux in the red wing of Hα at late phases in some +SNe (most notable in SNe 2018evt and 2020uem), which can +indicate formation of new dust in the post-shock CSM. S13 +claim to observe this for all non-PTF SNe Ia-CSM in their +sample starting at ∼75–100 days, while for the PTF SNe +Ia-CSM they do not have spectra available post that phase +range. For some BTS SNe Ia-CSM, we also do not have +spectra available post 100 days which limits any analysis of +this phenomenon for a large enough sample. + +14 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +Rest Wavelength (Å) +Normalized Flux + Constant +15 d +9 d +14 d +23 d +13 d +11 d +27 d +19 d ++23 d ++6 d ++11 d ++21 d +H +He I +H +Ca II +Mg II +Si II +Early time spectra comparison +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +SN2011jb +SN2005gj +PTF11kx +SN1991T +SN2010jl +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +Rest Wavelength (Å) +Normalized Flux + Constant +92 d +75 d +144 d +42 d +103 d +26 d +38 d +71 d +140 d +340 d +146 d +169 d ++185 d ++91 d ++328 d ++21 d ++83 d +H +He I +H +[O I] +O I +He I +Ca II IR +[Fe II/III] +[Fe III] +Late time spectra comparison +8000 +4000 +0 +4000 +8000 +Velocity km s +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Normalized flux + Constant +92.0 d +75.0 d +144.0 d +42.0 d +103.0 d +26.0 d +38.0 d +71.0 d +8000 +4000 +0 +4000 +8000 +Velocity km s +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Normalized flux + Constant +167.0 d +360.0 d +340.0 d +448.0 d +146.0 d +132.0 d +169.0 d +211.0 d +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +Figure 7. Top left: Early-time spectra of BTS SNe Ia-CSM with phases between 0 and 30 days since explosion compared to spectra of SNe +2011jb, 2005gj, 1991T and PTF11kx (phases in days since discovery). Top right: Late-time spectra of BTS SNe Ia-CSM (phases ranging from +40 to 370 days since explosion) compared to spectra of SNe 2011jb, 2005gj, 2010jl and PTF11kx (phases in days since discovery). +Bottom left and right: Hα line profiles (post continuum removal) with the blue side reflected across the peak flux, marked by dashed lines. +SNe 2020aekp, 2020abfe, 2020xtg and 2020uem in the right panel, and SNe 2018crl, 2018gkx, 2018evt, 2019agi, 2019ink, 2019rvb, 2020onv, +2020qxz in left panel. + +15 +The spectra were reduced and processed as outlined in §2.5 +for the emission line analysis, the results of which are de- +scribed in the next section. We used only good SNR SEDM +spectra and intermediate resolution spectra for line identifi- +cation and analysis. +3.2.1. Hα, Hβ and He I emission lines +To analyze the Hα line emission, we first fit the con- +tinuum level using the fit continuum function of the +specutils Python package, where the continuum is es- +timated by a cubic function fitted on regions on each side of +the line. We remove this continuum level and then fit the Hα +line with a broad and a narrow component Gaussian func- +tion using the fit lines function of specutils which +returns the best fit Gaussian model and the 1σ uncertainty +on the model parameters. We generate 1000 sample mod- +els within 1σ uncertainties of the parameters centered around +the best-fit values and calculate the intensity, flux and veloc- +ity (FWHM) of the broad and narrow components for each +model. Then we take the median and standard deviation of +the intensity, flux and velocity FWHM distributions to get +their final best value and 1σ uncertainty. +The equivalent +width was also calculated for the Hα line using the model +fit as well as directly from the data, and the difference be- +tween the values derived from model and data is reported as +the error on the EW. All values are reported in Table 5. For 3 +SNe in our sample, we have a series of intermediate resolu- +tion spectra through which we can trace the evolution of the +Hα line with phase. Figure 8 shows this trend of the Hα line +parameters (integrated flux in the top panel and equivalent +width in the bottom panel) versus phase for all SNe in our +sample. The un-filled markers represent the narrow emis- +sion while the filled markers represent the broad emission. +For SNe where this analysis could be done on multiple spec- +tra, we see that the Hα equivalent width generally increase +over time, with some SNe showing fluctuations up to 100 +days possibly due to interaction of ejecta with multiple CSM +shells of varying density. For SN 2018evt, Yang et al. (2022) +analyzed Hα line properties from a comprehensive spectral +series data, which are plotted in Figure 8 in gray circles and +seem to agree well with our analysis at comparable epochs. +From the Gaussian profile line fitting analysis of the Hα +emission line, we found that the broader component has ve- +locities ranging from ∼1000 to ∼4000 km s−1 (intermedi- +ate width) and the narrow component has velocities of about +∼200 km s−1 to ∼1000 km s−1 (see Figure 9). The narrow +component could only be resolved down to ∼300 km s−1 +limited by the mediocre resolution of the spectrographs used +(KeckI/LRIS R∼800, P200/DBSP R∼1000, NOT/ALFOSC +has R∼360). While we know that the narrow lines originate +in the unshocked ionized CSM, the exact origin of the inter- +mediate components is uncertain. They could arise from the +post-shock gas behind the forward shock or from the shocked +dense clumps in the CSM (Chugai & Danziger 1994). +The luminosities of the Hα line measured from the BTS +SNe Ia-CSM lie in the range 2.5–37×1040 erg s−1 which are +comparable to the values from S13 who reported most of +their SNe in the 1–10×1040 erg s−1 range except one object +that had a luminosity of 39×1040 erg s−1. From the broad +Hα luminosity, we did a simple estimate of the mass-loss rate +assuming spherically symmetric CSM deposited by a station- +ary wind ρ ∝ r−2 having velocity vw (Chugai 1991; Sala- +manca et al. 1998). The mass-loss rate ˙M can be related to +the broad Hα luminosity LBroad +Hα +as (Salamanca et al. 1998, +their Eq. 2) +LBroad +Hα += 1 +4ϵHα +˙M +vw +v3 +s +where vs is the shock velocity (obtained from the broad +component velocity of the Hα line). We used a value of +100 km s−1 considering previous high resolution spectral +studies of SNe Ia-CSM (Kotak & Meikle 2005; Aldering +et al. 2006; Dilday et al. 2012) for vw as we cannot fully +resolve the narrow component and a maximum value of 0.1 +for the efficiency factor ϵHα (Salamanca et al. 1998). The +mass-loss rates were estimated from the available spectra +and are shown in Figure 10 as a function of years before +explosion (tw = +vst +vw , where t is the phase of the spectra). +For most SNe in the sample, the mass-loss rates lie be- +tween 0.001–0.02 M⊙ yr−1, except for SN 2019rvb which +has ∼0.07 M⊙ yr−1 lost within 2 years prior the explosion. +These rates are much higher than what could be attained +from a red giant superwind (∼ 3 × 10−4 M⊙ yr−1) but are +comparable to previous estimates (calculated through mul- +tiple methods) for SNe Ia-CSM and require some unusual +mechanism to reach such persistently higher mass-loss rates +in the decades prior to explosion. Also to consider is that the +simplistic assumption of spherical symmetry likely does not +apply for SNe Ia-CSM. Evidence of multiple thin shells and +asymmetric CSM was observed for PTF11kx (Dilday et al. +2012) and light curve modeling of SNe 1997cy and 2002ic +suggested a better fit to a flat density profile rather than sta- +tionary wind (Chugai & Yungelson 2004). An asymmetric or +clumpy CSM might be the norm for SNe Ia-CSM (and some +SNe IIn) rather than the exception. +The same analysis as for the Hα line was also carried out +for Hβ and He I λ5876 with a one component Gaussian fit. +For cases where a Gaussian model could not fit the data, we +integrate the flux value in a 100 ˚A region centered at 5876 ˚A +for He I. The Na ID absorption lines are also prevalent in +some spectra and blend with the He I line, resulting in posi- +tive EWs for some SNe. The cumulative distributions of Hβ +and He I equivalent widths are shown in the top and bottom +panels of Figure 11 respectively. + +16 +Table 5. Summary of Hα line properties obtained from two-component Gaussian fitting. +SN Name +Phase +Broad Flux +Narrow Flux +Total Flux +Broad Velocity +Narrow Velocity +(days) +(10−16 erg s−1 cm−2) +(10−16 erg s−1 cm−2) +(10−16 erg s−1 cm−2) +FWHM (km s−1) +FWHM (km s−1) +SN 2018crl +92 +135.4±10.0 +32.8±2.0 +168.2±12.0 +4137±312 +< 214 +SN 2018gkx +75 +9.9±0.7 +3.9±0.2 +13.7±0.9 +2640±398 +< 375 +SN 2018evt +144 +2020.3±128.5 +1247.4±52.8 +3267.7±181.3 +6465±997 +1816±973 +SN 2019agi +42 +52.7±3.6 +23.7±1.1 +76.4±4.7 +3836±349 +464±301 +SN 2019ibk +103 +85.6±1.7 +17.0±0.5 +102.6±2.3 +2431±217 +272±214 +SN 2019rvb +26 +22.0±3.0 +10.4±1.0 +32.5±4.1 +2321±298 +374±216 +SN 2020onv +38 +32.8±5.2 +33.3±2.0 +66.1±7.2 +2714±879 +<834 +SN 2020qxz +26 +76.6±6.2 +13.8±1.7 +90.4±7.9 +11294±1106 +< 836 +SN 2020qxz +34 +55.1±5.0 +10.8±1.8 +65.9±6.8 +8252±1039 +1070±845 +SN 2020qxz +40 +12.9±1.7 +7.6±0.5 +20.5±2.2 +2049±284 +245±215 +SN 2020qxz +45 +20.7±1.6 +9.1±0.4 +29.8±2.1 +3429±419 +< 375 +SN 2020qxz +71 +39.1±1.3 +10.4±0.4 +49.5±1.7 +5013±395 +400±375 +SN 2020uem +51 +246.3±47.2 +151.1±16.8 +397.4±64.0 +6520±1163 +1178±840 +SN 2020uem +101 +655.2±28.9 +241.2±9.6 +896.4±38.4 +7456±309 +1066±217 +SN 2020uem +130 +552.9±17.6 +281.8±6.2 +834.8±23.8 +7465±265 +1269±215 +SN 2020uem +140 +545.4±20.0 +283.4±6.8 +828.8±26.7 +7457±275 +1308±216 +SN 2020uem +167 +424.3±19.0 +312.0±7.7 +736.3±26.6 +6852±854 +1439±834 +SN 2020uem +360 +179.8±4.0 +77.4±1.4 +257.2±5.4 +5377±382 +1170±375 +SN 2020xtg +340 +129.2±4.2 +52.1±1.6 +181.3±5.8 +4242±382 +1258±376 +SN 2020xtg +448 +131.7±7.7 +96.3±3.2 +228.0±10.9 +4452±395 +1566±377 +SN 2020abfe +146 +33.6±1.1 +3.0±0.3 +36.6±1.4 +4411±389 +< 376 +SN 2020aekp +132 +149.5±4.0 +33.0±1.0 +182.5±5.0 +7728±846 +< 833 +SN 2020aekp +169 +231.0±4.5 +32.3±1.3 +263.3±5.8 +6775±839 +< 834 +SN 2020aekp +211 +251.0±9.5 +58.6±3.4 +309.6±12.8 +7422±852 +1342±836 +The Hβ median EW measured from the BTS SN Ia-CSM +sample is 7.1 ˚A , close to the S13 value of ∼6 ˚A and quite +weak compared to what S13 measured for SNe IIn (∼13 ˚A ). +The overall cumulative distribution of Hβ EW is also compa- +rable to the S13 SNe Ia-CSM rather than to the S13 SNe IIn. +For the He I λ5876 line, the median EW measured for our +BTS SN Ia-CSM sample, considering only significant emis- +sion features, is 2.4 ˚A . This is close to the value of ∼2 ˚A +reported in S13, and again significantly different from their +SN IIn value of ∼6 ˚A (∼4 ˚A with upper limits), however +the overall distribution seems to be closer to the S13 SNe IIn +(but still weaker) rather than to the S13 SNe Ia-CSM. This +indicates that perhaps He I is not as good a discriminant be- +tween the populations compared to Hβ. Among the most He- +rich SNe in our sample are SNe 2019ibk, 2020uem, 2020xtg, +2020aekp and 2018evt, and these SNe also have the higher +Hα equivalent widths in the sample. +Figure 12 plots the cumulative distribution of the Balmer +decrements ( FHα +FHβ ) measured for our sample SNe. The higher +Balmer decrement values (>15) have large errors associated +to them because of low SNR of the spectra from which they +were derived, particularly near the Hβ line. Consistent with +the results of S13, the SNe Ia-CSM from this sample also +have a high median Balmer decrement value of ∼7 (∼5 in +S13), indicating that the emission line mechanism is prob- +ably collisional excitation or self-absorption rather than re- +combination, from which the expected Balmer decrement +value is ∼3. In the case of SNe Ia-CSM, if the CSM distri- +bution consists of multiple shells as suggested for PTF11kx, +moderately high densities could be created when fast moving +ejecta overtake slowly moving thin dense CSM shells creat- +ing large enough optical depth in the Hα line which results +in the Hβ transition decaying as Paα + Hα (Xu et al. 1992). +For some individual SNe where multiple spectra are avail- +able, the Balmer decrement is observed to first increase and +later on decrease with phase. +3.3. Host galaxies +We retrieved science-ready co-added images from the +Galaxy Evolution Explorer (GALEX) general release 6/7 +(Martin et al. 2005), the Sloan Digital Sky Survey DR 9 +(SDSS; Ahn et al. 2012), the Panoramic Survey Telescope +and Rapid Response System (Pan-STARRS, PS1) DR1 +(Chambers et al. 2016), the Two Micron All Sky Survey +(2MASS; Skrutskie et al. 2006), and preprocessed WISE im- + +17 +1040 +1041 +Line luminosity (erg s +1) +SN2018evt (Yang et al.) +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +100 +200 +300 +400 +Phase (days) +102 +Equivalent Width (Å) +Figure 8. Integrated fluxes and equivalent widths of Hα emission +line with respect to SN phases for the BTS SN Ia-CSM sample. +Broad component values are shown with filled markers and narrow +component values with un-filled markers. SN 2018evt Hα lumi- +nosities and EWs presented in Yang et al. (2022) are also shown in +gray circles. +100 +200 +300 +400 +Phase (days) +102 +103 +104 +Line Velocity (km s +1) +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +Figure 9. Velocity of Hα emission line with respect to SN phases +for the BTS SN Ia-CSM sample. +Broad component values are +shown with filled markers and narrow component values with un- +filled markers. +ages (Wright et al. 2010b) from the unWISE archive (Lang +2014b)9. +We used the software package LAMBDAR (Lambda +Adaptive Multi-Band Deblending Algorithm in R) (Wright +et al. 2016) and tools presented in Schulze et al. (2021), +to measure the brightness of the host galaxy. The spectral +energy distribution (SED) was modelled with the software +9 http://unwise.me +0 +10 +20 +30 +40 +50 +Years before explosion +10 +3 +10 +2 +10 +1 +Mass loss rate (M + yr +1) +SN2018crl +SN2018gkx +SN2018evt +SN2019agi +SN2019ibk +SN2019rvb +SN2020onv +SN2020qxz +SN2020uem +SN2020xtg +SN2020abfe +SN2020aekp +Figure 10. Mass-loss rates estimated from the luminosity of the +broad component of Hα for the BTS SNe Ia-CSM. A value of +100 km s−1 was assumed for the wind velocity. +package Prospector10 (Johnson et al. 2021). +We assumed +a linear-exponential star-formation history, the Chabrier +(2003) initial mass function, the Calzetti et al. (2000) at- +tenuation model, and the Byler et al. (2017) model for the +ionized gas contribution. The priors were set as described in +Schulze et al. (2021). +Figure 13 shows the log of star formation rate (SFR) as a +function of stellar mass for hosts of BTS SNe Ia-CSM. We +also use a Galaxy-zoo (Lintott et al. 2011) sample of ellip- +tical and spiral galaxies (randomly sampled in the redshift +range z = 0.015−0.05), and BTS SN Ia hosts as comparison +samples collected by and used for comparison in Irani et al. +(2022). We find the SN Ia-CSM host galaxy population to +be consistent with late-type spirals and irregulars with recent +star formation history. 4 out of 12 SNe have clearly spiral +hosts, 3 have edge-on host galaxies, 4 seem to have irregu- +lars as hosts and 1 has an unclear host type. Host galaxies +of 10 out of 12 SNe have w2 − w3 measurements available +which are all > 1 mag, putting them in late-type category +(Irani et al. 2022), 1 (SN 2019rvb) does not have W3 mea- +surement but has NUV − PS1r ∼ 1 mag again putting it +towards late-type and 1 (SN 2020abfe) does not have any of +the above information available except the PS1r band mag- +nitude of 20.766, which is the faintest host galaxy (absolute +SDSS r-band magnitude of −17.4) in our BTS SN Ia-CSM +sample. As noted in S13, the SN Ia-CSM hosts of their sam- +ple had generally low luminosities (−19.1 < Mr < −17.6) +except MW like spiral hosts. Our BTS SN Ia-CSM host lu- +minosities lie in the range of −21.8 < Mr < −17.4 covering +low to MW like luminosities. +3.4. Rates +Following the methodology for calculating the volumetric +rate of transients found in the Bright Transient Survey from +Perley et al. (2020), we use their equation 2 to calculate the +10 https://github.com/bd-j/prospector version 0.3 + +18 +0 +20 +40 +60 +80 +100 +H Equivalent Width (Å) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Fraction of objects +BTS Ia-CSM +S13 Ia-CSM +S13 IIn +MedianBTS +Ia +CSM = 7.1 +MedianS13 +Ia +CSM = 6.0 +MedianS13 +IIn = 13.0 +0 +10 +20 +30 +40 +50 +He I 5876 Equivalent Width (Å) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Fraction of objects +BTS Ia-CSM +S13 Ia-CSM +S13 IIn +MedianBTS +Ia +CSM = 2.4 +MedianS13 +Ia +CSM = 2 +MedianS13 +IIn = 4 +Figure 11. Cumulative distributions of equivalent width of Hβ and +He I λ5876 emission lines calculated from the BTS SNe Ia-CSM +(in grey) compared with the respective distributions presented in +S13 for SNe Ia-CSM (blue) and SNe IIn (red). Vertical dashed lines +mark the median EW of the distributions. +SN Ia-CSM rate: +R = 1 +T +N +� +i=1 +1 +( 4π +3 D3 +max,i)fskyfextfrecfcl,i +where T is the duration of the survey, N is the number of +transients that pass the quality cut, Dmax,i is the distance out +to which the ith transient with peak absolute magnitude Mi +0 +5 +10 +15 +20 +25 +Balmer Decrement +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Fraction of objects +BTS Ia-CSM +S13 Ia-CSM +S13 IIn +MedianBTS +Ia +CSM = 7.2 +MedianS13 +Ia +CSM = 5 +MedianS13 +IIn = 3 +Figure 12. +Cumulative distribution of Hα/Hβ intensity ratio +(Balmer decrement) calculated from intermediate resolution spec- +tra of BTS SN Ia-CSM sample (grey shaded region). The red line is +the distribution of Balmer decrement of SNe IIn measured in S13, +the blue line is the SN Ia-CSM Balmer decrement distribution from +S13. The black circles are a few representative points indicating the +high Balmer decrement values and the uncertainties on them. The +vertical dashed line is the median Balmer decrement measured from +BTS SNe Ia-CSM. +can be detected above the survey magnitude limit mlim (=19 +mag for BTS SNe Ia-CSM) at peak light without any extinc- +tion, fsky is the average active survey coverage as a fraction +of full sky, fext is average reduction in effective survey vol- +ume due to Galactic extinction, frec is the average recovery +efficiency for a detectable transient within the survey cover- +age area, and fcl,i is the classification efficiency dependent +on apparent magnitude. +The duration of the survey in which these 12 SNe Ia-CSM +were detected is from 2018-05-01 to 2021-05-01, i.e. T = 3 +years. We calculate fsky during this time period by averaging +the sky area coverage of the public MSIP survey consider- +ing 3 day cadence for ZTF Phase I (2018-05-01 to 2020-10- +31) and 2 day cadence for ZTF Phase II (since 2020-11-01), +which turns out to be 12505 deg2 for Phase I and 14831 deg2 +for Phase II, giving a mean fsky = 0.32. We use the same +value of 0.82 for fext as calculated in Perley et al. (2020) +given there has not been any change in the number and posi- +tions of ZTF fields. +To estimate frec, we consider SNe Ia-CSM brighter than +−18.5 peak absolute magnitude and brighter than 18 appar- +ent magnitude (total 5) of which 4 pass the quality cut, giving +an frec of 0.8. We take classification completeness of 0.75 at + +19 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log M/M +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +log(SFR (M +/yr)) +sSFR = 10 +8 M +yr +1 +sSFR = 10 +9 M +yr +1 +sSFR = 10 +10 M +yr +1 +sSFR = 10 +11 M +yr +1 +sSFR = 10 +12 M +yr +1 +Galaxy zoo ellipticals +Galaxy zoo spirals +BTS SN Ia +BTS SN Ia-CSM +Figure 13. Host galaxies of BTS SN Ia-CSM (black circles) on +SFR vs stellar mass plot with Galaxy-zoo spiral (blue contours) and +elliptical (red contours) galaxies for comparison. BTS SN Ia hosts +are also shown for comparison in green circles. Equal sSFR lines +are marked with grey dashed lines. +19 mag, 0.9 at 18.5 mag and 1 at 17.2 mag and linearly inter- +polate in between these values to get fcl,i. +Then using H0 = 70 km s−1 Mpc−1, ignoring cosmolog- +ical effects11 as in Perley et al. (2020) and applying a uni- +form K-correction (K = 2.5×log10(1 + z)), we get a rate of +29.35+27.53 +−21.37 Gpc−3 yr−1 for SNe Ia-CSM. We also calculate +a SN Ia rate of 2.88+0.28 +−0.25 × 104 Gpc−3 yr−1 from SNe Ia ob- +served in the same period following the same method, which +is close to the value of 2.35×104 Gpc−3 yr−1 calculated in +Perley et al. (2020). This puts SNe Ia-CSM to be 0.02–0.2% +of SNe Ia. However this rate estimate should be considered a +lower limit given various caveats in the correct identification +of SNe Ia-CSM (see discussion §4.3). If the ambiguous clas- +sification cases outlined in Appendix A are considered to be +SN Ia-CSM and included in the rate calculation, we obtain +a rate upper limit of 97.7+135.8 +−77.3 Gpc−3 yr−1, which is 0.07– +0.8% of SNe Ia. +3.5. Precursor rates +The ZTF precursor rates were calculated following the +method in Strotjohann et al. (2021) which studied the fre- +quency of precursors in interacting SNe found in ZTF. +11 Contraction of control time window approximately compensated by +increase in the star-formation rate density in the low redshift regime for red- +shift dependent SN rates. +Strotjohann et al. (2021) included 6 of the SNe Ia-CSM +presented in this paper in addition to 4 other SNe Ia-CSM +not in this paper (see Appendix A for details) for their search +but did not find any robust 5σ precursor detections. This +non-detection was concluded to be due to the small sample +size of SNe Ia-CSM (or that they are more distant) compared +to the SN IIn sample, so even if the precursors were as bright +or frequent as for SNe IIn, it would be difficult to detect +them. +The same search was here carried out for our larger sam- +ple by taking the ZTF forced photometry multi-band (g, r, i) +light curves generated by the pipeline outlined in Masci +et al. (2019) and stacking them in 1, 3 and 7-day long bins +to search for faint outbursts. There were 7389 total avail- +able pre-explosion epochs for BTS SNe Ia-CSM, the earliest +epoch being 1012 days prior to the explosion and the median +phase 340 days prior. Hence the results are valid for typical +SN Ia-CSM progenitors at about ∼1 year before the SN. We +did not find any robust 5σ precursor detections. The upper +limits for the precursor rates in different bands are shown in +Figure 14, where the solid lines indicate up to what fraction +of the time a precursor of a given brightness could have been +detected while being consistent with the ZTF non-detections. +A precursor of −15 magnitude could occur as frequently as +∼10% of the time given the ZTF non-detections. A continu- +ous search for the precursors as more SNe Ia-CSM are found +and classified and their sample size increases could yield a +detection if the precursors are as frequent and bright as for +SNe IIn. The dense and massive CSM around these objects +is close enough to have been deposited within decades prior +to the SN but the lack of precursors within 1 year indicates +that there is likely no violent event that ejects a lot of mass +in that period. Probing for precursors could potentially con- +strain the progenitor in at least some cases. For example, +Soker et al. (2013) predicts for their core degenerate (CD) +model for PTF11kx-like SNe release of significant energy +(∼1049 erg) before explosion over timescale of several years, +implying a precursor 3–7 magnitudes fainter than the SN ex- +plosion spread over several years, peaking in the near-IR. +4. DISCUSSION +4.1. Fraction of SNe Ia-CSM with delayed interaction +The fastest declining SNe in our sample (SNe 2018crl, +2020qxz and 2020aekp) are also the ones that develop a +plateau and show relatively stronger SN Ia-like absorption +features in their early spectra. They seem to have a delayed +start for the interaction like PTF11kx but not as fast a decline, +and thus bridge the gap between PTF11kx and the rest of the +strongly interacting SNe Ia-CSM. It remains to be seen how +many SNe Ia are weakly interacting where the CSM inter- +action starts in earnest at timescales of ∼year or more after +explosion, this requires searching for faint detections in care- + +20 +20 +19 +18 +17 +16 +15 +14 +13 +Absolute precursor magnitude +10 +3 +10 +2 +10 +1 +100 +Fraction of time +Type Ia-CSM SNe +Figure 14. +Precursor rate as a function of magnitude calcu- +lated from BTS SN Ia-CSM pre-explosion ZTF forced photometry +stacked in 7-day bins. The different colored shaded regions corre- +spond to different ZTF bands (r-red, g-green, i-grey). The solid +lines depict the upper limits on fraction of the time a precursor of +the corresponding magnitude would have been detected which is +consistent with the ZTF non-detections. +fully calibrated forced photometry light curves (stacked to +go fainter), a study currently undertaken by Terwel et al. (in +prep). From the current sample, it appears that in addition to +SNe Ia-CSM being intrinsically rare, delayed interaction SNe +Ia-CSM are even rarer and only constitute about a quarter of +all SNe Ia-CSM. This delayed interaction behaviour could +also be an effect of asymmetric or clumpy CSM wherein part +of the SN ejecta shine through depending on the viewing an- +gle. Observational campaigns that capture the inner bound- +ary of the CSM and the geometry robustly could shed light +on the distribution of the inner CSM radius and reveal if it is +a continuous distribution or if there are multiple progenitor +scenarios within the SN Ia-CSM class. +4.2. Implications for progenitor based on observed mass +loss +From Figure 10, the estimated mass-loss rates from a sim- +ple spherical treatment of the CSM and a stationary wind lie +between ∼ 10−3 to 10−1 M⊙ yr−1 over a period of less than +∼ 60 years before explosion. That gives a total mass loss of +∼ 0.1 to ∼ 1 M⊙. Dilday et al. (2012) estimated ∼ 5 M⊙ +of CSM around PTF11kx while Graham et al. (2017) revised +it to be ∼ 0.06 M⊙. Light curve modeling of SN 1997cy +and SN 2002ic by Chugai & Yungelson (2004) resulted in +∼ 5 M⊙ estimates for both SNe. Inserra et al. (2016) also fit +analytical models to some SNe Ia-CSM and found the CSM +mass to lie between 0.4 and 4.4 M⊙. Since from Figure 5, +the pseudo-bolometric luminosities of our SNe Ia-CSM lie +somewhere between PTF11kx and SNe 1997cy, 2002ic and +2005gj, with SN 1999E somewhere in the middle, we can +say that the total CSM mass in our sample of SN Ia-CSM +should also be several solar masses. A WD+AGB star sys- +tem has typically been suggested for historical SNe Ia-CSM +to explain this massive CSM. The WD could either gain +mass through Roche Lobe overflow (RLOF) from the com- +panion that drives an optically thick wind (OTW) or merge +with the core of the AGB star that then explodes in or soon +after the common envelope phase. Meng & Podsiadlowski +(2019) model WD+MS systems for their common envelope +wind (CEW) model and find ∼ 1 M⊙ CSM around SNe Ia- +CSM. Thus, given the large observed CSM mass range, the +nature of the companion cannot be solely determined from +total mass lost. High resolution spectroscopy that can resolve +the narrow unshocked CSM wind velocity is also needed to +determine the compactness of the companion. +4.3. Implications for progenitor based observed volumetric +rate +Robust observed rate estimates for SNe Ia-CSM have been +few and far between. Dilday et al. (2010) found 1 interacting +SN Ia (SN 2005gj) in a sample of 79 SNe Ia at z < 0.15 +in the SDSS-II SN survey, giving a rate of ∼1%. After the +PTF11kx discovery in the Palomar Transient Factory (PTF) +survey, the SN Ia-CSM rate was estimated to be ∼0.1% (1 +in 1000 classified SNe Ia; Dilday et al. 2012) but without +spectroscopic completeness determination. S13 identified 7 +more SNe Ia-CSM from the PTF SN IIn sample, bumping +up the estimate to ∼0.8%. With this sample we have im- +proved the rate estimate, providing a robust value (along with +an uncertainty estimate on that value) from an unbiased sur- +vey with high spectroscopic completeness up to 18.5 mag- +nitude. However this rate quite possibly still underestimates +the true value for two reasons. The first being possible ther- +monuclear SNe that are enshrouded so completely by CSM +interaction that they are misclassified as SNe IIn in the ab- +sence of good early time data. In the BTS SN IIn sample, +we found 6 SNe IIn to have ambiguous classifications which +could possibly be SNe Ia-CSM and these are described in +Appendix A. Including these ambiguous cases in rate esti- +mation results in a rate upper limit of 0.07–0.8% for strongly +interacting thermonuclear SNe, while excluding them gives +an underestimated rate of 0.02-0.2%. +The second issue with the rates is if there is indeed a con- +tinuum of delayed interaction SNe Ia-CSM like PTF11kx, in- +teraction in SNe Ia may present itself hundreds of days later +at magnitudes fainter than ZTF’s limit (∼20.5) resulting in +those SNe not being counted when they may be sharing the +same progenitor as the rest of the interacting SNe Ia-CSM. +Lastly in some rare cases, the SN might appear normal in its +light curve shape and duration (and thus would be missed by +the selection criteria used in this paper) but seem to have pe- +culiar narrow Hα in its spectrum or bright mid-IR flux (like +in the case of SN 2020aaym; Th´evenot et al. 2021). + +21 +Han & Podsiadlowski (2006) predicted a rate of 0.1–1% +for 02ic-like events for their delayed dynamical instability +SD model but could not naturally explain the delayed interac- +tion and multiple CSM shells in PTF11kx (which is relevant +for some SNe in our sample). A symbiotic nova-like progen- +itor was suggested by Dilday et al. (2012) for PTF11kx and +they quoted the theoretical rates for the same to lie between +1–30%, however the model could not explain the massive +CSM. Soker et al. (2013) suggested a core degenerate (CD) +scenario in which the explosion is set by the violent prompt +merger of the core of the giant companion on to the WD and +could naturally explain the massive CSM of PTF11kx (Livio +& Riess 2003). Soker et al. (2013) estimated the occurrence +of such SNe (Mcore+ MW D ≳ 2 M⊙ and Menv ≳ 4 M⊙) +through population synthesis and found it to be 0.002 per +1000 M⊙ stars formed. Assuming ∼1–2 SNe Ia occur per +1000 M⊙ stars formed (Maoz et al. 2012), this corresponds +to 0.1–0.2%, which compares well with our observed rate es- +timate. +The CEW model by Meng & Podsiadlowski (2019) pre- +dicts that the SNe Ia-CSM like objects could arise in the SD +CEE scenario when CONe White Dwarfs (WD) steadily ac- +crete material at the base of the CE without quickly spiral- +ing in due to the driving of a CEW wind (10–100 km s−1). +The WD explodes when it reaches the Chandrasekhar mass +(1.38 M⊙) and could possibly explode within the CE before it +is ejected. The CEW model predicts that 25–40% of the SNe +Ia from CONe WD in Common envelope evolution with a +Main Sequence (MS) companion will show SN Ia-CSM like +properties. Meng & Podsiadlowski (2019) also give the ratio +of SNe Ia from CONe WDs to normal SNe Ia from CO WDs +to be between 1/9 and 1/5 (considering normal SNe Ia only +come from CO WD + MS systems). Combining that with the +estimate that roughly 10–20% of all SNe Ia may come from +the SD scenario (Hayden et al. 2010; Bianco et al. 2011), +SNe Ia-CSM from CONe WD according to the CEW model +should be 0.28% to 1.6% of all SNe Ia. A spin-down be- +fore explosion of the WD (Justham 2011; Di Stefano & Kilic +2012) could also explain the time delay between explosion +and interaction. +Soker (2022) estimated the common envelope to explo- +sion delay time distribution (CEEDTD) shortly after the CEE +(tCEED < 104 yr) from SN in planetary nebula rates and +SN Ia-CSM observed rates to be roughly constant rather than +having a t−1 dependence, that is the SN explosion could oc- +cur very soon after the CEE as well. Our observed rates are +on the lower side compared to these theoretical model esti- +mates but compare well within the observational uncertain- +ties, though the CEW model seems to best account for the +overall SNe Ia-CSM properties. +5. SUMMARY +In this paper, we have presented optical and mid-IR pho- +tometry, optical spectra and detailed analysis of 12 new SNe +Ia-CSM identified in the Zwicky Transient Facility Bright +Transient Survey, nearly doubling the total number of such +objects discussed previously by Silverman et al. (2013). The +properties of the sample extracted in this paper agree very +well with similar analysis conducted in S13, particularly the +median EW of Hβ is found to be significantly weaker in +SNe Ia-CSM compared with SNe IIn and consequently the +Balmer decrements are ubiquitously higher in SNe Ia-CSM. +The brightness of SNe Ia-CSM in mid-IR is comparable to +SNe IIn and observations of reduced flux in the red side of +the Hα wing together with the mid-IR brightness points to +formation of new dust in the cooling post-shock gas. The +host galaxies of SNe Ia-CSM lie towards late-type galaxies +with recent star formation. Unlike SNe IIn, no precursors +were found within ∼1000 days before explosion for SNe Ia- +CSM, which could be an observational bias (less number of +SNe Ia-CSM compared to SNe IIn). We provide a robust +rate estimate of 0.02–0.2% of all SNe Ia for SNe Ia-CSM on +account of the BTS survey being unbiased and spectroscop- +ically highly complete. The simple mass-loss rate estimates +from broad Hα luminosity of ∼ 10−2 M⊙ yr−1 are similar to +previous estimates from various methods and indicate several +solar masses of CSM around these SNe. The observed rate +agrees well within the observational uncertainties with the +CEW model by Meng & Podsiadlowski (2019) which can +also explain the interaction delay and massive CSM. +There are still many unanswered questions about the nature +of the progenitors and if we are accurately identifying all po- +tential members of this class. As ZTF Phase II continues, we +are identifying more and more SNe Ia-CSM (interacting with +hydrogen rich and helium rich CSM) and looking further to +the future, if ZTF continues for a Phase III and when LSST +survey operations begins, a larger sample would further im- +prove upon the observed rate calculation. However, individ- +ual object studies are as important and detailed spectroscopic +and multi-wavelength follow-up is essential to capture the +CSM configuration and mass. +6. ACKNOWLEDGMENT +Based on observations obtained with the Samuel Oschin +Telescope 48-inch and the 60-inch Telescope at the Palo- +mar Observatory as part of the Zwicky Transient Facility +project. ZTF is supported by the National Science Founda- +tion under Grants No. AST-1440341 and AST-2034437 and +a collaboration including current partners Caltech, IPAC, +the Weizmann Institute of Science, the Oskar Klein Cen- +ter at Stockholm University, the University of Maryland, +Deutsches Elektronen-Synchrotron and Humboldt Univer- +sity, the TANGO Consortium of Taiwan, the University of +Wisconsin at Milwaukee, Trinity College Dublin, Lawrence + +22 +Livermore National Laboratories, IN2P3, University of +Warwick, Ruhr University Bochum, Northwestern Uni- +versity and former partners the University of Washington, +Los Alamos National Laboratories, and Lawrence Berke- +ley National Laboratories. +Operations are conducted by +COO, IPAC, and UW. The ZTF forced-photometry ser- +vice was funded under the Heising-Simons Foundation grant +#12540303 (PI: Graham). This work was supported by the +GROWTH project (Kasliwal et al. 2019) funded by the Na- +tional Science Foundation under PIRE Grant No 1545949. +The Oskar Klein Centre was funded by the Swedish Re- +search Council. Partially based on observations made with +the Nordic Optical Telescope, operated by the Nordic Op- +tical Telescope Scientific Association at the Observatorio +del Roque de los Muchachos, La Palma, Spain, of the In- +stituto de Astrofisica de Canarias. +Some of the data pre- +sented here were obtained with ALFOSC. Some of the data +presented herein were obtained at the W. M. Keck Observa- +tory, which is operated as a scientific partnership among +the California Institute of Technology, the University of +California, and NASA; the observatory was made possi- +ble by the generous financial support of the W. M. Keck +Foundation. +The SED Machine is based upon work sup- +ported by the National Science Foundation under Grant No. +1106171. This work has made use of data from the Aster- +oid Terrestrial-impact Last Alert System (ATLAS) project. +The Asteroid Terrestrial-impact Last Alert System (ATLAS) +project is primarily funded to search for near earth asteroids +through NASA grants NN12AR55G, 80NSSC18K0284, and +80NSSC18K1575; byproducts of the NEO search include +images and catalogs from the survey area. The ATLAS sci- +ence products have been made possible through the contri- +butions of the University of Hawaii Institute for Astronomy, +the Queen’s University Belfast, the Space Telescope Sci- +ence Institute, the South African Astronomical Observatory, +and The Millennium Institute of Astrophysics (MAS), Chile. +This research has made use of the NASA/IPAC Infrared Sci- +ence Archive, which is funded by the National Aeronautics +and Space Administration and operated by the California In- +stitute of Technology. The Liverpool Telescope is operated +on the island of La Palma by Liverpool John Moores Univer- +sity in the Spanish Observatorio del Roque de los Muchachos +of the Instituto de Astrofisica de Canarias with financial sup- +port from the UK Science and Technology Facilities Council. +Y. Sharma thanks the LSSTC Data Science Fellowship +Program, which is funded by LSSTC, NSF Cybertraining +Grant #1829740, the Brinson Foundation, and the Moore +Foundation; her participation in the program has benefited +this work. +S. Schulze acknowledges support from the G.R.E.A.T re- +search environment, funded by Vetenskapsr˚adet, the Swedish +Research Council, project number 2016-06012. +This work has been supported by the research project +grant “Understanding the Dynamic Universe” funded by +the Knut and Alice Wallenberg Foundation under Dnr KAW +2018.0067, +The research of Y. 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AMBIGUOUS SN IA-CSM/IIN IN BTS +To identify potential SNe Ia-CSM hiding in the SN IIn sample classified by BTS, we rechecked all SNe IIn classifications (total +142) using SuperNova IDentification (SNID; Blondin & Tonry 2007) software. SNe IIn spectra were processed through SNID, +and any SN having ≥ 3 matches to a SN Ia-CSM in the top 10 matches were manually checked. The SNe having ambiguous +classifications are described below. +A.1. SN 2019smj +Discovered by ZTF and reported to TNS by ALeRCE (F¨orster et al. 2021) on 2019-10-13 11:28:42.000, SN 2019smj +(ZTF19aceqlxc) was classified as a Type IIn by BTS at z = 0.06. It peaked at apparent magnitude 17.1 in r band (∼ −20.1) +and then developed a weaker but broader bump. The spectra showed very weak Hβ, barely any He I λ5876, no O I λ7774 +or [O I] lines but showed some iron group lines, Ca NIR emission and [Ca II]. SNID best matches were to SNe 1997cy and +2005gj. The early spectra from P60/SEDM have some matches to SN 2005gj but are too noisy and of ultra-low resolution to +conclusively provide a Ia-CSM classification. From these observations, SN 2019smj is most likely a Type Ia-CSM but given the +lack of confirmation we have excluded it from the main sample. +A.2. SN 2018dfa +Discovered and reported to TNS by ATLAS on 2018-07-05 08:51:21.000, SN 2018dfa was classified initially as a Type IIP +by BTS but later spectra revealed it to be a Type IIn at z = 0.128. It peaked at apparent magnitude of 17.5 in r band (−20.2) +and showed a minor bump before main peak in the light curve. The spectra showed weak Hβ and He I λ5876, no O I λ7774 or +[O I] lines. SNID best matches were to SNe 2002ic and 2005gj along with SNe Ia-norm/91T. The earliest spectra with good SNR +from P200/DBSP had one match to SN 2005gj but could not provide a robust Ia-CSM classification. From these observations, +SN 2018dfa is most likely a Type Ia-CSM but given the lack of confirmation we have excluded it from the main sample. +A.3. SN 2019vpk +Discovered by ZTF and reported to TNS by ALeRCE on 2019-11-25 06:33:38.000, SN 2019vpk was classified as a Type IIn +by BTS at z = 0.1. It peaked at apparent magnitude of ∼ 18 in r band (∼ −20.5). The early spectra were too noisy and the +only spectrum with good SNR was obtained with P200/DBSP nearly 6 weeks after discovery which showed weak Hβ, no clear +He I emission but possibly Si II λ5958 emission (which is unlike any other SN Ia-CSM). SNID top matches were to SN 2005gj +but visually did not look entirely convincing, and some matches were also to Type IIn. We conclude SN 2019vpk does not have +enough data for a robust Ia-CSM classification. +A.4. SN 2019wma +Discovered by ZTF and reported to TNS by ALeRCE on 2019-12-13 13:35:26.000, SN 2019wma was classified as a Type IIn +by BTS at z = 0.088. It peaked at apparent magnitude of ∼ 18.5 in r band (∼ −19.5). The spectra obtained were either from +P60/SEDM or LT/SPRAT hence of low resolution and showed weak Hβ and He I emission. SNID top matches to earliest SEDM +spectrum were to SN 2005gj at the correct redshift but given the lack of intermediate resolution spectra and absence of late time +follow-up we did not assign a Type Ia-CSM classification to SN 2019wma and excluded it from the main sample. +A.5. SN 2019kep +Discovered and reported to TNS by ATLAS on 2019-07-02 14:13:55.000, SN 2019kep was classified as a Type IIn by BTS +at z = 0.02388. It peaked at apparent magnitude of 18.2 in r band (−17). Most early spectra were too noisy for classification +but matched to SN 2005gj. A good SNR P200/DBSP spectrum showed narrow P-Cygni Hα with absorption minimum at ∼ +2500 km s−1 but overall matched to a Type II SN. From these observations, we could not determine a robust classification for SN +2019kep and excluded it from the main sample. +A.6. SN 2018ctj +Discovered and reported to TNS by ZTF on 2018-04-21 08:36:57.000, SN 2018ctj was classified as a Type IIn by BTS at +z = 0.0378. It peaked at apparent magnitude of 18.4 in r band (−17.8) and was also detected in unWISE data. Only one +P60/SEDM spectrum was obtained that matched well to SNe 1997cy and 2005gj. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' AlbaNova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 10691 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Sweden 6School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Trinity College Dublin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the University of Dublin,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' USA 11Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' University of Hawai’i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' HI 96822,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' USA 12IPAC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' California Blvd, Pasadena, CA 91125, USA 13Arkansas Tech University, Russellville, AR 72801, USA 14Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA 15Department of Astronomy, University of California, Berkeley, Berkeley, CA 94720 16Lawrence Berkeley National Laboratory, 1 Cyclotron Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Berkeley, CA 94720 17Universit´e Clermont Auvergne, CNRS/IN2P3, Laboratoire de Physique de Clermont, 63000 Clermont-Ferrand, France 18Isaac Newton Group (ING), Apt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' de correos 321, E-38700, Santa Cruz de La Palma, Canary Islands, Spain ABSTRACT Among the supernovae (SNe) that show strong interaction with the circumstellar medium, there is a rare subclass of Type Ia supernovae, SNe Ia-CSM, that show strong narrow hydrogen emission lines much like SNe IIn but on top of a diluted over-luminous Type Ia spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the only previous systematic study of this class (Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013), 16 objects were identified, 8 historic and 8 from the Palomar Transient Factory (PTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Now using the successor survey to PTF, the Zwicky Transient Facility (ZTF), we have classified 12 additional objects of this type through the systematic Bright Transient Survey (BTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In this study, we present and analyze the optical and mid-IR light curves, optical spectra and host galaxy properties of this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Consistent with previous studies, we find the objects to have slowly evolving light curves compared to normal SNe Ia with peak absolute magnitudes between −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 and −21, spectra having weak Hβ, large Balmer decrements of ∼ 7 and strong Ca NIR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Out of 10 SNe from our sample observed by NEOWISE, 9 have 3σ detections, along with some showing a clear reduction in red-wing of Hα, indicative of newly formed dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We do not find our SN Ia-CSM sample to have significantly different distribution of equivalent width of He I λ5876 than SNe IIn as observed in Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The hosts tend to be late-type galaxies with recent star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We also derive a rate estimate of 29+27 −21 Gpc−3 yr−1 for SNe Ia-CSM which is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2% of the SN Ia rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This work nearly doubles the sample of well studied Ia-CSM objects in Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013), increasing the total number to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Corresponding author: Yashvi Sharma yssharma@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='04637v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='HE] 11 Jan 2023 2 Keywords: circumstellar matter – supernovae: general – supernovae: individual (SN 1997cy, SN 2002ic, SN 2005gj, SN 2005ip, SN 2006jc, SN 2008J, SN 2009ip, SN 2010jl, PTF11kx, SN 2012ca, SN 2013dn, SN 2018crl, SN 2018gkx, SN 2018evt, SN 2019agi, SN 2019ibk, SN 2019rvb, SN 2020onv, SN 2020qxz, SN 2020uem, SN 2020xtg, SN 2020abfe, SN 2020aekp) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' INTRODUCTION When it comes to supernovae (SNe) interacting with cir- cumstellar material (CSM), a number of sub-types of core- collapse SNe (CCSNe) show signs of strong interaction, like SNe IIn (Schlegel 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Filippenko 1997), SNe Ibn (Pas- torello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Foley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Chugai 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Hos- seinzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2017) and most recently SNe Icn (Gal-Yam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN IIn progenitors are generally thought to be massive stars (like Luminous Blue Variables, LBVs) that lose their hydrogen envelopes to wind- driven mass loss and outbursts (Gal-Yam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Gal- Yam & Leonard 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Kiewe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Taddia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Smith 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Helium-rich but hydrogen-deficient CSM in the case of SNe Ibn (Pastorello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Foley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Chugai 2009) and both hydrogen and helium deficient CSM in SNe Icn (Gal-Yam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Pel- legrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022) are thought to arise from high-velocity wind mass loss or stripping of the envelope in binary con- figurations of massive Wolf-Rayet (WR) like stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SNe IIn in most cases, the mass-loss rate derived from the CSM velocity is consistent with estimates from LBV-like eruptive mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However, there exists a rare sub-type of thermonuclear su- pernovae (SNe Ia) which also interacts strongly with CSM i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This class poses a challenge to the progen- itor debate of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There is some consensus on there being at least two major progenitor channels for SNe Ia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the double- degenerate (DD) channel (Webbink 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Iben & Tutukov 1984) which is the merging of two C/O white dwarfs and the single-degenerate (SD) channel (Whelan & Iben 1973) where the white dwarf accretes enough material from a non- degenerate companion to explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Although there are more arguments for the DD scenario from observations of nearby SNe Ia (Nugent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bloom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011), the strongest observational evidence for the SD scenario are SNe Ia with CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Indications of CSM around SNe Ia ranges from detec- tion of time varying narrow Na ID absorption lines (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2009) in high- resolution spectra (found in at least 20% of SNe Ia in spiral hosts, Sternberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Maguire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021), to strong intermediate and narrow Balmer emission features in the spectra and large deviations of the light curves from the standard shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The latter phenomena have been named SNe Ia-CSM (Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013), but were ear- lier referred to as “SNe IIna” or “SNe Ian” due to the strong similarity between their spectra and those of SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The first two examples of this class studied in detail were SNe 2002ic (Hamuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Wood-Vasey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Kotak & Meikle 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Chugai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004) and 2005gj (Aldering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007), but for a long time there was ambiguity regarding their thermonuclear nature (Benetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These SNe were dominated by interaction from the first spectrum and were quite over-luminous compared to normal SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The first clear example of a thermonuclear SN Ia-CSM was PTF11kx (Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It looked like a luminous SN Ia (99aa-like) at early phases but started show- ing interaction at ∼ 60 days from explosion and thereafter strongly resembled SNe 2002ic and 2005gj at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Higher resolution spectra taken at early times indicated mul- tiple shells of CSM with some evacuated regions in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2012) suggested a symbiotic nova progenitor involving a WD and a red giant (similar to RS Ophiuchi) could produce such CSM distribution, however later studies argued that the massive CSM of PTF11kx was inconsistent with the mass-loss rates from symbiotic nova systems (Sil- verman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Ever since, a handful of SNe of this class have been stud- ied in detail to investigate their progenitors and to distinguish them from their spectroscopic cousins, the Type IIn SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Both SN Ia-CSM and SN IIn spectra share a blue quasi- continuum, a strong Hα feature with an intermediate and a narrow component, and often a broad Ca NIR triplet fea- ture, but they differ with regards to the line strength of Hβ, strength/presence of helium and presence of emission lines from intermediate mass elements often found in CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There are some individual SNe with unclear type often re- ferred to as SN Ia-CSM/IIn, like SN 2012ca for which some papers argue for core-collapse (Inserra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2014, 2016) and others for a thermonuclear origin (Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This am- biguity becomes more dominant as the underlying SN flux gets smaller compared to the interaction power (Leloudas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013, hereafter S13) is the only study to analyze a sample of SNe Ia-CSM, 16 objects in total including 6 previously known, 3 re-discovered (re- classified SNe IIn) and 7 new from the Palomar Transient Factory (PTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Their paper presents the common properties of optical light curves, spectra and host galaxies and contrast them against SN IIn properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In this paper, we present 12 new SNe Ia-CSM discovered as part of the Zwicky Tran- sient Facility’s (ZTF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3 Dekany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) Bright Transient Survey (BTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) and analyze their optical light curves, spectra, hosts and rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Throughout this paper, we have compared the results derived from our sample to the ones in S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This paper is organised as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' we first discuss the sam- ple selection criteria, the photometric and spectroscopic data collection in §2, then the analysis of light- and color-curves and the bolometric luminosities is done in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The analysis of early and late-time spectra and emission line identification is presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2, and analysis of the host galaxies is pro- vided in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The rates are estimated from the BTS survey in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We end with a discussion about the nature of SN Ia-CSM progenitors and a summary in §4 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION In this section, we outline our selection criteria, and present the optical photometry and spectroscopic observa- tions of the 12 SNe Ia-CSM in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Selection Criteria To carefully curate our sample of SNe Ia-CSM, we used the BTS sample and its publicly available BTS Sample Ex- plorer1 website to obtain the list of all classified Type Ia sub- types during the period 2018-05-01 to 2021-05-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We then filter out oddly behaving Type Ia SNe based on their light- curve properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We used two criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the primary being rest-frame duration considering flux above 20% of peak flux, and the second being change in magnitude after 30 days from peak (∆m30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We calculated these two properties from either g or r-band light curves (whichever had maximum number of detections) grouped into 3-day bins and used Gaussian Pro- cess Regression2 to interpolate the light curves where cov- erage was missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For the first filtering, we calculated the mean (µ ≈ 35 days) and standard deviation (σ ≈ 16 days) of the duration distribution and selected everything that had a duration greater than µ + 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Given the large sample size (N = 3486), the standard error on the mean is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 days, hence our duration cut of 3σ is suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This filtering se- lected 41 out of 3486 BTS SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Then from these 41 SNe, we calculated the mean and standard deviation of the ∆m30 distribution and removed SNe that were more than 1σ away from the mean on the higher side to reject the relatively steeply declining long SNe, which resulted in 35 SNe being kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Again, the mean and standard deviation of ∆m30 dis- tribution of these 41 long duration SNe are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='48 mag and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='27 mag respectively and the standard error on mean is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='04, making our 1σ cut suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Finally, we manually inspected 1 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='edu/ztf/bts/explorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='php 2 Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2011) https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='org/stable/modules/gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='html the 35 selected SNe Ia to confirm their classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 20 out of the 35 SNe that passed the above filtering criteria were just normal SNe Ia either caught late or missing some post- peak coverage in ZTF or had spurious detections that resulted in long duration estimates, 2 had incorrect duration estimate due to an interpolation error and were recalculated and 1 (AT2020caa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soraisam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021) had some detections be- fore the SN explosion which could be connected to a different SN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' a sibling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The remaining 12 long-duration SNe Ia all turned out to be spectroscopically classified SNe Ia-CSM in BTS, and none of the classified BTS SNe Ia-CSM were missed in this fil- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' No other SNe apart from these stood out in particu- lar, indicating the classification reliability of the BTS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' During the same period, 9 SNe Ia-CSM were reported to the Transient Name Server (TNS), out of which 7 are already in our sample, 1 was detected by ZTF but did not meet the BTS criteria, and 1 was not detected in ZTF as the transient lo- cation fell too close to the field edges and was masked by the automated image subtraction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2019) presented early photometric observations of one SN Ia-CSM in our sample, SN 2018crl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Table 1 summarizes the coor- dinates, redshifts, peak absolute magnitudes, durations, host galaxy information and Milky Way extinction for the 12 SNe Ia-CSM in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Furthermore, we re-checked the classifications of 142 SNe IIn classified in BTS during the same period as above, in case any SN Ia-CSM was masquerading among them and found 6 to have ambiguous classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These are discussed fur- ther in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Discovery All SNe Ia-CSM were detected by the ZTF (Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dekany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) and passed the criteria for the BTS (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) automatic filtering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' extra-galactic real transients with peak magnitudes brighter than 19 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These were saved and classified as part of BTS which aims to classify all transients brighter than 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 magnitude, and reported to the Transient Name Server3 (TNS) during the period 2018-05- 01 to 2021-05-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Out of the 12 SNe, 6 were first reported to TNS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' discovered) by ZTF (AMPEL, Nordin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soumagnac & Ofek 2018 and BTS), 3 were first reported by GaiaAlerts (Hodgkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021), 2 by ATLAS (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) and 1 by ASAS-SN (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For classi- fication, 9 were classified by the ZTF group, 1 by ePESSTO (Smartt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018a), 1 by SCAT (Tucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019) and 1 by the Trinity College Dublin (TCD) group (Prentice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The follow-up spectral series for these SNe were obtained as part of the 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='wis-tns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='org/ 4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Properties of the 12 BTS SNe Ia-CSM ZTF Name IAU Name z M peak r Duration1 Host Name Host Mag2 (mag) (days) (mr) ZTF18aaykjei SN 2018crl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='097 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='66 130 SDSS J161938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='90+491104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='89 ZTF18abuatfp SN 2018gkx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1366 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07 322 SDSS J135219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='22+553830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='23 ZTF18actuhrs SN 2018evt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02378 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='10 447 MCG-01-35-011 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07 ZTF19aaeoqst SN 2019agi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0594 <-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='76 >303 SDSS J162244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='06+240113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='82 ZTF19abidbqp SN 2019ibk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='04016 <-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='55 >576 SDSS J014611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='93-161701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='55 ZTF19acbjddp SN 2019rvb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1835 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='74 172 WISEA J163809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='90+682746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='44 ZTF20abmlxrx SN 2020onv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='095 <-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='36 >154 WISEA J231646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='31-231839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='95 ZTF20abqkbfx SN 2020qxz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0964 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='00 166 WISEA J180400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='99+740050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='65 ZTF20accmutv SN 2020uem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='041 <-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='17 >279 WISEA J082423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='32-032918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='88 ZTF20aciwcuz SN 2020xtg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0612 <-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='60 >336 SDSS J153317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='64+450022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='42 ZTF20acqikeh SN 2020abfe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='093 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='24 171 SDSS J200003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='30+100904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='18 ZTF21aaabwzx SN 2020aekp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='046 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='62 458 SDSS J154311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='45+174843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='41 1 Rest frame duration above 20% of r-band peak flux, uncertainty of ±2 − 3 days from ZTF cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2 Corrected for Galactic extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' BTS classification campaign as many were difficult to clas- sify with the ultra-low resolution spectrograph P60/SEDM (Blagorodnova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018) and hence were followed up with intermediate resolution spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The SEDM spectra were helpful in determining an initial redshift but the tem- plate matches were unclear (matched to SN IIn as well as SN Ia-CSM and SN Ia-pec templates, some matched poorly to SN Ia/Ic at early times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2019agi (classification and spectrum taken from TNS), 2019rvb, 2020onv, 2020qxz and 2020uem were classified as Ia-CSM ∼ 1 − 2 month after discovery using spectra at phases of 42, 26, 38, 45 and 51 days respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2018crl, 2018gkx and 2019ibk were classified ∼ 2 − 3 months after discovery using spectra at phases of 92, 75 and 103 days respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2018evt, 2020abfe and 2020aekp were classified ∼ 4 − 5 months af- ter discovery using the spectra at phases of 144, 146 and 132 days respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2020xtg immediately went behind the sun after its first detection in ZTF therefore its first spectrum (using SEDM) was taken at 91 days since explosion which was dominated by strong Hα emission, and thus SN 2020xtg was initially classified as a Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' As this SN was exhibiting a long lasting light curve, an intermediate resolution spec- trum was taken at 340 days which matched very well to SN Ia-CSM and therefore its classification was updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2020uem and 2020aekp showed peculiar features and were followed up for more optical spectroscopy for single object studies (to be presented in future papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Optical photometry To assemble our sample light curves, we obtained forced PSF photometry via the ZTF forced-photometry service (Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' IRSA 2022) in g, r and i bands and also added data from ATLAS (Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) forced-photometry service in c and o bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The high cadence ZTF partnership survey in i band contributed some photometry to SNe 2018crl, 2018gkx, 2019agi, 2019ibk and 2019rvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The ZTF and ATLAS data were supplemented with data from the Rainbow camera (RC, Ben-Ami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012) on the robotic Palomar 60-inch telescope (P60, Cenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006) and the Optical wide field camera (IO:O) on the Liver- pool telescope (LT, Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The P60 data was pro- cessed with the automatic image subtraction pipeline FPipe (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016) using reference images from SDSS when available, and otherwise from Pan-STARRS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The IO:O data was initially reduced with their standard pipeline4 then image subtraction was carried out using the method outlined in Taggart (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SN 2018evt, some early time data available from ASAS-SN (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2017) in the V band was obtained through their Sky Patrol5 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We corrected all photometry for Milky Way extinction with the Python package extinction (Barbary 2016) us- ing the dust extinction function from Fitzpatrick (1999), the Schlafly & Finkbeiner (2011) dust map, and an RV of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Then we converted all measurements into flux units for anal- ysis and considered anything less than a 3σ detection an up- per limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There is moderate to good coverage in g, r, c and o bands for all SNe in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 1 shows a multi- paneled figure of the light curves of the objects in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Mid-IR photometry 4 https://telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='livjm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='uk/TelInst/Pipelines/ 5 https://asas-sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='edu/ 5 0 80 160 240 320 400 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2018crl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='11 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='36 mag/100 d 0 100 200 300 400 500 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2018gkx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='18 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='42 mag/100 d 0 150 300 450 600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 Co decay SN 2018evt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51 mag/100 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='36 mag/100 d 0 80 160 240 320 400 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2019rvb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='93 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 mag/100 d 0 100 200 300 400 500 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2019agi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='96 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='45 mag/100 d 0 150 300 450 600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 Co decay SN 2019ibk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='61 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='18 mag/100 d 0 80 160 240 320 400 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2020qxz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='75 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='29 mag/100 d 0 100 200 300 400 500 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 D Co decay SN 2020onv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='26 mag/100 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='06 mag/100 d 0 150 300 450 600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 Co decay SN 2020uem* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 mag/100 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='27 mag/100 d 0 80 160 240 320 400 D 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2020abfe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51 mag/100 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51 mag/100 d 0 100 200 300 400 500 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Co decay SN 2020xtg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='62 mag/100 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='21 mag/100 d 0 150 300 450 600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 Co decay SN 2020aekp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='52 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07 mag/100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='66 mag/100 d Rest-frame days since explosion Absolute magnitude P48:ZTF: g P48:ZTF: r P48:ZTF: i ATLAS: c ATLAS: o P60:RC: g P60:RC: r P60:RC: i LT:IOO: g LT:IOO: r LT:IOO: i LT:IOO: z LCO: sdssg LCO: sdssr LCO: sdssi ASASSN: V Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Optical light curves of the ZTF BTS SN Ia-CSM sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The SNe Ia-CSM have longer duration than the average SN Ia, with some variety like bumpy light curves or long plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The one SN marked with an asterisk (SN 2020uem) has an unconstrained explosion time estimate (∼ ±50 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The decline rate from Cobalt decay is marked with black dashed line, the light curve decline rates measured from r-band data are shown in the subplot legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 6 The transients were observed during the ongoing NEO- WISE all-sky mid-IR survey in the W1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 µm) and W2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 µm) bands (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2010a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Mainzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We retrieved time-resolved coadded images of the field cre- ated as part of the unWISE project (Lang 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' To remove contamination from the host galax- ies, we used a custom code (De et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) based on the ZOGY algorithm (Zackay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016) to perform image sub- traction on the NEOWISE images using the full-depth coadds of the WISE and NEOWISE mission (obtained during 2010- 2014) as reference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Photometric measurements were obtained by performing forced PSF photometry at the tran- sient position on the subtracted WISE images until the epoch of the last NEOWISE data release (data acquired until De- cember 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Further analysis of the mid-IR photometry is presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Optical spectroscopy The main instruments used for taking spectra and the soft- ware used to reduce the data are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Ad- ditionally, the spectrum Reguitti (2020) obtained using the Asiago Faint Object Spectrograph and Camera (AFOSC) on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 m telescope at Cima Ekar, and the spectrum Stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2018b) obtained using the ESO Faint Object Spectro- graph and Camera version 2 (EFOSC2) on ESO New Tech- nology Telescope (NTT) were taken from TNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The details for all optical spectra (61 for the sample in to- tal) presented in this paper are provided in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Further- more, all spectra were corrected for Milky Way extinction using extinction and the same procedure as for the pho- tometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The SN redshifts were derived using narrow host lines for the objects which did not already have a host red- shift available in the NASA/IPAC Extragalactic Database6 (NED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Photometric calibration was done for all spectra i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' they were scaled such that the synthetic photometry from the spectrum matched the contemporaneous host-subtracted ZTF r-band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SN 2018crl, a host galaxy spectrum taken using P200/DBSP was available, which was subtracted from the P200/DBSP SN spectrum taken at +92 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SN 2020aekp, more spectra beyond ∼ 350 days were obtained but will be presented in a future study of the object (34 addi- tional spectra up to ∼600 day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These processed spectra were used for the rest of the anal- ysis as detailed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 and will be available on WISeREP7 (Yaron & Gal-Yam 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ANALYSIS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Photometry 6 https://ned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='edu/ 7 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='wiserep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='org/ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Description of spectrographs used for follow-up and the corresponding data reduction pipelines Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Telescope Reduction Software SEDM1 Palomar 60-inch (P60) pySEDM2 ALFOSC3 Nordic Optical Telescope IRAF4, PyNOT14, pypeit DBSP5 Palomar 200-inch (P200) IRAF6, DBSP DRP7 KAST8 Shane 3-m IRAF LRIS9 Keck-I LPipe10 SPRAT11 Liverpool Telescope Barnsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2012) DIS12 APO13 IRAF 1 Spectral Energy Distribution Machine (Blagorodnova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018) 2 Rigault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2019) 3 Andalucia Faint Object Spectrograph and Camera 4 Tody (1986, 1993) 5 Double Beam Spectrograph (Oke & Gunn 1982) 6 Standard pipeline by Bellm & Sesar (2016) used prior to Fall 2020 7 pypeit (Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020) based pipeline (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' com/finagle29/dbsp drp) used since Fall 2020 8 Kast Double Spectrograph (Miller & Stone 1987) 9 Low Resolution Imaging Spectrometer (Oke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1995) 10 IDL based automatic reduction pipelinea (Perley 2019) 11 Spectrograph for the Rapid Acquisition of Transients (Piascik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2014) 12 Dual Imaging Spectrograph 13 Astrophysics Research Consortium telescope at the Apache Point Observatory 14 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='com/jkrogager/PyNOT ahttps://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='edu/∼dperley/programs/lpipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='html 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Explosion epoch estimates For the purpose of this paper, the ‘explosion time’ simply refers to the time when optical flux rises above the zero-point baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' first light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We used pre-peak g, r, i-band ZTF photometry and c, o-band ATLAS photometry (binned in 1- day bins), when available, for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For each SN, the light curve was interpolated using Gaussian process regres- sion to obtain the peak flux epoch, then a power-law (PL) model was fit using epochs from baseline to 60% of peak brightness in each band following Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The PL fits converged in at least one band for 6 out of 12 BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For the rest, we simply took the middle point between the first 5σ detection and the last upper limit before this detection as the explosion epoch with half of the separa- tion between these two points as the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The explosion time estimates, light curve bands used for the PL fits and the 1σ uncertainties on explosion times are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The unfilled ‘PL fit filters’ column in the table are the SNe for which the PL fit did not converge and averages were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='399 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='use only the last non-detection the uncertainty range is typ- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='ically less than 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Only for SN 2020uem is the date of explosion virtually unconstrained (±57 days) as it was be- hind the sun at the time of explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Although for SN 2019ibk the explosion time is formally constrained with a ±3 day uncertainty, this estimate was derived using only ATLAS o-band data right after the SN emerges from behind the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There is not a clear rise ob- served over a few epochs but two non-detections before a 5σ detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It is possible that the actual peak of this SN occurred earlier while it was behind the sun and the rising o-band points after it emerged are due to a second peak or bump (similar to SN 2018evt, in that case the actual rise was caught before the SN went behind the sun in ASAS-SN data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' If the former explosion epoch estimate from o-band is to be believed then SN 2019ibk would be the most sub-luminous among the SNe Ia-CSM, peaking at −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Duration and absolute magnitudes Figure 2 shows the SNe Ia-CSM (colored squares) in our sample in the duration-luminosity and duration-∆m30 phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the top panel, the x-axis is duration above half-max and the y-axis is the peak absolute magnitude (see Table 1) when we have photometric coverage both pre-peak and post- peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SNe missing the pre-peak coverage, their discov- ery magnitude is taken to be the upper limit to peak absolute magnitude and the duration from discovery the lower limit 8 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Explosion time epoch estimates derived from pre-peak multi-band light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For 6 out of 12 SNe Ia-CSM, we were able to fit a power-law model to multi-band data following Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For the remaining 6 SNe, the explosion epoch was estimated by taking the mean of the first 5σ detection and last upper-limit before the first detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' IAU Name PL fit filters to 1σ interval (MJD) (days) SN 2018crl g, r, o 58271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='83 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='48,+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='38] SN 2018gkx r, o 58371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='34 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='64,+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='53] SN 2018evt 58334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='26 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='00,+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='00] SN 2019agi 58502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='48 [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51,+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51] SN 2019ibk 58654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='61 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='99,+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='99] SN 2019rvb g, r, i, o 58749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='16 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='79,+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='60] SN 2020onv o 59032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='75 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='49,+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='10] SN 2020qxz g, r, o 59063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='05 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='51,+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='45] SN 2020uem 59117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='03 [−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='63,+56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='63] SN 2020xtg 59130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='14 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='04,+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='04] SN 2020abfe g, r, o 59159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='36 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='16,+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='23] SN 2020aekp 59204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='53 [−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='50,+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='50] to duration above half-max (marked by arrows in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The BTS SN Ia sample is shown in gray points, and we also show the SNe Ia-CSM presented in S13 with empty trian- gles for comparison in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the bottom panel, the x-axis is duration above 20% of peak flux (∆t20) and the y-axis is ∆m30, the two parameters used in the selection cri- teria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Most of the SNe Ia-CSM lie on the longer duration and brighter luminosity side, and are even more distinctly separated in the ∆t20-∆m30 phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This makes the SN initial decline rate and duration useful tools for identify- ing thermonuclear SNe potentially interacting with CSM, if they have not revealed themselves already in their early time spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The gray points lying in the same phase space as SNe Ia-CSM are the false positive cases described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Also worth noting is that the duration calculated by taking the flux above half of peak flux value does not capture the true duration of the light curve when the plateau phase falls below half-max as is the case for SN 2020aekp (> 500 days light curve) but ∆t20 and ∆m30 do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Light and color curves We have good pre-peak coverage in ZTF data for 8 of the 12 SNe in our sample8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2018evt was discovered by ASAS-SN on JD 2458341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='91 (Nicholls & Dong 2018) and classified by ePESSTO the next day (Stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2018a), around 115 days before the first detection in ZTF when the SN came back from behind the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Hence we have only one 8 except for SNe 2018evt, 2019ibk, 2020onv and 2020uem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 0 25 50 75 100 125 150 175 Rest-Frame duration above half-max (days) 22 21 20 19 18 17 16 15 Peak absolute magnitude Ia SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp Silverman 2013 0 100 200 300 400 500 Rest-frame duration above 20% of max (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 mpeak + 30d mpeak (r-band) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Top: Location of our 12 SNe Ia-CSM in the peak absolute magnitude vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' rest-frame duration above half max phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The colored points are the BTS SNe Ia-CSM and the gray points are the rest of the BTS SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Also shown with empty triangles are the SNe Ia-CSM from S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The vertical arrows mark the upper limits to peak absolute magnitudes and horizontal arrows mark the lower limits to durations of SNe not having pre-peak coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bottom: Change in magnitude 30 days after peak (∆m30) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' rest-frame duration above 20% of peak-flux for BTS SNe Ia and SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These criteria were used to filter out potential SNe Ia-CSM from all SNe Ia and demonstrate that SNe Ia-CSM occupy a distinct portion in this phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However some gray points (not SN Ia-CSM) remain on the longer duration side and are the false positive cases described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' epoch of pre-peak photometry and one early spectrum for SN 2018evt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Our mixed bag of SNe Ia-CSM show post-maximum de- cline rates ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 mag 100d−1 in the r band from peak to ∼ 100 days post peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The median de- cline rate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07 mag 100d−1, which is much slower than the decline rates of normal SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We see a variety of changes in decline rates after around 100 days from peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Two SNe (2020onv and 2020abfe) show no change and have a constant slow decline throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Four SNe (2018gkx, 2019agi, 2019ibk and 2019rvb) evolve to a shallower slope going from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6–1 mag 100d−1 to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 mag 100d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Three SNe (2018crl, 2020qxz and 2020aekp) show a ma- jor change in decline rate with the light curves becoming 9 almost flat, and SN 2020aekp shifts back to a slow de- cline from this plateau after ∼ 200 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In three cases, the decline rate actually becomes steeper, SN 2018evt goes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='52 mag 100d−1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 mag 100d−1, SN 2020uem goes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='52 mag 100d−1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='25 mag 100d−1 and SN 2020xtg seems to go from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='61 mag 100d−1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='35 mag 100d−1 (even though there is only one epoch at late times to measure this change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The 3 SNe with fastest initial decline rates (≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 mag 100d−1 in the r band) are similar to SN 2002ic (initial decline of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='66 mag 100d−1 in V ) and PTF11kx (ini- tial decline of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 mag 100d−1 in R) and coincidentally are also the ones that evolve into a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The rest of the sample have initial decline rates comparable to SN 1997cy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='75 mag 100d−1) and SN 2005gj (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='88 mag 100d−1) (In- serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From these observations, we can conclude that SNe Ia-CSM exhibit a range of slow evolution indicat- ing that there exists a continuum of phases at which strong CSM interaction begins to dominate the powering of the light curves for these SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It is, however, difficult to pinpoint the exact phase when interaction starts from the light curve without modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' CSM interaction could be affecting the peak brightness significantly even in cases where interaction only appears to dominate after a few weeks (SNe 2018crl, 2020qxz 2020aekp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Considering the average peak phase to be ∼ 20 days past explosion from the light curves and as- suming an ejecta velocity of ∼ 20000 km s−1, the CSM is located at ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 × 1015 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This estimate can be refined by considering the phase of the earliest spectrum that shows interaction signatures (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' At late times, all the de- cline rates are slower than that expected from Cobalt decay (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='98 mag 100d−1), confirming that the power from CSM in- teraction dominates the light curve behaviour for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 3 shows the g − r color evolution of our sam- ple SNe as a function of phase (rest-frame days from r- band maximum), comparing them with some famous SNe Ia-CSM (SNe 2005gj, 1997cy, 1999E), and SNe 2012ca (Ia- CSM/IIn), 2010jl (IIn) and 1991T (over-luminous Type Ia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The color evolution of normal SNe Ia from ZTF (Dhawan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022) is shown in grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We use g − r colors when available, otherwise we estimate the g − r color by fitting Planck functions to estimate the black-body tempera- tures from the V − R colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Our SNe Ia-CSM show simi- lar color evolution as the older Type Ia-CSM/IIn interacting SNe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the g − r color increases gradually for about 100 days and then settles onto a plateau or slowly declines, and one object (SN 2019ibk) becomes redder at late times similar to SN 2012ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The interacting SNe are redder at late times compared to the normal SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Mid-IR brightness comparison Out of 12 SNe in our sample, only one observed (SN 2020abfe) did not have 3σ detections post explosion in the unWISE difference photometry light curves and two (SNe 2019rvb and 2020qxz) did not have coverage post explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The unWISE light curves for the rest of the SNe Ia-CSM having > 3σ detections in W1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 µm) and W2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 µm) bands are shown in Figure 4 (black and red stars) along with Spitzer IRAC survey data of SN 2008cg (indigo and ma- genta empty triangles), SN 2008J (indigo and magenta empty squares) (both Ia-CSM) and some SNe IIn (blue and orange crosses) taken from Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The most nearby SN in our sample, SN 2018evt, is among the brightest (∼ 17 AB mag) in MIR at least until ∼1000 days after explosion and has a bumpy light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2019ibk and 2018crl however are the most luminous with an absolute magnitude of −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 mag in the W1 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The brightness of the BTS SNe Ia-CSM is comparable with other interacting SNe and span a similar range (−16 to −19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However, SNe IIn have been detected until even later epochs (up to 1600 days) than SNe Ia-CSM, probably due to the larger number of SNe IIn at closer dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2020abfe has upper limits around ∼ −18 in W1 band and ∼ −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 in W2 band up to ∼300 days post explo- sion shown with upside down filled triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' As the mid-IR luminosity can be fainter than these limits for SNe Ia-CSM (as can be seen for other nearby SNe in this sample) and SN 2020abfe is at a redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='093, it might just be out of reach for WISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This brightness of SNe Ia-CSM in mid-IR can be indica- tive of existing or newly formed dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A clear signature of new dust is reduced flux in the red wing of the Hα emission line at late phases as the new dust formed in the cold dense shell behind forward shock absorbs the far-side (redshifted) intermediate and narrow line emission (see bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For our sample, this reduction in Hα red wing is the most pronounced for SN 2018evt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bolometric luminosity As the SN Ia-CSM luminosity is dominated by CSM inter- action, their spectra comprise of a pseudo-continuum on the blue side and strong Hα emission on the red side, hence a blackbody fit to multi-band photometric data is not appropri- ate to estimate the bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Instead we calcu- late a pseudo-bolometric luminosity from the available multi- band optical data by linearly interpolating the flux between the bands and integrating over the optical wavelength range spanned by the ATLAS and ZTF bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The individual band light curves are first interpolated using Gaussian process re- gression to fill in the missing epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This estimate places a strict lower limit on the bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In Figure 5 we show the pseudo-bolometric luminosity of our SN Ia-CSM sample in comparison with SN 1991T (Type Ia), SNe 1997cy, 1999E, 2002ic, 2005gj, 2013dn and PTF11kx (Ia-CSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Multi-band photometric data were taken from the Open Supernova Catalog (Guillochon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2017) 10 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2018crl 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2018gkx 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2018evt 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2019agi 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2019ibk 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2019rvb 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 D SN 2020onv 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2020qxz 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2020uem 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2020xtg 0 150 300 450 600 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2020abfe 0 150 300 450 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 SN 2020aekp Rest-frame days from r-band maximum g r color (mag) SN2005gj SN2012ca SN1997cy SN1999E SN1991T SN2010jl SN Ia Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Color evolution (g − r) of BTS SNe Ia-CSM from r-band maximum (plotted in black) compared with SNe 2005gj, 1997cy, 1999E (Ia-CSM), SN 2012ca (IIn/Ia-CSM), SN 2010jl (IIn), SN 1991T (SN Ia) and ZTF SNe Ia (gray lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' As can be seen for up to ∼ 150 days, our SNe Ia-CSM tend to be redder than SNe Ia and at late times develop a plateau similar to other interacting SNe (IIn/Ia-CSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 11 0 250 500 750 1000 1250 1500 1750 Days since explosion 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 Absolute Magnitude (AB) BTS SN Ia-CSM: W1 BTS SN Ia-CSM: W2 SN 2008J: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 m SN 2008J: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 m SN 2008cg: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 m SN 2008cg: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 m SN IIn: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 m SN IIn: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 m Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' unWISE detections in the W1 and W2 bands of BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The W1 and W2 points are marked with black and red filled stars respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Spitzer IRAC photometry of SNe IIn (blue and orange crosses) and two SNe Ia-CSM from Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2011) (SNe 2008cg and 2008J in empty triangle and square) are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 9 out of 12 BTS SNe Ia-CSM are as bright in mid-IR as other interacting SNe (∼ −16 to ∼ −19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The upper limits for SN 2020abfe are shown in black and red filled up- side down triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' for SN 1991T (Filippenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1994) to generate the bolometric luminos- ity light curve through black body fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The pseudo- bolometric luminosity light curve for SN 1997cy was ob- tained from Germany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2000), for SN 2013dn from Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2015) and for SNe 2002ic, 2005gj, 1999E and PTF11kx from Inserra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' All BTS SNe Ia-CSM show a slow evolution in bolomet- ric luminosity, inconsistent with the decay of 56Co to 56Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The sample’s overall luminosity decline rates are comparable to those of SNe 1997cy and 2013dn, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Only SNe 2018crl and 2020aekp seem to show early decline in their pseudo-bolometric light curves similar to SN 1991T for about 40 days after peak like SN 2002ic and PTF11kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Another BTS interacting SN Ia, ZTF20aatxryt (Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022), was found to follow the PTF11kx light-curve evo- lution very closely and as its light curve fell into a plateau the SN started showing signs of interaction with a helium- rich CSM and evolved into a helium-rich SN Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We have excluded ZTF20aatxryt from the sample as we focus on typical SNe Ia-CSM interacting with hydrogen-rich CSM in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' At late phases (∼ 300 days), the SNe Ia-CSM are approximately 100 times brighter than normal SNe Ia at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Therefore, at these late phases, the lu- minosity and spectral features of SNe Ia-CSM are entirely dominated by CSM-interaction with little emergent SN flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From the pseudo-bolometric light curves, we place a lower limit on the total radiated energy for SNe Ia-CSM to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 ×1050erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This is well below the thermonuclear budget (Ekin ∼ 1051 erg), but as this is a lower limit and some SNe in the sample have unconstrained peaks, the true total radia- tive energy might come close to the thermonuclear budget, requiring high conversion efficiency to achieve their lumi- nosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 0 80 160 240 320 400 Rest-frame days since explosion 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 log(Lopt (erg/s)) SN2013dn SN1997cy SN1991T PTF11kx SN2002ic SN2005gj SN1999E SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Pseudo-bolometric luminosity light curves of BTS SNe Ia-CSM compared with pseudo-bolometric light curves of SNe 1991T, 1997cy, 1999E, 2002ic, 2005gj, 2013dn, and PTF11kx from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The light curves in each filter having more than 10 epochs were interpolated using Gaussian process regression to fill in the missing epochs, and at each epoch the fluxes between the bands were linearly interpolated and integrated over the optical wave- length range spanned by ZTF and ATLAS filters to get the pseudo- bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For BTS SNe, the phases are with respect to the estimated explosion epochs, while for comparison SNe the phases are with respect to discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Spectroscopy Figure 6 displays the spectral series obtained for the BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Most of the early time spectra were taken with the SEDM, the BTS workhorse instrument (R ∼100), which is not able to resolve the narrow CSM lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There- fore, these SNe were followed up with higher resolution in- struments to get more secure classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For each spec- trum in Figure 6, the phase is provided with respect to the explosion epoch estimate given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We have spec- tra ranging from a few to around 470 days from explo- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Considering the well constrained explosion times of SN 2018evt, presence of narrow Hα in its first spectrum at 8 days since explosion and assuming a typical ejecta veloc- ity of ∼20000 km s−1, this implies that the CSM interaction start as close as ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4×1015 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 7 shows the early time (left) and late time (right) spectral behaviour of the BTS SNe Ia-CSM together with a few historical SNe for comparison, namely SNe Ia-CSM SN 2011jb (Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013), SN 2005gj and PTF11kx, the Type Ia SN 1991T and the well-observed Type IIn SN 2010jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Spectral series of all SNe Ia-CSM presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The rest-frame phases are shown alongside the spectra in each subplot and have been calculated using the explosion epoch estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The colors depict different instruments used to obtain this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Major emission lines are marked with vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 13 and [Fe II/III] line regions, and vertical dashed lines mark the Balmer emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The sample spectra have been mul- tiplied by a constant factor to magnify relevant spectral fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the following paragraphs, we compare the observa- tions of some of the spectral features with previous analysis of this class (Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Inserra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A few of our early time SNe Ia-CSM show underlying SN Ia absorption features like PTF11kx and SN 2002ic (most are, however, quite diluted and also affected by the low res- olution and signal-to-noise ratio (SNR) of the SEDM spec- tra), the most notable being SNe 2018evt, 2020qxz and 2020aekp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2020qxz and 2020aekp also have among the fastest initial post-peak decline rates in the sample, similar to PTF11kx, while coverage around peak is not available for SN 2018evt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' On the other hand, SNe with slower decline rates similar to SN 1997cy and SN 2005gj have more SN IIn-like early time spectra dominated by blue pseudo-continuum and Balmer emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The faster decline rate suggests we are still seeing some of the emission from the ejecta at those phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' To unveil the nature of the progenitor of interacting SNe, it is therefore necessary to obtain some spectroscopic follow-up before peak light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Spectroscopic data at the phase of tran- sition to interaction-dominated luminosity would also help in deducing the extent and density structure of the optically thick CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Late time spectra of SNe Ia-CSM look very similar to those of SNe IIn, heavily dominated by Hα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The CSM in- teraction masks the underlying SN signature and we instead see late-time spectra riddled with photoionized CSM lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In some cases, the photosphere might lie in an optically thick cold dense shell (CDS) formed between the forward and re- verse shocks which obscures the ejecta completely (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Chugai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The continuum is also en- shrouded under a blue quasi-continuum from a forest of iron- group element lines (S13) as identified and analyzed for SNe 2012ca and 2013dn by Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The blue quasi-continuum blend of iron lines ([Fe III] lines around ∼4700 ˚A and [Fe II] around ∼5200 ˚A) in the spectra of the BTS SN Ia-CSM sample (see Figure 7 top right panel) is the dominant feature blue-ward of 5500 ˚A but the ratio of [Fe III]/[Fe II] is much weaker compared to for SNe Ia (like SN 1991T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This feature is more apparent in the SNe Ia- CSM like PTF11kx and SN 2002ic that became interaction- dominated later than for other SNe Ia-CSM such as SNe 1997cy, 1999E and SN 2012ca (SN Ia-CSM/IIn, for which a clear type has not been established).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Inserra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2014) argues for a core-collapse origin for SN 2012ca given this low amount of [Fe III] along with the detection of blueshifted Carbon and Oxygen lines (which however, were later argued to be [Fe II] lines by Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' S13 instead argues in favor of a thermonuclear origin given the presence of this blue quasi-continuum, despite [Fe III] being weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2015) points out that a similarly suppressed ratio of [Fe III]/[Fe II] is observed in some SNe Ia, particularly the super-Chandra candidate SN 2009dc, for which the expla- nation was suggested to be a low ionization nebular phase owing to high central ejecta density and low expansion ve- locities (Taubenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2015) argue that in the case of SNe Ia-CSM, a lower ionization state could arise owing to the deceleration of ejecta by the dense CSM explaining the Fe line ratio suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Since Ca has lower first and second ionization potentials than Fe, the detection of [Ca II] λλ7291, 7324 would be consistent with this low ionization, which Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2015) confirms for SNe 2012ca and 2013dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Indeed, we find clear evidence of [Ca II] emis- sion for 8 out of 12 SNe in our sample and moderate to weak signal for the remaining 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Although this does favor the argument for a thermonuclear origin, a similar blue quasi- continuum is also observed in other interacting SN types like SNe Ibn (SN 2006jc, Foley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007) and SNe IIn (SNe 2005ip and 2009ip), making Fe an incomplete indicator of the progenitor nature (see detailed discussion in Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We do not find strong evidence of O I λ7774 or [O I] λλ6300, 6364 emission in our sample, although they might be present at very weak levels in some SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2020uem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2020uem has strong emission lines at 6248, 7155 and 7720 ˚A which are consistent with being iron lines and were also observed in SNe 2012ca, 2013dn and 2008J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' S13 note that the very broad emission around 7400 ˚A can be due to a blend of [Ca II] λλ7291, 7324 and [O II] λλ7319, 7330, however we note that this broad emission is likely to be from calcium as O II is harder to excite than O I which is either very weak or absent in our spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The broad Ca NIR triplet feature resulting from electron scattering is the next strongest feature after the Balmer emission and is present in all mid to late-time spectra of the SNe in our sample where the wavelength coverage is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We observe it increas- ing in relative strength with phase, at least for a year, after which we no longer have spectral coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The bottom panel of Figure 7 shows the line profile of Hα, with the blue side reflected over the red side at the maxi- mum flux after continuum removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We do see evidence of diminished flux in the red wing of Hα at late phases in some SNe (most notable in SNe 2018evt and 2020uem), which can indicate formation of new dust in the post-shock CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' S13 claim to observe this for all non-PTF SNe Ia-CSM in their sample starting at ∼75–100 days, while for the PTF SNe Ia-CSM they do not have spectra available post that phase range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For some BTS SNe Ia-CSM, we also do not have spectra available post 100 days which limits any analysis of this phenomenon for a large enough sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='14 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='O I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='He I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='Ca II IR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='[Fe II/III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='[Fe III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='Late time spectra comparison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='Velocity km s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='Normalized flux + Constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 8000 4000 0 4000 8000 Velocity km s 1 0 1 2 3 4 5 6 7 8 Normalized flux + Constant 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 d SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Top left: Early-time spectra of BTS SNe Ia-CSM with phases between 0 and 30 days since explosion compared to spectra of SNe 2011jb, 2005gj, 1991T and PTF11kx (phases in days since discovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Top right: Late-time spectra of BTS SNe Ia-CSM (phases ranging from 40 to 370 days since explosion) compared to spectra of SNe 2011jb, 2005gj, 2010jl and PTF11kx (phases in days since discovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bottom left and right: Hα line profiles (post continuum removal) with the blue side reflected across the peak flux, marked by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe 2020aekp, 2020abfe, 2020xtg and 2020uem in the right panel, and SNe 2018crl, 2018gkx, 2018evt, 2019agi, 2019ink, 2019rvb, 2020onv, 2020qxz in left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 15 The spectra were reduced and processed as outlined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 for the emission line analysis, the results of which are de- scribed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We used only good SNR SEDM spectra and intermediate resolution spectra for line identifi- cation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Hα, Hβ and He I emission lines To analyze the Hα line emission, we first fit the con- tinuum level using the fit continuum function of the specutils Python package, where the continuum is es- timated by a cubic function fitted on regions on each side of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We remove this continuum level and then fit the Hα line with a broad and a narrow component Gaussian func- tion using the fit lines function of specutils which returns the best fit Gaussian model and the 1σ uncertainty on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We generate 1000 sample mod- els within 1σ uncertainties of the parameters centered around the best-fit values and calculate the intensity, flux and veloc- ity (FWHM) of the broad and narrow components for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Then we take the median and standard deviation of the intensity, flux and velocity FWHM distributions to get their final best value and 1σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The equivalent width was also calculated for the Hα line using the model fit as well as directly from the data, and the difference be- tween the values derived from model and data is reported as the error on the EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' All values are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For 3 SNe in our sample, we have a series of intermediate resolu- tion spectra through which we can trace the evolution of the Hα line with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 8 shows this trend of the Hα line parameters (integrated flux in the top panel and equivalent width in the bottom panel) versus phase for all SNe in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The un-filled markers represent the narrow emis- sion while the filled markers represent the broad emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SNe where this analysis could be done on multiple spec- tra, we see that the Hα equivalent width generally increase over time, with some SNe showing fluctuations up to 100 days possibly due to interaction of ejecta with multiple CSM shells of varying density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For SN 2018evt, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2022) analyzed Hα line properties from a comprehensive spectral series data, which are plotted in Figure 8 in gray circles and seem to agree well with our analysis at comparable epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From the Gaussian profile line fitting analysis of the Hα emission line, we found that the broader component has ve- locities ranging from ∼1000 to ∼4000 km s−1 (intermedi- ate width) and the narrow component has velocities of about ∼200 km s−1 to ∼1000 km s−1 (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The narrow component could only be resolved down to ∼300 km s−1 limited by the mediocre resolution of the spectrographs used (KeckI/LRIS R∼800, P200/DBSP R∼1000, NOT/ALFOSC has R∼360).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' While we know that the narrow lines originate in the unshocked ionized CSM, the exact origin of the inter- mediate components is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' They could arise from the post-shock gas behind the forward shock or from the shocked dense clumps in the CSM (Chugai & Danziger 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The luminosities of the Hα line measured from the BTS SNe Ia-CSM lie in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5–37×1040 erg s−1 which are comparable to the values from S13 who reported most of their SNe in the 1–10×1040 erg s−1 range except one object that had a luminosity of 39×1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From the broad Hα luminosity, we did a simple estimate of the mass-loss rate assuming spherically symmetric CSM deposited by a station- ary wind ρ ∝ r−2 having velocity vw (Chugai 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Sala- manca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The mass-loss rate ˙M can be related to the broad Hα luminosity LBroad Hα as (Salamanca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1998, their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2) LBroad Hα = 1 4ϵHα ˙M vw v3 s where vs is the shock velocity (obtained from the broad component velocity of the Hα line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We used a value of 100 km s−1 considering previous high resolution spectral studies of SNe Ia-CSM (Kotak & Meikle 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Aldering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012) for vw as we cannot fully resolve the narrow component and a maximum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 for the efficiency factor ϵHα (Salamanca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The mass-loss rates were estimated from the available spectra and are shown in Figure 10 as a function of years before explosion (tw = vst vw , where t is the phase of the spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For most SNe in the sample, the mass-loss rates lie be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='001–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02 M⊙ yr−1, except for SN 2019rvb which has ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07 M⊙ yr−1 lost within 2 years prior the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' These rates are much higher than what could be attained from a red giant superwind (∼ 3 × 10−4 M⊙ yr−1) but are comparable to previous estimates (calculated through mul- tiple methods) for SNe Ia-CSM and require some unusual mechanism to reach such persistently higher mass-loss rates in the decades prior to explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Also to consider is that the simplistic assumption of spherical symmetry likely does not apply for SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Evidence of multiple thin shells and asymmetric CSM was observed for PTF11kx (Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012) and light curve modeling of SNe 1997cy and 2002ic suggested a better fit to a flat density profile rather than sta- tionary wind (Chugai & Yungelson 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' An asymmetric or clumpy CSM might be the norm for SNe Ia-CSM (and some SNe IIn) rather than the exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The same analysis as for the Hα line was also carried out for Hβ and He I λ5876 with a one component Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For cases where a Gaussian model could not fit the data, we integrate the flux value in a 100 ˚A region centered at 5876 ˚A for He I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The Na ID absorption lines are also prevalent in some spectra and blend with the He I line, resulting in posi- tive EWs for some SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The cumulative distributions of Hβ and He I equivalent widths are shown in the top and bottom panels of Figure 11 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 16 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Summary of Hα line properties obtained from two-component Gaussian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN Name Phase Broad Flux Narrow Flux Total Flux Broad Velocity Narrow Velocity (days) (10−16 erg s−1 cm−2) (10−16 erg s−1 cm−2) (10−16 erg s−1 cm−2) FWHM (km s−1) FWHM (km s−1) SN 2018crl 92 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 4137±312 < 214 SN 2018gkx 75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 2640±398 < 375 SN 2018evt 144 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4±52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 3267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 6465±997 1816±973 SN 2019agi 42 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 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+page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 2431±217 272±214 SN 2019rvb 26 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 2321±298 374±216 SN 2020onv 38 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 2714±879 <834 SN 2020qxz 26 76.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 2049±284 245±215 SN 2020qxz 45 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1±0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 5013±395 400±375 SN 2020uem 51 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} 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52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 4242±382 1258±376 SN 2020xtg 448 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 4452±395 1566±377 SN 2020abfe 146 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 4411±389 < 376 SN 2020aekp 132 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 7728±846 < 833 SN 2020aekp 169 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 6775±839 < 834 SN 2020aekp 211 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 7422±852 1342±836 The Hβ median EW measured from the BTS SN Ia-CSM sample is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 ˚A , close to the S13 value of ∼6 ˚A and quite weak compared to what S13 measured for SNe IIn (∼13 ˚A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The overall cumulative distribution of Hβ EW is also compa- rable to the S13 SNe Ia-CSM rather than to the S13 SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For the He I λ5876 line, the median EW measured for our BTS SN Ia-CSM sample, considering only significant emis- sion features, is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 ˚A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This is close to the value of ∼2 ˚A reported in S13, and again significantly different from their SN IIn value of ∼6 ˚A (∼4 ˚A with upper limits), however the overall distribution seems to be closer to the S13 SNe IIn (but still weaker) rather than to the S13 SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This indicates that perhaps He I is not as good a discriminant be- tween the populations compared to Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Among the most He- rich SNe in our sample are SNe 2019ibk, 2020uem, 2020xtg, 2020aekp and 2018evt, and these SNe also have the higher Hα equivalent widths in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 12 plots the cumulative distribution of the Balmer decrements ( FHα FHβ ) measured for our sample SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The higher Balmer decrement values (>15) have large errors associated to them because of low SNR of the spectra from which they were derived, particularly near the Hβ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Consistent with the results of S13, the SNe Ia-CSM from this sample also have a high median Balmer decrement value of ∼7 (∼5 in S13), indicating that the emission line mechanism is prob- ably collisional excitation or self-absorption rather than re- combination, from which the expected Balmer decrement value is ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the case of SNe Ia-CSM, if the CSM distri- bution consists of multiple shells as suggested for PTF11kx, moderately high densities could be created when fast moving ejecta overtake slowly moving thin dense CSM shells creat- ing large enough optical depth in the Hα line which results in the Hβ transition decaying as Paα + Hα (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For some individual SNe where multiple spectra are avail- able, the Balmer decrement is observed to first increase and later on decrease with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Host galaxies We retrieved science-ready co-added images from the Galaxy Evolution Explorer (GALEX) general release 6/7 (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2005), the Sloan Digital Sky Survey DR 9 (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS, PS1) DR1 (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016), the Two Micron All Sky Survey (2MASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006), and preprocessed WISE im- 17 1040 1041 Line luminosity (erg s 1) SN2018evt (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=') SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp 100 200 300 400 Phase (days) 102 Equivalent Width (Å) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Integrated fluxes and equivalent widths of Hα emission line with respect to SN phases for the BTS SN Ia-CSM sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Broad component values are shown with filled markers and narrow component values with un-filled markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2018evt Hα lumi- nosities and EWs presented in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2022) are also shown in gray circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 100 200 300 400 Phase (days) 102 103 104 Line Velocity (km s 1) SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Velocity of Hα emission line with respect to SN phases for the BTS SN Ia-CSM sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Broad component values are shown with filled markers and narrow component values with un- filled markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ages (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2010b) from the unWISE archive (Lang 2014b)9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We used the software package LAMBDAR (Lambda Adaptive Multi-Band Deblending Algorithm in R) (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016) and tools presented in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2021), to measure the brightness of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The spectral energy distribution (SED) was modelled with the software 9 http://unwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='me 0 10 20 30 40 50 Years before explosion 10 3 10 2 10 1 Mass loss rate (M yr 1) SN2018crl SN2018gkx SN2018evt SN2019agi SN2019ibk SN2019rvb SN2020onv SN2020qxz SN2020uem SN2020xtg SN2020abfe SN2020aekp Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Mass-loss rates estimated from the luminosity of the broad component of Hα for the BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A value of 100 km s−1 was assumed for the wind velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' package Prospector10 (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We assumed a linear-exponential star-formation history, the Chabrier (2003) initial mass function, the Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2000) at- tenuation model, and the Byler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2017) model for the ionized gas contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The priors were set as described in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Figure 13 shows the log of star formation rate (SFR) as a function of stellar mass for hosts of BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We also use a Galaxy-zoo (Lintott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011) sample of ellip- tical and spiral galaxies (randomly sampled in the redshift range z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='015−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='05), and BTS SN Ia hosts as comparison samples collected by and used for comparison in Irani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We find the SN Ia-CSM host galaxy population to be consistent with late-type spirals and irregulars with recent star formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 4 out of 12 SNe have clearly spiral hosts, 3 have edge-on host galaxies, 4 seem to have irregu- lars as hosts and 1 has an unclear host type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Host galaxies of 10 out of 12 SNe have w2 − w3 measurements available which are all > 1 mag, putting them in late-type category (Irani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022), 1 (SN 2019rvb) does not have W3 mea- surement but has NUV − PS1r ∼ 1 mag again putting it towards late-type and 1 (SN 2020abfe) does not have any of the above information available except the PS1r band mag- nitude of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='766, which is the faintest host galaxy (absolute SDSS r-band magnitude of −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4) in our BTS SN Ia-CSM sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' As noted in S13, the SN Ia-CSM hosts of their sam- ple had generally low luminosities (−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 < Mr < −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6) except MW like spiral hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Our BTS SN Ia-CSM host lu- minosities lie in the range of −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 < Mr < −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 covering low to MW like luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Rates Following the methodology for calculating the volumetric rate of transients found in the Bright Transient Survey from Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020), we use their equation 2 to calculate the 10 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='com/bd-j/prospector version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 18 0 20 40 60 80 100 H Equivalent Width (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Cumulative Fraction of objects BTS Ia-CSM S13 Ia-CSM S13 IIn MedianBTS Ia CSM = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 MedianS13 Ia CSM = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 MedianS13 IIn = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0 10 20 30 40 50 He I 5876 Equivalent Width (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Cumulative Fraction of objects BTS Ia-CSM S13 Ia-CSM S13 IIn MedianBTS Ia CSM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 MedianS13 Ia CSM = 2 MedianS13 IIn = 4 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Cumulative distributions of equivalent width of Hβ and He I λ5876 emission lines calculated from the BTS SNe Ia-CSM (in grey) compared with the respective distributions presented in S13 for SNe Ia-CSM (blue) and SNe IIn (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Vertical dashed lines mark the median EW of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN Ia-CSM rate: R = 1 T N � i=1 1 ( 4π 3 D3 max,i)fskyfextfrecfcl,i where T is the duration of the survey, N is the number of transients that pass the quality cut, Dmax,i is the distance out to which the ith transient with peak absolute magnitude Mi 0 5 10 15 20 25 Balmer Decrement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 Cumulative Fraction of objects BTS Ia-CSM S13 Ia-CSM S13 IIn MedianBTS Ia CSM = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 MedianS13 Ia CSM = 5 MedianS13 IIn = 3 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Cumulative distribution of Hα/Hβ intensity ratio (Balmer decrement) calculated from intermediate resolution spec- tra of BTS SN Ia-CSM sample (grey shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The red line is the distribution of Balmer decrement of SNe IIn measured in S13, the blue line is the SN Ia-CSM Balmer decrement distribution from S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The black circles are a few representative points indicating the high Balmer decrement values and the uncertainties on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The vertical dashed line is the median Balmer decrement measured from BTS SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' can be detected above the survey magnitude limit mlim (=19 mag for BTS SNe Ia-CSM) at peak light without any extinc- tion, fsky is the average active survey coverage as a fraction of full sky, fext is average reduction in effective survey vol- ume due to Galactic extinction, frec is the average recovery efficiency for a detectable transient within the survey cover- age area, and fcl,i is the classification efficiency dependent on apparent magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The duration of the survey in which these 12 SNe Ia-CSM were detected is from 2018-05-01 to 2021-05-01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' T = 3 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We calculate fsky during this time period by averaging the sky area coverage of the public MSIP survey consider- ing 3 day cadence for ZTF Phase I (2018-05-01 to 2020-10- 31) and 2 day cadence for ZTF Phase II (since 2020-11-01), which turns out to be 12505 deg2 for Phase I and 14831 deg2 for Phase II, giving a mean fsky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We use the same value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='82 for fext as calculated in Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020) given there has not been any change in the number and posi- tions of ZTF fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' To estimate frec, we consider SNe Ia-CSM brighter than −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 peak absolute magnitude and brighter than 18 appar- ent magnitude (total 5) of which 4 pass the quality cut, giving an frec of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We take classification completeness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='75 at 19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 log M/M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 log(SFR (M /yr)) sSFR = 10 8 M yr 1 sSFR = 10 9 M yr 1 sSFR = 10 10 M yr 1 sSFR = 10 11 M yr 1 sSFR = 10 12 M yr 1 Galaxy zoo ellipticals Galaxy zoo spirals BTS SN Ia BTS SN Ia-CSM Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Host galaxies of BTS SN Ia-CSM (black circles) on SFR vs stellar mass plot with Galaxy-zoo spiral (blue contours) and elliptical (red contours) galaxies for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' BTS SN Ia hosts are also shown for comparison in green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Equal sSFR lines are marked with grey dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 19 mag, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='9 at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 mag and 1 at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 mag and linearly inter- polate in between these values to get fcl,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Then using H0 = 70 km s−1 Mpc−1, ignoring cosmolog- ical effects11 as in Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020) and applying a uni- form K-correction (K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5×log10(1 + z)), we get a rate of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='35+27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='53 −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='37 Gpc−3 yr−1 for SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We also calculate a SN Ia rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='25 × 104 Gpc−3 yr−1 from SNe Ia ob- served in the same period following the same method, which is close to the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='35×104 Gpc−3 yr−1 calculated in Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This puts SNe Ia-CSM to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2% of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However this rate estimate should be considered a lower limit given various caveats in the correct identification of SNe Ia-CSM (see discussion §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' If the ambiguous clas- sification cases outlined in Appendix A are considered to be SN Ia-CSM and included in the rate calculation, we obtain a rate upper limit of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='7+135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8 −77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3 Gpc−3 yr−1, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8% of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Precursor rates The ZTF precursor rates were calculated following the method in Strotjohann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2021) which studied the fre- quency of precursors in interacting SNe found in ZTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 11 Contraction of control time window approximately compensated by increase in the star-formation rate density in the low redshift regime for red- shift dependent SN rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Strotjohann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2021) included 6 of the SNe Ia-CSM presented in this paper in addition to 4 other SNe Ia-CSM not in this paper (see Appendix A for details) for their search but did not find any robust 5σ precursor detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This non-detection was concluded to be due to the small sample size of SNe Ia-CSM (or that they are more distant) compared to the SN IIn sample, so even if the precursors were as bright or frequent as for SNe IIn, it would be difficult to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The same search was here carried out for our larger sam- ple by taking the ZTF forced photometry multi-band (g, r, i) light curves generated by the pipeline outlined in Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2019) and stacking them in 1, 3 and 7-day long bins to search for faint outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There were 7389 total avail- able pre-explosion epochs for BTS SNe Ia-CSM, the earliest epoch being 1012 days prior to the explosion and the median phase 340 days prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Hence the results are valid for typical SN Ia-CSM progenitors at about ∼1 year before the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We did not find any robust 5σ precursor detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The upper limits for the precursor rates in different bands are shown in Figure 14, where the solid lines indicate up to what fraction of the time a precursor of a given brightness could have been detected while being consistent with the ZTF non-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A precursor of −15 magnitude could occur as frequently as ∼10% of the time given the ZTF non-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A continu- ous search for the precursors as more SNe Ia-CSM are found and classified and their sample size increases could yield a detection if the precursors are as frequent and bright as for SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The dense and massive CSM around these objects is close enough to have been deposited within decades prior to the SN but the lack of precursors within 1 year indicates that there is likely no violent event that ejects a lot of mass in that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Probing for precursors could potentially con- strain the progenitor in at least some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' For example, Soker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013) predicts for their core degenerate (CD) model for PTF11kx-like SNe release of significant energy (∼1049 erg) before explosion over timescale of several years, implying a precursor 3–7 magnitudes fainter than the SN ex- plosion spread over several years, peaking in the near-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fraction of SNe Ia-CSM with delayed interaction The fastest declining SNe in our sample (SNe 2018crl, 2020qxz and 2020aekp) are also the ones that develop a plateau and show relatively stronger SN Ia-like absorption features in their early spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' They seem to have a delayed start for the interaction like PTF11kx but not as fast a decline, and thus bridge the gap between PTF11kx and the rest of the strongly interacting SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It remains to be seen how many SNe Ia are weakly interacting where the CSM inter- action starts in earnest at timescales of ∼year or more after explosion, this requires searching for faint detections in care- 20 20 19 18 17 16 15 14 13 Absolute precursor magnitude 10 3 10 2 10 1 100 Fraction of time Type Ia-CSM SNe Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Precursor rate as a function of magnitude calcu- lated from BTS SN Ia-CSM pre-explosion ZTF forced photometry stacked in 7-day bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The different colored shaded regions corre- spond to different ZTF bands (r-red, g-green, i-grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The solid lines depict the upper limits on fraction of the time a precursor of the corresponding magnitude would have been detected which is consistent with the ZTF non-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' fully calibrated forced photometry light curves (stacked to go fainter), a study currently undertaken by Terwel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From the current sample, it appears that in addition to SNe Ia-CSM being intrinsically rare, delayed interaction SNe Ia-CSM are even rarer and only constitute about a quarter of all SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This delayed interaction behaviour could also be an effect of asymmetric or clumpy CSM wherein part of the SN ejecta shine through depending on the viewing an- gle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Observational campaigns that capture the inner bound- ary of the CSM and the geometry robustly could shed light on the distribution of the inner CSM radius and reveal if it is a continuous distribution or if there are multiple progenitor scenarios within the SN Ia-CSM class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Implications for progenitor based on observed mass loss From Figure 10, the estimated mass-loss rates from a sim- ple spherical treatment of the CSM and a stationary wind lie between ∼ 10−3 to 10−1 M⊙ yr−1 over a period of less than ∼ 60 years before explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' That gives a total mass loss of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 to ∼ 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2012) estimated ∼ 5 M⊙ of CSM around PTF11kx while Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2017) revised it to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='06 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Light curve modeling of SN 1997cy and SN 2002ic by Chugai & Yungelson (2004) resulted in ∼ 5 M⊙ estimates for both SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Inserra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2016) also fit analytical models to some SNe Ia-CSM and found the CSM mass to lie between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Since from Figure 5, the pseudo-bolometric luminosities of our SNe Ia-CSM lie somewhere between PTF11kx and SNe 1997cy, 2002ic and 2005gj, with SN 1999E somewhere in the middle, we can say that the total CSM mass in our sample of SN Ia-CSM should also be several solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A WD+AGB star sys- tem has typically been suggested for historical SNe Ia-CSM to explain this massive CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The WD could either gain mass through Roche Lobe overflow (RLOF) from the com- panion that drives an optically thick wind (OTW) or merge with the core of the AGB star that then explodes in or soon after the common envelope phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Meng & Podsiadlowski (2019) model WD+MS systems for their common envelope wind (CEW) model and find ∼ 1 M⊙ CSM around SNe Ia- CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Thus, given the large observed CSM mass range, the nature of the companion cannot be solely determined from total mass lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' High resolution spectroscopy that can resolve the narrow unshocked CSM wind velocity is also needed to determine the compactness of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Implications for progenitor based observed volumetric rate Robust observed rate estimates for SNe Ia-CSM have been few and far between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2010) found 1 interacting SN Ia (SN 2005gj) in a sample of 79 SNe Ia at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='15 in the SDSS-II SN survey, giving a rate of ∼1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' After the PTF11kx discovery in the Palomar Transient Factory (PTF) survey, the SN Ia-CSM rate was estimated to be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1% (1 in 1000 classified SNe Ia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012) but without spectroscopic completeness determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' S13 identified 7 more SNe Ia-CSM from the PTF SN IIn sample, bumping up the estimate to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' With this sample we have im- proved the rate estimate, providing a robust value (along with an uncertainty estimate on that value) from an unbiased sur- vey with high spectroscopic completeness up to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 mag- nitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However this rate quite possibly still underestimates the true value for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The first being possible ther- monuclear SNe that are enshrouded so completely by CSM interaction that they are misclassified as SNe IIn in the ab- sence of good early time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' In the BTS SN IIn sample, we found 6 SNe IIn to have ambiguous classifications which could possibly be SNe Ia-CSM and these are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Including these ambiguous cases in rate esti- mation results in a rate upper limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='07–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8% for strongly interacting thermonuclear SNe, while excluding them gives an underestimated rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The second issue with the rates is if there is indeed a con- tinuum of delayed interaction SNe Ia-CSM like PTF11kx, in- teraction in SNe Ia may present itself hundreds of days later at magnitudes fainter than ZTF’s limit (∼20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5) resulting in those SNe not being counted when they may be sharing the same progenitor as the rest of the interacting SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Lastly in some rare cases, the SN might appear normal in its light curve shape and duration (and thus would be missed by the selection criteria used in this paper) but seem to have pe- culiar narrow Hα in its spectrum or bright mid-IR flux (like in the case of SN 2020aaym;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Th´evenot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 21 Han & Podsiadlowski (2006) predicted a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1–1% for 02ic-like events for their delayed dynamical instability SD model but could not naturally explain the delayed interac- tion and multiple CSM shells in PTF11kx (which is relevant for some SNe in our sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A symbiotic nova-like progen- itor was suggested by Dilday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2012) for PTF11kx and they quoted the theoretical rates for the same to lie between 1–30%, however the model could not explain the massive CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013) suggested a core degenerate (CD) scenario in which the explosion is set by the violent prompt merger of the core of the giant companion on to the WD and could naturally explain the massive CSM of PTF11kx (Livio & Riess 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013) estimated the occurrence of such SNe (Mcore+ MW D ≳ 2 M⊙ and Menv ≳ 4 M⊙) through population synthesis and found it to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='002 per 1000 M⊙ stars formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Assuming ∼1–2 SNe Ia occur per 1000 M⊙ stars formed (Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012), this corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2%, which compares well with our observed rate es- timate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The CEW model by Meng & Podsiadlowski (2019) pre- dicts that the SNe Ia-CSM like objects could arise in the SD CEE scenario when CONe White Dwarfs (WD) steadily ac- crete material at the base of the CE without quickly spiral- ing in due to the driving of a CEW wind (10–100 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The WD explodes when it reaches the Chandrasekhar mass (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='38 M⊙) and could possibly explode within the CE before it is ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The CEW model predicts that 25–40% of the SNe Ia from CONe WD in Common envelope evolution with a Main Sequence (MS) companion will show SN Ia-CSM like properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Meng & Podsiadlowski (2019) also give the ratio of SNe Ia from CONe WDs to normal SNe Ia from CO WDs to be between 1/9 and 1/5 (considering normal SNe Ia only come from CO WD + MS systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Combining that with the estimate that roughly 10–20% of all SNe Ia may come from the SD scenario (Hayden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011), SNe Ia-CSM from CONe WD according to the CEW model should be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='28% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6% of all SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A spin-down be- fore explosion of the WD (Justham 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Di Stefano & Kilic 2012) could also explain the time delay between explosion and interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Soker (2022) estimated the common envelope to explo- sion delay time distribution (CEEDTD) shortly after the CEE (tCEED < 104 yr) from SN in planetary nebula rates and SN Ia-CSM observed rates to be roughly constant rather than having a t−1 dependence, that is the SN explosion could oc- cur very soon after the CEE as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Our observed rates are on the lower side compared to these theoretical model esti- mates but compare well within the observational uncertain- ties, though the CEW model seems to best account for the overall SNe Ia-CSM properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SUMMARY In this paper, we have presented optical and mid-IR pho- tometry, optical spectra and detailed analysis of 12 new SNe Ia-CSM identified in the Zwicky Transient Facility Bright Transient Survey, nearly doubling the total number of such objects discussed previously by Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The properties of the sample extracted in this paper agree very well with similar analysis conducted in S13, particularly the median EW of Hβ is found to be significantly weaker in SNe Ia-CSM compared with SNe IIn and consequently the Balmer decrements are ubiquitously higher in SNe Ia-CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The brightness of SNe Ia-CSM in mid-IR is comparable to SNe IIn and observations of reduced flux in the red side of the Hα wing together with the mid-IR brightness points to formation of new dust in the cooling post-shock gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The host galaxies of SNe Ia-CSM lie towards late-type galaxies with recent star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Unlike SNe IIn, no precursors were found within ∼1000 days before explosion for SNe Ia- CSM, which could be an observational bias (less number of SNe Ia-CSM compared to SNe IIn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We provide a robust rate estimate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2% of all SNe Ia for SNe Ia-CSM on account of the BTS survey being unbiased and spectroscop- ically highly complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The simple mass-loss rate estimates from broad Hα luminosity of ∼ 10−2 M⊙ yr−1 are similar to previous estimates from various methods and indicate several solar masses of CSM around these SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The observed rate agrees well within the observational uncertainties with the CEW model by Meng & Podsiadlowski (2019) which can also explain the interaction delay and massive CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' There are still many unanswered questions about the nature of the progenitors and if we are accurately identifying all po- tential members of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' As ZTF Phase II continues, we are identifying more and more SNe Ia-CSM (interacting with hydrogen rich and helium rich CSM) and looking further to the future, if ZTF continues for a Phase III and when LSST survey operations begins, a larger sample would further im- prove upon the observed rate calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' However, individ- ual object studies are as important and detailed spectroscopic and multi-wavelength follow-up is essential to capture the CSM configuration and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ACKNOWLEDGMENT Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palo- mar Observatory as part of the Zwicky Transient Facility project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' ZTF is supported by the National Science Founda- tion under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' AST-1440341 and AST-2034437 and a collaboration including current partners Caltech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' IPAC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the Weizmann Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the Oskar Klein Cen- ter at Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Deutsches Elektronen-Synchrotron and Humboldt Univer- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the TANGO Consortium of Taiwan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the University of Wisconsin at Milwaukee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Trinity College Dublin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Lawrence 22 Livermore National Laboratories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' University of Warwick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Ruhr University Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Northwestern Uni- versity and former partners the University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Los Alamos National Laboratories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' and Lawrence Berke- ley National Laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Operations are conducted by COO, IPAC, and UW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The ZTF forced-photometry ser- vice was funded under the Heising-Simons Foundation grant #12540303 (PI: Graham).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This work was supported by the GROWTH project (Kasliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019) funded by the Na- tional Science Foundation under PIRE Grant No 1545949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The Oskar Klein Centre was funded by the Swedish Re- search Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Partially based on observations made with the Nordic Optical Telescope, operated by the Nordic Op- tical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the In- stituto de Astrofisica de Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Some of the data pre- sented here were obtained with ALFOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Some of the data presented herein were obtained at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Keck Observa- tory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' the observatory was made possi- ble by the generous financial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The SED Machine is based upon work sup- ported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1106171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This work has made use of data from the Aster- oid Terrestrial-impact Last Alert System (ATLAS) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The Asteroid Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to search for near earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' byproducts of the NEO search include images and catalogs from the survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The ATLAS sci- ence products have been made possible through the contri- butions of the University of Hawaii Institute for Astronomy, the Queen’s University Belfast, the Space Telescope Sci- ence Institute, the South African Astronomical Observatory, and The Millennium Institute of Astrophysics (MAS), Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This research has made use of the NASA/IPAC Infrared Sci- ence Archive, which is funded by the National Aeronautics and Space Administration and operated by the California In- stitute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores Univer- sity in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial sup- port from the UK Science and Technology Facilities Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Sharma thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant #1829740, the Brinson Foundation, and the Moore Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' her participation in the program has benefited this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Schulze acknowledges support from the G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='T re- search environment, funded by Vetenskapsr˚adet, the Swedish Research Council, project number 2016-06012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' This work has been supported by the research project grant “Understanding the Dynamic Universe” funded by the Knut and Alice Wallenberg Foundation under Dnr KAW 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0067, The research of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Yang is supported through a Bengier- Winslow-Robertson Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Fritz (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Duev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019) and GROWTH marshal (Kasliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019) (dynamic collab- orative platforms for time-domain astronomy) were used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Software: LAMBDAR (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016), Prospector (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021), pySEDM (Rigault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019), IRAF (Tody 1986, 1993), pyNOT (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='com/jkrogager/ PyNOT), LPipe (Perley 2019), pypeit (Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2020), extinction (Barbary 2016), pyraf-dbsp (Bellm & Sesar 2016), FPipe (Fremling et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2011, ApJ, 744, L17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1088/2041-8205/744/2/l17 Brown, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Dawson, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Moon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2006, PASP, 118, 1396 Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2003, PASP, 115, 763 Chambers, K.' metadata={'source': 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021, MNRAS, 507, 4367 De, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Hankins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Kasliwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1993, AJ, 106, 1101 F¨orster, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Cabrera-Vives, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Castillo-Navarrete, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021, AJ, 161, 242 Fox, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Silverman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2015, MNRAS, 447, 772 Fremling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3847/1538-4357/ab8943 Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', & Leonard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2009, Nature, 458, 865 Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Yaron, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Pastorello, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2007, ApJ, 656, 372 Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Bruch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Schulze, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2022, Nature, 601, 201 Germany, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019, PASP, 131, 078001 Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Harris, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Fox, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', van Santen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019, A&A, 631, A147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Oliva, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', & Randich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 1992, ApJ, 386, 181 Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Baade, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Hoeflich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2019, ApJ, 886, 152 Yaron, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', & Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2012, PASP, 124, 668 Zackay, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', Ofek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=', & Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2016, ApJ, 830, 27 26 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' AMBIGUOUS SN IA-CSM/IIN IN BTS To identify potential SNe Ia-CSM hiding in the SN IIn sample classified by BTS, we rechecked all SNe IIn classifications (total 142) using SuperNova IDentification (SNID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Blondin & Tonry 2007) software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNe IIn spectra were processed through SNID, and any SN having ≥ 3 matches to a SN Ia-CSM in the top 10 matches were manually checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The SNe having ambiguous classifications are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2019smj Discovered by ZTF and reported to TNS by ALeRCE (F¨orster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' 2021) on 2019-10-13 11:28:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2019smj (ZTF19aceqlxc) was classified as a Type IIn by BTS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1 in r band (∼ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1) and then developed a weaker but broader bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The spectra showed very weak Hβ, barely any He I λ5876, no O I λ7774 or [O I] lines but showed some iron group lines, Ca NIR emission and [Ca II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNID best matches were to SNe 1997cy and 2005gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The early spectra from P60/SEDM have some matches to SN 2005gj but are too noisy and of ultra-low resolution to conclusively provide a Ia-CSM classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From these observations, SN 2019smj is most likely a Type Ia-CSM but given the lack of confirmation we have excluded it from the main sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2018dfa Discovered and reported to TNS by ATLAS on 2018-07-05 08:51:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2018dfa was classified initially as a Type IIP by BTS but later spectra revealed it to be a Type IIn at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 in r band (−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2) and showed a minor bump before main peak in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The spectra showed weak Hβ and He I λ5876, no O I λ7774 or [O I] lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNID best matches were to SNe 2002ic and 2005gj along with SNe Ia-norm/91T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The earliest spectra with good SNR from P200/DBSP had one match to SN 2005gj but could not provide a robust Ia-CSM classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From these observations, SN 2018dfa is most likely a Type Ia-CSM but given the lack of confirmation we have excluded it from the main sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2019vpk Discovered by ZTF and reported to TNS by ALeRCE on 2019-11-25 06:33:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2019vpk was classified as a Type IIn by BTS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude of ∼ 18 in r band (∼ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The early spectra were too noisy and the only spectrum with good SNR was obtained with P200/DBSP nearly 6 weeks after discovery which showed weak Hβ, no clear He I emission but possibly Si II λ5958 emission (which is unlike any other SN Ia-CSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNID top matches were to SN 2005gj but visually did not look entirely convincing, and some matches were also to Type IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' We conclude SN 2019vpk does not have enough data for a robust Ia-CSM classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2019wma Discovered by ZTF and reported to TNS by ALeRCE on 2019-12-13 13:35:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2019wma was classified as a Type IIn by BTS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude of ∼ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5 in r band (∼ −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' The spectra obtained were either from P60/SEDM or LT/SPRAT hence of low resolution and showed weak Hβ and He I emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SNID top matches to earliest SEDM spectrum were to SN 2005gj at the correct redshift but given the lack of intermediate resolution spectra and absence of late time follow-up we did not assign a Type Ia-CSM classification to SN 2019wma and excluded it from the main sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2019kep Discovered and reported to TNS by ATLAS on 2019-07-02 14:13:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2019kep was classified as a Type IIn by BTS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='02388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='2 in r band (−17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Most early spectra were too noisy for classification but matched to SN 2005gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A good SNR P200/DBSP spectrum showed narrow P-Cygni Hα with absorption minimum at ∼ 2500 km s−1 but overall matched to a Type II SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' From these observations, we could not determine a robust classification for SN 2019kep and excluded it from the main sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' SN 2018ctj Discovered and reported to TNS by ZTF on 2018-04-21 08:36:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='000, SN 2018ctj was classified as a Type IIn by BTS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='0378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' It peaked at apparent magnitude of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='4 in r band (−17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content='8) and was also detected in unWISE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Only one P60/SEDM spectrum was obtained that matched well to SNe 1997cy and 2005gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} +page_content=' Given the lack of intermediate resolution spectra this SN remains classified as Type IIn and excluded from the main sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfowri/content/2301.04637v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..3e9829c53ea4be5325d340586dbef5a61a15b9e6 --- /dev/null +++ b/69E2T4oBgHgl3EQfkweU/content/tmp_files/2301.03982v1.pdf.txt @@ -0,0 +1,2383 @@ +Exploring the Use of WebAssembly in HPC +Mohak Chadha, Nils Krueger, Jophin John, +Anshul Jindal, Michael Gerndt +{firstname.lastname}@tum.de +Chair of Computer Architecture and Parallel Systems, +Technische Universität München, Germany +Shajulin Benedict +shajulin@iiitkottayam.ac.in +Department of Computer Science and Engg., Indian +Institute of Information Technology Kottayam, Kerala +Abstract +Containerization approaches based on namespaces offered +by the Linux kernel have seen an increasing popularity in the +HPC community both as a means to isolate applications and +as a format to package and distribute them. However, their +adoption and usage in HPC systems faces several challenges. +These include difficulties in unprivileged running and build- +ing of scientific application container images directly on HPC +resources, increasing heterogeneity of HPC architectures, and +access to specialized networking libraries available only on +HPC systems. These challenges of container-based HPC appli- +cation development closely align with the several advantages +that a new universal intermediate binary format called We- +bAssembly (Wasm) has to offer. These include a lightweight +userspace isolation mechanism and portability across oper- +ating systems and processor architectures. In this paper, we +explore the usage of Wasm as a distribution format for MPI- +based HPC applications. To this end, we present MPIWasm, a +novel Wasm embedder for MPI-based HPC applications that +enables high-performance execution of Wasm code, has low- +overhead for MPI calls, and supports high-performance net- +working interconnects present on HPC systems. We evaluate +the performance and overhead of MPIWasm on a production +HPC system and AWS Graviton2 nodes using standardized +HPC benchmarks. Results from our experiments demonstrate +that MPIWasm delivers competitive native application per- +formance across all scenarios. Moreover, we observe that +Wasm binaries are 139.5x smaller on average as compared +to the statically-linked binaries for the different standardized +benchmarks. +CCS Concepts: • Software and its engineering → Process +management. +Keywords: WebAssembly, Wasmer, Wasm, MPI, HPC +Permission to make digital or hard copies of all or part of this work for +personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear +this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with +credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request +permissions from permissions@acm.org. +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +© 2023 Association for Computing Machinery. +ACM ISBN 979-8-4007-0015-6/23/02...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +ACM Reference Format: +Mohak Chadha, Nils Krueger, Jophin John, Anshul Jindal, Michael +Gerndt and Shajulin Benedict. 2023. Exploring the Use of We- +bAssembly in HPC. In The 28th ACM SIGPLAN Annual Sympo- +sium on Principles and Practice of Parallel Programming (PPoPP +’23), February 25-March 1, 2023, Montreal, QC, Canada. ACM, +New York, NY, USA, 16 pages. https://doi.org/10.1145/nnnnnnn. +nnnnnnn +1 +Introduction +Linux containers, due to their portability and high availability, +have become the de-facto standard for developing, testing, +and deploying a wide range of applications from enterprise +to web services in cloud environments [29]. This is because +containers enable users to package their application along +with its custom software dependencies as a single unit into +easy-to-deploy images. Motivated by their popularity in the +cloud, containers have also seen a growing interest in the HPC +community [30, 74, 90]. For HPC systems, containers provide +flexibility to users and allow them to define custom software +stacks, i.e., user-defined software stack (UDSS) for their large- +scale scientific applications. Moreover, they enable easy, reli- +able, and verifiable environments that can be reproduced in +the future. To this end, several HPC-focused containerization +solutions, such as Charliecloud [72], Shifter [49], Singular- +ity [58], Podman [48], and Sarus [33] have been introduced. +In contrast to previous approaches, this paper investigates us- +ing a new novel technology called WebAssembly (Wasm) [52], +dubbed as an alternative to Linux containers [81], for packag- +ing and distributing HPC applications. +Despite their increasing popularity, the adoption and us- +age of containers in HPC systems is still significantly lim- +ited [36]. This can be attributed to the several challenges +commonly faced by users in running and building container +images for their applications on HPC systems. For execut- +ing containers, most containerization solutions require root +privileges which are not possible for normal HPC users due +to shared filesystems and their UNIX permissions in HPC. +While HPC-focused containerization solutions such as Sin- +gularity [58] and Podman [48] support rootless-containers +through fakeroot [61], their current implementations do +not support distributed filesystems such as GPFS commonly +found on HPC systems [70, 80]. Moreover, as argued by [71], +building Open Container Initiative (OCI) [69] compliant con- +tainer images on HPC resources by unprivileged (normal) +arXiv:2301.03982v1 [cs.DC] 10 Jan 2023 + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +users where the applications will eventually run is signifi- +cantly hard and requires support from the supercomputing +center. This is because most container building solutions such +as Docker [43] also require root privileges. As a result, most +users use their own local systems for building/developing their +application container images and then transfer the built image +to a login/front-end node of an HPC system for execution. +However, this scenario leads to several problems in container- +based HPC application development. First, HPC nodes are +becoming more heterogeneous [11] with different processor +architectures such as x86_64 or aarch64 and have special- +ized accelerators such as GPUs. As application performance +is critical in HPC, compiling an application using the specific +microarchitectural features of a particular processor is signifi- +cantly important. While building container images for mul- +tiple platforms either by cross-compiling HPC applications +or by emulation with QEMU is possible with plugins such as +build-x [44], it is not widely supported by HPC application +build procedures and requires the presence of specific Linux +kernel features (binfmt_misc [62]). Moreover, testing and +developing HPC applications offers insights only on the target +system. In addition, most container images can range from +several MiBs to several GiBs. As a result, frequent network +transfers from the local to the HPC system can be cumber- +some. Second, building HPC applications requires access to +specialized networking libraries and licenses to compilers +that are not available on the local user systems. Finally, while +different containerization solutions have almost no impact on +the performance of the containerized application [72, 75, 84], +building high-performant HPC application container images +is non-trivial, involves a steep learning curve, and requires +knowledge about specific MPI library versions (e.g., Open- +MPI [10] 4.0) and high performance network interconnect +hardware (e.g., Intel OmniPath [4]) and libraries (e.g., Intel +Performance Scaled Messaging [5]) present on the target sys- +tem. These challenges of container-based HPC application +development closely align with the several advantages and +core problems that Wasm [52] aims to solve. +Wasm is a low-level, statically typed universal binary in- +struction format for memory-safe, sandboxed execution in a +virtual machine. It offers portability across modern proces- +sor architectures and operating systems, fast execution, and +a low-level memory model [52]. Although originally meant +for execution in Web browsers, due to its simplicity and gen- +erality, Wasm has seen widespread adoption and usage in +non-Web domains such as serverless computing [78], edge +computing [47, 53], and Internet of Things [51]. It does not +require garbage collection and is designed to be a universal +compilation target with mature support for programming lan- +guages with an LLVM [59] front-end such as C, C++, C#, +and Rust [32, 35, 40, 45, 77]. +Figure 1 demonstrates a general workflow for using Wasm +in HPC. Developers can compile their HPC applications to +Wasm once on their local systems ahead-of-time (AoT) and +HPC Application +WebAssembly +x86_64 +aarch64 +WebAssembly embedder +Compile +Figure 1. An HPC application can be compiled to WebAssem- +bly and distributed to multiple platforms where it can be +executed efficiently by a supporting WebAssembly embedder. +distribute it across multiple platforms instead of distribut- +ing source code or building application containers. Typically, +Wasm binaries have a smaller size as compared to native +x86_64 binaries [52, 56, 91]. Following this, the resulting +binary can be executed on any platform using a standalone +Wasm embedder [52]. The Wasm embedder serves two major +purposes. First, it provides an isolated execution environ- +ment for running a Wasm binary on a platform. In contrast to +container-based approaches that utilize different Linux names- +paces [71] for isolation and security, Wasm provides light- +weight isolation at the application level based on software +fault isolation (SFI) [85] and control flow integrity (§2.2). +Second, it is responsible for compiling Wasm binaries to +native machine code, either by using Just-in-Time (JIT) en- +gines at the time of execution, or AoT by using the same +JIT engines or AoT compilers. Note that, Wasm binaries +can be executed by normal users and are completely unpriv- +ileged. Several open-source standalone embedders such as +Wasmer [87], Wasmtime [37], and Wasm3 [79] are currently +available. However, none of them support the execution of +HPC applications. +As the first step towards bringing Wasm to the HPC ecosys- +tem, we only focus on MPI-based [6] HPC applications in +this paper. We chose MPI due to its understanding and in- +fluence in the HPC community [34]. Towards this, our key +contributions are: +• We implement and present MPIWasm, a novel Wasm +embedder for MPI-based HPC applications based on +Wasmer [87]. MPIWasm enables high performance exe- +cution of Wasm code, has low-overhead for MPI calls +through zero-copy memory operations, and supports +high-performance networking interconnects such as +Intel OmniPath [4]. +• We demonstrate with extensive experiments the low- +overhead and performance of MPIWasm using standard- +ized HPC benchmarks on a production HPC system and +AWS Graviton2 [1] nodes based on the x86_64 and the +aarch64 architectures respectively. +• We elaborate on the different possible future directions +for using Wasm in the HPC ecosystem. + +WA口口Exploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +The rest of this paper is structured as follows. §2 provides a +detailed overview on Wasm. In §3, we describe our embedder +MPIWasm in detail. Our experimental results are presented +in §4. In §5, we describe the different possible directions +for using Wasm in the HPC ecosystem. §6 describes some +previous approaches related to our work. In §7, we conclude +the paper and present an outlook. +2 +A primer on WebAssembly +2.1 +WebAssembly Overview +WebAssembly (Wasm) was introduced in 2015 as an alter- +native to JavaScript for web-browser based applications. It +superseded asm.js [67], a previous attempt by Mozilla which +focused on a subset of Javascript code that can be optimized +AoT. +When an application is compiled to Wasm, the resulting +binary is called a module. Wasm modules contain function +definitions, declarations of global variables, tables, and a lin- +ear memory address space. All of the application code in +Wasm is organized in functions. The conceptual machine in +Wasm is stack-based and does not contain registers, there- +fore all instructions pop their operands from the stack of +the machine. However, since application control flow is an +explicit part of the module and Wasm operations are typed, +it is possible to statically predict the layout of the stack at +any point in the program which allows compilers to trans- +late the stack semantics to a register-based instruction set. +Similar to other higher-level programming languages, Wasm +allows the definition of global variables that are not scoped +to a specific function or block. Tables in Wasm modules are +used for storing references to functions [52]. The Wasm ISA +currently supports only four data types for variables: (i) i32, +32-bit integers, (ii) i64, 64-bit integers, (iii) f32, 32-bit IEEE +754 floating point numbers, and (iv) f64 64-bit IEEE 754 +floating point numbers. For constructing, complex types a +combination of these basic types is commonly used. +Wasm provides the capability for data and code to be shared +between the module and its embedder using the import/export +system. All of the function definitions that can occur in a +Wasm module can be imported from the embedder instead of +being defined within it. Similarly, function definitions that are +present in the module can be exported so that the embedder +can utilize them (§2.3). +2.2 +WebAssembly Security and Sandboxing Model +Wasm utilizes software fault isolation techniques (SFI) [85] +to sandbox the executing Wasm module. By default, a Wasm +module cannot interact with the host system or perform I/O +operations of any kind. Any system interaction that is to be +initiated by the Wasm module’s code must be done through +the functions imported from the embedder (§2.1). As a result, +the embedder can act both as a translation layer and as an +arbiter to enforce isolation requirements. As a translator, it +1 +(type (;1;) (func (param i32) (result i32))) +2 +... +3 +(type (;5;) (func (param i32 i32) (result i32))) +4 +... +5 +(type (;14;) (func (param i32 i32 i32 i32 i32 i32) +6 +(result i32))) +7 +(type (;15;) (func (param i32 i32 i32 i32) (result i32 ))) +8 +... +9 +(import "wasi_snapshot_preview1" "path_open" +10 +(func $__wasi_path_open (type 22))) +11 +(import "wasi_snapshot_preview1" "fd_close" +12 +(func $__wasi_fd_close (type 1))) +13 +(import "wasi_snapshot_preview1" "fd_seek" +14 +(func $__wasi_fd_seek (type 23))) +15 +(import "wasi_snapshot_preview1" "fd_read" +16 +(func $__wasi_fd_read (type 15))) +17 +(import "wasi_snapshot_preview1" "proc_exit" +18 +(func $__wasi_proc_exit (type 0))) +19 +... +20 +(export "_start" (func $_start )) +21 +(export "memory" (memory 0)) +Listing 1. Example representation of a compiled C++ appli- +cation’s Wasm module using the WASI-SDK in WebAssembly +text format (WAT) [68]. Ellipses signify sections that are +omitted for brevity. +is possible for the embedder to provide a common interface +to the Wasm module even though the underlying system may +have different native interfaces, while as an arbiter it is possi- +ble for the embedder to restrict access of the Wasm module +to system resources based on an application-level security +policy. For instance, it is possible for the embedder to allow +file I/O only to files that reside in a specific directory to iso- +late the Wasm module from the rest of the filesystem. While +in principle similar to kernel-level system call filtering tech- +niques such as Seccomp-BPF [83] on Linux, performing such +filtering on the application level allows to define semantically +more meaningful policies. +In Wasm, all memory access is confined to a module’s lin- +ear memory which is separate from the code space. Currently, +the Wasm specification [52] supports 32-bit addresses to in- +dex the memory that a module has access to. While this limits +a single module’s memory to 4GiB, it also enables hardware +accelerated bound checks of memory accesses at runtime [42]. +If an embedder is a process with a 64-bit memory address +space, it can safely execute an untrusted Wasm module in its +memory space without requiring additional isolation by re- +serving a continuous range of virtual memory for the module +to use. Not all pages in this range need to be mapped to phys- +ical memory, it is sufficient to only map the required number +of pages to fit the amount of memory used by the module at +a given point in time. This ensures that a Wasm module can +only operate in its own execution environment and cannot +corrupt the memory of the embedder, since any out-of-bounds +memory access will result in a page fault which can then be +handled by it. Moreover, since the memory instructions in +Wasm’s specification [52] work with offsets, it is not possible +to read and write to arbitrary memory locations in Wasm. +In the assembly produced by C programs, where a func- +tion call is expressed as a jump instruction to the address of +the function’s first instruction, a typical exploit is to change +this address to take control of the program’s control flow. + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +1 +typedef int MPI_Comm; +2 +typedef int MPI_Datatype; +3 +... +4 +int MPI_Init(int* argc , char *** argv); +5 +int MPI_Finalize(void); +6 +int MPI_Send( +7 +const void* buf , int count , MPI_Datatype datatype , +8 +int dest , int tag , MPI_Comm comm +9 +); +10 +int MPI_Recv( +11 +void* buf , int count , MPI_Datatype datatype , +12 +int source , int tag , MPI_Comm comm , MPI_Status* status +13 +); +Listing 2. Excerpt of the custom MPIWasm mpi.h header +file. +However, such exploits are not possible with Wasm since it +features control flow integrity by enforcing structured pro- +gram control flow. This is because of two reasons. First, in +Wasm, a function is represented as an index in a table (§2.1) +which adds an additional level of indirection to express the +function address. Second, the Wasm specification prevents +constructing arbitrary memory addresses [42] and the sepa- +ration of the embedder and the module’s memory prevents +overwriting function instructions. +2.3 +WebAssembly System Interface +Since Wasm was originally designed for web browsers, a sys- +tem interface that targets POSIX environments and enables +execution of Wasm modules on them was not part of the orig- +inal specification [52]. To overcome this, the WebAssembly +System Interface (WASI) specification [89] was designed. +WASI specifies the interface an embedder needs to implement +to execute most POSIX applications. Embedders that imple- +ment the WASI specification will be able to run any generic +application compiled with the WASI-SDK [28]. The WASI-SDK +includes the clang compiler and its own C library based on +musl libc that call WASI systemcalls imported from the em- +bedder instead of relying on Linux systemcalls [22]. Note that, +due to the ubiquity of glibc [26] on Linux systems, some ap- +plications have come to depend on glibc-specific functions or +behavior. Such applications will require modifications before +they can be compiled to a WASI-compliant Wasm module. +Listing 1 shows a compiled Wasm module of a C++ appli- +cation using the WASI-SDK in the WebAssembly text format +(WAT). WAT is a human readable format that enables de- +velopers to examine the source code of a Wasm module. It +can be observed that the module contains several functions +with integers as parameter and return types (Lines 1-7) (§2.1), +imports WASI functions (Lines 9-18), and exports its _start +(main function) and memory (Lines 20-21). Exporting these +two definitions allows the embedder that executes this module +to call its entrypoint function and to read from and write to +the module’s linear memory. While the import statements on +Lines 9-16 enable the Wasm module to open and read from +a file, the function proc_exit is used by the embedder to +handle the termination of the application, e.g., by deallocat- +ing the memory reserved for the module. For the module to +1 +(import "env" "MPI_Init" ( +2 +func $MPI_Init (param i32 i32) (result i32) +3 +)) +4 +(import "env" "MPI_Finalize" (func $MPI_Finalize (result i32 ))) +5 +(import "env" "MPI_Send" ( +6 +func $MPI_Send (param i32 i32 i32 i32 i32 i32) (result i32) +7 +)) +8 +(import "env" "MPI_Recv" ( +9 +func $MPI_Recv (param i32 i32 i32 i32 i32 i32 i32) +10 +(result i32) +11 +)) +Listing 3. WAT representation of module imports that corre- +spond to the functions shown in Listing 2. +execute, the imported functions need to be implemented by +the embedder. +3 +MPIWasm +In this section, we describe MPIWasm, our embedder for +executing Wasm modules that utilize functions from the MPI +standard in detail. +3.1 +Overview +The purpose of MPIWasm is to support the execution of MPI +applications compiled to Wasm on HPC systems. To facili- +tate its adoption and suitability in HPC environments, it (i) +supports high-performance execution of MPI-based HPC ap- +plications compiled to Wasm (§3.3), (ii) has low-overhead +for MPI calls through zero-copy memory operations (§3.6), +and (iii) supports high-performance interconnects such as +Infiniband [64] and Intel OmniPath [4]. These network inter- +connects are utilized by MPI libraries on HPC systems for +high-performance inter-rank communication. To enable the +immediate support for network interconnects present on mod- +ern HPC systems, MPIWasm links against the MPI library on +the target HPC system at runtime and provides a translation +layer between the Wasm module and the host1 MPI library. +As a result, the developer doesn’t need to be aware about +the particular networking libraries or network interconnects +present on the target HPC system. Depending on the partic- +ular host MPI library such as OpenMPI [10] or MPICH [8], +MPIWasm needs to be built separately. Both of these libraries +are currently supported by MPIWasm. +Our embedder currently supports the execution of MPI +applications written in C/C++ and conforming to the MPI-2.2 +standard [65]. Integrating the support for MPI-3.1 [66] is of +our interest for the future but is out of scope for this work. +We chose to focus on C/C++ applications due to the stability +and maturity of the Wasm backend in the LLVM/Clang [59] +project since llvm-8. As the base for MPIWasm, we use the +open-source Wasm embedder called Wasmer [87]. Wasmer +supports the execution of Wasm modules on three major plat- +forms, i.e., Linux, Windows, and macOS, and supports both +x86_64 and aarch64 instruction set architectures. Moreover, +it implements the WASI specification (§2.3) and provides +1We use the term target and host interchangeably for the system on which +the Wasm module is executing. + +Exploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +ergonomic mechanisms to define additional functions that +are provided to the module. This dynamic extension of the +embedder’s functionality enables the addition of MPI func- +tions to the functionality it provides to the Wasm module. For +implementing MPIWasm, we use the Rust programming lan- +guage. This is because of two reasons. First, it provides high +performance comparable to C/C++ with memory-safety [82]. +Second, it has extensive support and documentation for em- +bedding Wasmer and using it as a library. +3.2 +Compiling C/C++ MPI applications to Wasm +Most MPI applications expect POSIX functionality to be +available in their execution environment, for instance the abil- +ity to read from and write to file descriptors. WASI (§2.3) +defines the WebAssembly exports that enable Wasm mod- +ules that target it to call most of the functions defined in the +C standard libraries shipped on POSIX systems. Towards +this, the WASI-SDK [28] combines the clang compiler and +the wasi-libc C library to enable the compilation of C/C++ +applications that only make use of POSIX functions and no +additional libraries to Wasm. The compilation of C/C++ MPI +applications is not supported by the stock WASI-SDK. To this +end, we implement a custom mpi.h MPI header file and add +it to the WASI-SDK. The header file includes the definitions +for the different MPI types such as MPI_Op, MPI_Comm, and +MPI_Datatype and the definition for the MPI_Status struc- +ture. Moreover, it defines the signatures for the MPI functions +according to the MPI-2.2 [65] standard. An excerpt from the +header file is shown in Listing 2. It is a reduced version of +a traditional header file found with MPI libraries with most +types defined as integers (§3.6). By combining our header file +with the WASI-SDK, a C/C++ MPI application conforming to +the MPI-2.2 standard can be compiled to Wasm. Moreover, +to facilitate the ease-of-use and enable adoption, we imple- +ment a custom python-based tool that simplifies the entire +compilation process for MPI applications. Listing 3 shows +the different MPI-specific imports present in a Wasm module +corresponding to the functions shown in Listing 2. MPIWasm +provides definitions for these imports to enable the execution +of MPI-based HPC applications. In addition, it supports the +WASI specification which enables the POSIX functionality +for MPI applications. +3.3 +Executing Wasm Code with High Performance +There exist several strategies for executing Wasm modules. +These include using an interpreter [79], Ahead-of-Time (AoT) +compilation [37], or Just-in-Time (JIT) compilation [37]. +However, for HPC systems the most useful approach is trans- +lating the Wasm instructions (Wasm ISA) to the native instruc- +tion set of the host machine before the application is executed, +i.e., AoT. Towards this, MPIwasm builds on the code genera- +tion infrastructure provided by Wasmer [87]. Wasmer currently +supports three compiler backends, i.e., Singlepass [86], +Cranelift [3], and LLVM [59]. The SinglePass compiler +Table 1. Comparing compile duration and performance for +the different compiler backends supported by Wasmer [87] +for the HPCG [73] Wasm module. The Wasm module was +generated using our WASI-SDK (§3.3). The Wasm module is +executed using MPIWasm on an x86_64 system. +Compiler +Compile Duration (ms) +Single-Core Performance (GFLOP/s) +Singlepass [86] +52 +0.3769 +Cranelift [3] +150 +1.3240 +LLVM [59] +2811 +1.5426 +is designed to emit machine code in linear time and does +not perform many code optimizations. The Cranelift com- +piler is completely based on Rust and is similar to LLVM. +With Cranelift, the WASM instructions are first translated +to the intermediate representation (IR) of Cranelift, i.e., +(Cranelift-IR) which are then translated to the native in- +struction set of the host machine by taking microarchitecture- +specific optimizations into account. On the other hand, with +LLVM the Wasm ISA is first translated to LLVM-IR followed +by the generation of native machine code. Cranelift-IR is +similar to LLVM-IR but at a lower level of abstraction which +hinders mid-level code optimizations2. At the end of the com- +pilation process, all three compilers produce a shared ob- +ject, which can be loaded with a fast dlopen call using the +libloading library [27]. +Table 1 shows a comparison of the compile-time and run- +time performance of the three different compilers supported +by Wasmer for the HPCG benchmark. While LLVM is the slow- +est to compile the Wasm module, it also results in the fastest +runtime performance for the HPCG application. As a result, +we chose LLVM as the compiler backend in MPIWasm. To +offset the longer compilation times required by LLVM as com- +pared to the other two compilers, we implement a caching +mechanism for the generated machine code. Our caching +mechanism builds on the FileSystemCache [46] provided +by Wasmer. In our implementation, we generate a hash for +each Wasm module using the Blake-3 hash function [2]. +Moreover, we store the generated shared object from LLVM +as the generated hash in the local filesystem. As a result, any +changes to the Wasm module lead to the generation of a new +hash which triggers the recompilation of the module. To this +end, repeated execution of the same application on a system +with MPIWasm will not lead to recompilation overhead for +execution. +3.4 +Filesystem Isolation with MPIWasm +Since in Wasm all system interactions by the application have +to be performed by calling functions implemented by the +embedder (§2.2,§2.3), it enables the embedder to place addi- +tional restrictions on their use and to employ checks on the +arguments supplied to them. In Wasmer, all exported functions +2A more detailed discussion between Cranelift-IR and LLVM-IR can be +found here [25]. + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +Figure 2. Memory address space of MPIWasm with an instan- +tiated Wasm module. All memory access instructions to the +Wasm module’s linear address space are given offsets relative +to the base address. +that handle file I/O perform their own permission handling +that is separate from the one employed by the OS. This in- +process indirection of filesystem accesses allows Wasmer to +present a virtual directory tree to the Wasm module that only +contains directories that the module is allowed to access. In +addition, access rights to individual directories can be more +restrictive than the permissions granted to the user that is exe- +cuting the embedder. For instance, a user can have read and +write access to their home directory and all of its subdirecto- +ries, but grant read-only access to one specific subdirectory +to a Wasm module executed by the embedder. MPIWasm ex- +poses this isolation functionality with its -d flag that grants +read-write access to the given directory to the Wasm module. +Note that the full absolute path to the exposed directories +is not presented to the Wasm module. In the virtual direc- +tory tree presented to it, all of the subdirectories it has been +given access to are direct children of the root directory. This +approach to mapping directory paths avoids exposure of in- +formation contained in the full path to the directories, such as +a username in the case of a home directory. +3.5 +Translating from Wasm to Host Memory Address +A major part of the Wasm security model is the separation +of the host and the module’s linear memory address space +(§2.2). Since it is the responsibility of the Wasm embedder to +uphold capability restrictions, protecting it’s data structures +from unintended or malicious access by the modules’ code is +significantly important. However, this separation presents a +challenge for supporting MPI applications, because the MPI +API is based on the library being able to read and write di- +rectly to the memory of the application. The executing Wasm- +based MPI application can only provide memory addresses +in its own linear memory address space, while the target MPI +library requires addresses in the host memory address space. +For executing Wasm modules, MPIWasm reserves a part +of its own address space for use by the Wasm module. As a +result, every byte contained in this range can be addressed +either with a memory address in the module’s memory space +or with a memory address in the embedder’s (host’s) memory +space. Moreover, while instantiating the module’s linear mem- +ory, MPIWasm records its base address. Following this, it is +possible to convert an address from the linear address space of +the Wasm module to the embedder’s address space and vice- +versa by treating the address in the linear address space as +an offset relative to the module’s base address. This is shown +in Figure 2. In particular, MPIWasm directly converts 32-bit +Wasm pointers that refer to the module’s linear address space +to 64-bit pointers that refer to the embedder’s address space +and vice-versa. To this end, MPIWasm directly utilizes the +MPI library present on the host system without copying any +data from the module’s address space to a different location, +i.e., it supports zero-copy memory operations. +Our mechanism for memory address translation does not +violate memory-safety because: (i) a malicious Wasm module +cannot violate control flow integrity (§2.2) and (ii) since the +size of the linear memory is always known, MPIWasm can +perform runtime bound checks for all memory accesses. As a +result, a module cannot access the memory of the embedder +or the memory of the underlying operating system unless +explicitly given access to it. +3.6 +Translating MPI Datatypes +MPI is implemented as a library with the most common being +OpenMPI [10], MPICH [8], and MVAPICH [9]. Hence, it +does not guarantee an Application Binary Interface (ABI) +and interoperability between libraries. This means that chang- +ing the MPI implementation requires recompilation of the +entire application code. One of the reasons for ABI incom- +patibility is that the MPI standard does not specify explicit +types for its datatypes such as MPI_Op and their implemen- +tation is completely up to the MPI library. However, since +Wasm modules are designed to be portable not just between +the different MPI libraries but also between different CPU +architectures, it becomes necessary to add an abstraction be- +tween the datatypes used by the host’s MPI library and the +datatypes exposed to the Wasm module by MPIWasm. An +abstraction is possible since most MPI datatypes such as +MPI_Comm, MPI_Datatype and MPI_Op are opaque to the ap- +plication and only used as arguments to MPI functions. MPI- +Wasm defines most MPI datatypes as 32-bit integers from +the perspective of the Wasm module (Listing 2) and transpar- +ently translates these datatypes to the host equivalents (§3.7). +We use integers as datatypes since MPIWasm internally uses +IDs to identify data structures that it creates on behalf of the +module in order to communicate with the host MPI library. +3.7 +Implementing MPI Functions in MPIWasm +Wasm imports are referred to by namespace and name of +the definition to import. By default, any symbols that are +not defined while compiling C/C++ applications to Wasm +will be resolved by making them imports of the module in +the env namespace. This is also demonstrated in Listing 3 +with the function imports related to the MPI standard. MPI- +Wasm provides definitions for all these functions with the +same name as the original MPI function and exports them +in the env namespace. For implementing these functions, we + +0x0 +OxFFFF_FFFFExploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +combine the memory address and MPI datatype translations +as described in §3.5 and §3.6 respectively. Towards this, we +maintain a structure called Env that stores the global state +required by these translations. This structure includes infor- +mation about the memory allocated to the Wasm module, it’s +base pointer (§3.5) and information about the different used +datatypes such as MPI_Comm by the module. For directly utiliz- +ing the host MPI library, we use the project rsmpi [7] in MPI- +Wasm. rsmpi provides MPI bindings for Rust and supports +OpenMPI [10] and MPICH [8]. It utilizes the rust-bindgen +project to generate foreign function interfaces tailored to spe- +cific MPI libraries. Each MPI function in MPIWasm directly +calls the equivalent function in rsmpi with the appropriate +arguments. +While for most functions in the MPI-2.2 standard MPI- +Wasm directly defers the execution to the host MPI library, +the implementation of the MPI functions MPI_Alloc_mem and +MPI_Free_mem is done differently. With these functions, it is +possible to allocate memory for use with other MPI functions. +When a Wasm module calls MPI_Alloc_mem, it expects a 32- +bit memory address in the module’s address space, while call- +ing the MPI_Alloc_mem function of the host MPI library re- +turns a 64-bit memory address in the embedder’s memory ad- +dress space which is not inside the chunk of memory reserved +for the Wasm module. To overcome this, MPIWasm only +supports MPI_Alloc_mem and MPI_Free_mem if the Wasm +module defines and exports the functions malloc and free. +When MPI_Alloc_mem is called, MPIWasm simply invokes +the exported malloc and receives a suitable 32-bit module +memory address. This address can then be used as the return +value for MPI_Alloc_mem. We implement MPI_Free_mem in +a similar way. +3.8 +Limitations +The Wasm specification currently assumes little-endian byte +order for multi-byte values [52] in execution environments. +By giving direct access to the Wasm module’s memory to the +host MPI library, we assume that the byte order of values in +the module address space and embedder address space is the +same. As a result, MPIWasm does not support big-endian CPU +architectures. This is not a disadvantage since most processor +architectures in HPC systems are little-endian. Moreover, due +to the current linear 32-bit memory space for a Wasm module, +HPC applications compiled to Wasm cannot have more than +4GiB of memory. The support for 64-bit memory addresses +is an important milestone for the Wasm specification and is +highlighted in the Wasm Memory64 proposal [20], but is out +of scope for this work. +4 +Experimental Results +In this section, we present performance results for our embed- +der MPIWasm across different processor architectures. For +1 +mpirun -np ./ mpiWasm mpi -app.wasm +Listing 4. Executing MPI applications compiled to Wasm +with MPIWasm. +Table 2. Comparing the size of native dynamically-linked, +statically-linked, and Wasm binaries for the different MPI +applications. The native applications are compiled for the +x86_64 architecture. +Application +Native Size Dynamic (KiB) +Native Size Static (MiB) +Wasm Size (KiB) +Intel MPI Benchmarks [55]. +1087 +27 +893 +HPCG [73]. +164 +26 +722 +IOR [60]. +364 +16 +315.32 +IS [31]. +36 +15 +57.88 +DT [31]. +40 +15 +49.51 +all our experiments, we follow best practices while reporting +results [54]. +4.1 +System Description +For analyzing the performance of our implemented Wasm +embedder, we use two systems. First, a production HPC clus- +ter located at our institute, i.e., SuperMUC-NG. Second, an +AWS EC2 virtual machine (VM) instance with the Gravi- +ton2 processor [1]. Our HPC cluster contains eight islands +comprising a total of 6480 compute nodes based on the Intel +Skylake-SP architecture. Each compute node has two sockets, +comprising two Intel Xeon Platinum 8174 processors, with +24 cores each and a total of 96GiB of main memory. The +nominal operating core frequency for each core is 3.10 GHz. +Hyper-Threading and Turbo Boost are disabled on the system. +The internal interconnect on our system is a fast Intel Omni- +Path [4] network with a bandwidth of 100 Gbit/s. Moreover, +our cluster provides a general parallel filesystem based on +the Lenovo DSS-G for IBM Spectrum Scale [19] with an +aggregate bandwidth of 200 GiB/s. For our experiments, we +use up to 128 nodes of the HPC system, i.e., 6144 cores. On +the other hand, the AWS Graviton2 processor based on the +64-bit ARMv8-A Neoverse-N1 [24] architecture consists of 32 +cores each with a nominal frequency of 2.50 GHz and a total +main memory of 64GiB. We limit our experiments to one +node for the Graviton2 processor. +4.2 +HPC Benchmarks +For our experiments with MPIWasm, we use the Intel MPI +Benchmarks [55], two benchmarks from the the NASA Ad- +vanced Supercomputing (NAS) Parallel Benchmark (NPB) +suite [31], the IOR benchmark [60], and the High Perfor- +mance Compute Gradient (HPCG) benchmark [73]. +The Intel MPI benchmarks perform a set of MPI perfor- +mance measurements for point-to-point and global communi- +cation operations for a range of message sizes. We use them +since they characterize the performance of a cluster and are +an indication of the efficiency of the used MPI implementa- +tion. The NPB suite includes a set of benchmarks that aim +to evaluate the overall performance of HPC clusters. Due to +the support for compiling Fortran to Wasm being in the early +stages (§5), only the Integer Sort (IS) and Data Transfer (DT) + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +20 +22 +24 +26 +28 +210 +1 +1.5 +Bytes +Iteration Time (usec) +PingPong (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +101 +102 +Bytes +Iteration Time (usec) +PingPong (time) > 1024 Bytes +Native +WASM +(a) PingPong. +20 +22 +24 +26 +28 +210 +2 +4 +Bytes +Iteration Time (usec) +Sendrecv 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +101 +102 +103 +104 +Bytes +Iteration Time (usec) +Sendrecv 6144 Ranks (time) > 1024 Bytes +Native +WASM +(b) SendRecv. +20 +22 +24 +26 +28 +210 +0 +20 +40 +Bytes +Iteration Time (usec) +Bcast 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +100 +102 +104 +Bytes +Iteration Time (usec) +Bcast 6144 Ranks (time) > 1024 Bytes +Native +WASM +(c) Broadcast. +22 +24 +26 +28 +210 +40 +60 +Bytes +Iteration Time (usec) +Allreduce 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +102 +104 +Bytes +Iteration Time (usec) +Allreduce 6144 Ranks (time) > 1024 Bytes +Native +WASM +(d) AllReduce. +20 +22 +24 +26 +28 +210 +0 +1 +2 +·104 +Bytes +Iteration Time (usec) +Allgather 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216 217 +105 +106 +Bytes +Iteration Time (usec) +Allgather 6144 Ranks (time) > 1024 Bytes +Native +WASM +(e) AllGather. +20 +22 +24 +26 +28 +210 +0 +1 +2 +·105 +Bytes +Iteration Time (usec) +Alltoall 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216 +106 +Bytes +Iteration Time (usec) +Alltoall 6144 Ranks (time) > 1024 Bytes +Native +WASM +(f) Alltoall. +22 +24 +26 +28 +210 +0 +10 +20 +30 +Bytes +Iteration Time (usec) +Reduce 768 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +100 +102 +104 +Bytes +Iteration Time (usec) +Reduce 768 Ranks (time) > 1024 Bytes +Native +WASM +22 +24 +26 +28 +210 +0 +20 +40 +60 +Bytes +Iteration Time (usec) +Reduce 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +100 +102 +104 +Bytes +Iteration Time (usec) +Reduce 6144 Ranks (time) > 1024 Bytes +Native +WASM +(g) Reduce. +20 +22 +24 +26 +28 +210 +0 +50 +100 +150 +Bytes +Iteration Time (usec) +Gather 768 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216217218219220 +100 +102 +104 +Bytes +Iteration Time (usec) +Gather 768 Ranks (time) > 1024 Bytes +Native +WASM +20 +22 +24 +26 +28 +210 +0 +200 +400 +600 +Bytes +Iteration Time (usec) +Gather 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216 217 +100 +102 +104 +Bytes +Iteration Time (usec) +Gather 6144 Ranks (time) > 1024 Bytes +Native +WASM +(h) Gather. +20 +22 +24 +26 +28 +210 +0 +100 +200 +300 +Bytes +Iteration Time (usec) +Scatter 768 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216217218219220 +101 +103 +105 +Bytes +Iteration Time (usec) +Scatter 768 Ranks (time) > 1024 Bytes +Native +WASM +20 +22 +24 +26 +28 +210 +0 +1,000 +2,000 +Bytes +Iteration Time (usec) +Scatter 6144 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 +214 +216 217 +101 +103 +105 +Bytes +Iteration Time (usec) +Scatter 6144 Ranks (time) > 1024 Bytes +Native +WASM +(i) Scatter. +Figure 3. Performance comparison of the Intel MPI benchmarks for MPIWasm and their native execution on our HPC system. +20 +22 +24 +26 +28 +210 +0.4 +0.6 +0.8 +1 +Bytes +Iteration Time (usec) +PingPong (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +100 +101 +102 +Bytes +Iteration Time (usec) +PingPong (time) > 1024 Bytes +Native +WASM +(a) PingPong. +20 +22 +24 +26 +28 +210 +0.5 +1 +1.5 +Bytes +Iteration Time (usec) +Sendrecv 32 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +100 +101 +102 +103 +Bytes +Iteration Time (usec) +Sendrecv 32 Ranks (time) > 1024 Bytes +Native +WASM +(b) SendRecv. +22 +24 +26 +28 +210 +2 +4 +6 +8 +Bytes +Iteration Time (usec) +Allreduce 32 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +101 +102 +103 +104 +Bytes +Iteration Time (usec) +Allreduce 32 Ranks (time) > 1024 Bytes +Native +WASM +(c) AllReduce. +20 +22 +24 +26 +28 +210 +0 +10 +20 +Bytes +Iteration Time (usec) +Allgather 32 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +102 +103 +104 +105 +Bytes +Iteration Time (usec) +Allgather 32 Ranks (time) > 1024 Bytes +Native +WASM +(d) AllGather. +20 +22 +24 +26 +28 +210 +20 +40 +Bytes +Iteration Time (usec) +Alltoall 32 Ranks (time) ≤ 1024 Bytes +Native +WASM +212 214 216 218 220 222 +102 +104 +Bytes +Iteration Time (usec) +Alltoall 32 Ranks (time) > 1024 Bytes +Native +WASM +(e) Alltoall. +12 4 +8 +16 +32 +0 +10 +20 +Ranks +GFLOP/s +HPCG GFLOPS +Native +WASM +12 4 +8 +16 +32 +0 +50 +100 +150 +Ranks +GB/s +HPCG Bandwidth +Native +WASM +(f) HPCG +Figure 4. Performance comparison of selected Intel MPI benchmarks and HPCG for MPIWasm against their native execution on +the AWS Graviton2 Processor. +benchmarks from this suite were used since they are written +in pure C. The IS benchmark performs bucketed parallel sort- +ing of integers across all participating processes, while the +DT benchmark tests the communication and the performance +of 64-bit floating point operations of a HPC cluster by send- +ing data through a topology of nodes. We use the topologies +Black-Hole (bh), White-Hole (wh), and Shuffle (sh) for +the DT benchmark. For our experiments, we use the classes +C and B for the IS and DT benchmarks respectively. The IOR +Benchmark measures the filesystem I/O performance avail- +able to MPI processes. It supports multiple backends that +utilize different APIs to perform system I/O. For our experi- +ments with MPIWasm, we use the POSIX API backend since +the POSIX filesystem APIs are included in the WASI specifi- +cation (§2.3, §3.2). The HPCG benchmark aims to evaluate +the real-world performance of HPC systems by solving a sys- +tem of linear equations with the conjugate gradient method. +For our experiments, we use the default available problem + +Exploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +size for HPCG. Note that, in our experiments we use the ver- +sions 2019 Update 6 and 3.3.1 for the Intel MPI and NAS +parallel benchmarks respectively. +4.3 +Experiment Setup +For all our experiments, we execute the benchmarks in a pure- +MPI configuration without shared memory parallelization +with OpenMP, as it is currently not supported by MPIWasm. +We use OpenMPI-4.0 as the MPI library since it is available +on our HPC system and can be easily installed on the AWS +Graviton2 nodes. For compiling the native applications on +our HPC system (§4.1), we use the clang-11 compiler, while +for the AWS Graviton2 node, we use the gcc7.1 compiler. +In both cases, the applications were compiled with the -O3 +optimization flag. For compiling the different benchmarks +to Wasm, we use the clang-11 compiler along with our cus- +tomized WASI-SDK with -O3 -msimd128 flags for both test +systems (§3.2). The -msimd128 flag enables the generation +of SIMD instructions in Wasm. We compile the applications +to Wasm only once on our local systems and execute them +directly with MPIWasm on the different test systems. We +build MPIWasm on our local system for the different plat- +forms, i.e., x86_64 and aarch64 with OpenMPI to generate +bindings for rsmpi (§3.7). Following this, we directly exe- +cute the applications compiled to Wasm on the test systems +as shown in Listing 4. Each MPI rank corresponds to one +instance of the embedder with it’s own Wasm module. The +native applications were executed directly using mpirun. +4.4 +Comparing Wasm Binary Size +Table 2 shows the comparison between the absolute binary +sizes for the different applications. The static versions of the +binaries are generated by supplying the -static flag to the +clang-11 compiler (§4.3) and linking the different applica- +tions with the static versions of the required libraries such +as libmpi.a, libopen-rte.a, and libz.a. To this end, we +made necessary changes to the Make [50] and CMake [39] +files used by the different applications (§4.2). While the stack- +based instruction set and compact binary format give Wasm +the potential to produce smaller binaries for the same appli- +cations as compared to the native dynamically-linked bina- +ries [52], three out of five applications that we used had a +bigger binary size when compiled to WebAssembly in com- +parison to the equivalent dynamically-linked native binary. +While Wasm can benefit from a smaller representation on +a function-by-function basis, in practice dynamically-linked +native binaries can offset that advantage by being able to rely +on commonly used libraries to be present on the system. For +instance, a native binary can dynamically link against glibc, +while a Wasm binary must statically include functions from +wasi-libc (§2.3). However, in contrast to containers, Wasm +binaries are significantly smaller making them more feasible +for application distribution in HPC environments. In addi- +tion, Wasm binaries are 139.5x smaller on average than the +statically-linked binaries of the different applications. This is +because the linker, i.e., lld copies all library routines from +the different libraries used by an application into the binary +during static linking. +4.5 +Benchmarking MPIWasm +Figure 3 and Figure 4 show the iteration times for the different +Intel MPI benchmarks for their native execution as compared +to their execution with MPIWasm on our HPC system and +the AWS Graviton2 processor respectively. For execution +with MPIWasm, the iteration times don’t include the time +required for compiling the Wasm modules to native machine +code (§3.3). To avoid repetition, we omit some results for the +Graviton2 processor. Error bars in the graphs represent mini- +mum and maximum values for iteration timings as reported +by the Intel MPI Benchmarks, while points in the graphs +represent the average timings as reported by the benchmarks. +For the PingPong benchmark using MPIWasm leads to a +geometric mean (GM) average slowdown of 0.05x for the +x86_64 system and a GM average speedup of 1.01x for the +aarch64 system across all message sizes (Figures 3a, 4a). We +calculate this value by dividing the metric t_avg_us reported +by the benchmarks for their native execution by the value +reported for execution with MPIWasm, followed by a GM of +the obtained values. For computing slowdown, we subtract +one from the obtained GM value. We observe a maximum +bandwidth of 12.80 GiB/s and 10.98 GiB/s for the native exe- +cution of the PingPong benchmark on the HPC and Graviton2 +processor respectively. On the other hand, with MPIWasm, +we observe a maximum bandwidth of 13.44 GiB/s and 10.61 +GiB/s on the two systems. For the SendRecv benchmark, +we observe a GM average slowdown of 0.06x and 0.07x +with MPIWasm across all message sizes on the x86_64 and +aarch64 systems respectively (Figures 3b, 4b). For the native +version of the benchmark, we observe a maximum bandwidth +of 7.24 GiB/s and 11.01 GiB/s on the two systems, while +with MPIWasm, we observe a maximum bandwidth of 7.50 +GiB/s and 10.83 GiB/s. For the collective communication +Broadcast routine, we observe an average GM slowdown of +0.13x with MPIWasm across all message sizes for 128 nodes +as shown in Figure 3c. We observe an average GM slow- +down of 0.06x and 0.10x with MPIWasm across all message +sizes for the collective communication AllReduce routine +as shown in Figures 3d and 4c. For AllGather with MPI- +Wasm, we observe an average GM slowdown of 0.06x and +0.09x across all message sizes for the HPC system and Gravi- +ton2 processor respectively (Figure 3e, 4d). Similarly, for the +Alltoall collective communication routine, we observe an +average GM slowdown of 0.10x for the two systems across all +message sizes with MPIWasm as shown in Figures 3f and 4e. +For 16 nodes of our HPC system, we observe an average GM +slowdown of 0.12x, 0.14x, and 0.05x across message sizes +for the routines Reduce, Gather, and Scatter as shown in +Figures 3g, 3h, and 3i. On the other hand, for 128 nodes, + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +64128 +256 +512 +1,024 +0.4 +0.6 +0.8 +1 +1.2 +·104 +Ranks +Mop/s +IS Total Operations/s +Native +WASM +bh +wh +sh +0 +500 +1,000 +1,500 +Topology +MB/s +DT Total Throughput +Native +WASM w/o SIMD +WASM w SIMD +(a) NPB. +1 +4 +8 +12 +16 +2.4 +2.6 +2.8 +3 +3.2 +·104 +Block Size (MiB) +MiB/s +IOR Bandwidth (Read) +Native +WASM +1 +4 +8 +12 +16 +3 +4 +·104 +Block Size (MiB) +MiB/s +IOR Bandwidth (Write) +Native +WASM +(b) IOR. +48 16 +48 +96 +144 +0 +20 +40 +60 +80 +Ranks +GFLOP/s +HPCG GFLOPS +Native +WASM +48 16 +48 +96 +144 +0 +200 +400 +600 +Ranks +GB/s +HPCG Bandwidth +Native +WASM +192 768 1,536 +3,072 +6,144 +0 +2,000 +4,000 +Ranks +GFLOP/s +HPCG GFLOPS +Native +WASM +192 768 1,536 +3,072 +6,144 +0 +1 +2 +3 +·104 +Ranks +GB/s +HPCG Bandwidth +Native +WASM +(c) HPCG. +Figure 5. Performance comparison of standardized HPC benchmarks for MPIWasm against their native execution on our HPC +system. +we observe an average GM slowdown of 0.05x, 0.10x, and +0.08x for the three routines. The results for testing MPIWasm +with the Intel MPI Benchmarks compiled to Wasm demon- +strate that neither the mechanism for calling host functions +in Wasmer [87] nor the translation layer implemented in MPI- +Wasm induce significant overhead for MPI communication +(§3.5,§3.6,§3.7). We expand on the translation overhead in +MPIWasm in §4.6. Overall, our results indicate that MPI- +Wasm delivers close to native performance for the different +MPI routines on both x86_64 and aarch64 architectures. +The performance of MPIWasm on our HPC system for +the IS and DT benchmarks is shown in Figure 5a. For the IS +benchmark with MPIWasm, we observe 8260 average mega +operations per second across all processes as compared to +8546 average mega operations per second for the native exe- +cution. For the DT benchmark with different topologies (§4.2), +execution with MPIWasm leads to decreased throughput as +compared to the native execution. The DT benchmark per- +forms a significant number of pairwise comparison opera- +tions which benefit greatly from vectorization with SIMD +instructions. To demonstrate the effect of SIMD for the DT +benchmark, we compile it to Wasm by disabling and enabling +the generation of SIMD instructions. The Wasm version of the +DT benchmark with SIMD leads to 1.36x better throughput +as compared to the Wasm version without SIMD (Figure 5a). +The difference in performance as compared to the native ver- +sion of the DT benchmark can be attributed to the support +for only 128-bit SIMD instructions in the Wasm specifica- +tion [52] as compared to 512-bit SIMD instructions present +in modern Intel processors [38, 57, 76] (§3.3). Support for +higher-width SIMD in Wasm is an important milestone in its +road-map but out of scope for this work (§5). +Figure 5b shows total aggregated read and write band- +width available to all MPI processes for the IOR benchmark. +Points in the graph represent the average bandwidth reported +by the benchmark, while error bars in the graph represent +the maximum and minimum bandwidth observed over all +iterations of the benchmark with the same block size. With +four nodes of our HPC system, the upper bound for achiev- +able bandwidth in our setup (§4.1) with IOR is 400 GBit/s +(≈ 47684 MiB/s). With MPIWasm, we observe similar read +(29411 MiB/s) and write (40206 MiB/s) bandwidth averaged +across all block sizes as compared to the native execution +of the benchmark. Testing the filesystem I/O performance +of the MPIWasm demonstrates that the userspace permission +handling and virtual directory tree implemented by Wasmer +to provide filesystem isolation (§3.4) has no significant im- +pact on the achievable bandwidth when performing I/O with +the POSIX filesystem API. For the HPCG benchmark, we ob- +serve similar performance when executed with MPIWasm as +compared to it’s native execution on the HPC system and +the Graviton2 processor up to 192 MPI processes (Figures 5c +and 4f). On increasing the number of processes, the native exe- +cution of the HPCG benchmark outperforms the execution with +MPIWasm as shown in Figure 5c. For 6144 MPI processes, we +observe a 14% reduction in GFLOP/s on execution with MPI- +Wasm. This behavior can be attributed to the significantly fre- +quent amount of communication required by the HPCG bench- +mark [63]. HPCG repeatedly uses the Allreduce routine to +reduce a single variable of size double over all MPI processes +to finalize vector-vector dot operations. With increasing num- +ber of processes, the number of times the Allreduce routine +is called also increases. For instance, executing HPCG with +768 processes results in four times more calls to Allreduce +as compared to the execution with 192 processes. As a re- +sult, the repeated datatype translations in MPIWasm increase +the cost for invoking the collective communication routine +leading to performance degradation (§4.6). +4.6 +Analyzing Datatype Translation Overhead +To measure the datatype translation overhead in MPIWasm, +we implement a custom PingPong application that sends/re- +ceives messages of varying sizes between two processes and +iterates over the different MPI datatypes, i.e., BYTE, CHAR, +INT, FLOAT, DOUBLE, and LONG. Following this, we instru- +ment the Send routine in MPIWasm to determine the latency +for translating the different datatypes. Finally, we execute the +application on our HPC system. Figure 6 shows the trans- +lation overhead for different datatypes and message sizes +in MPIWasm. We observe an average overhead of 85.44ns, + +Exploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +8 +64 +256 +1024 +32768 +262144 1048576 2097152 4194304 +Bytes +0 +50 +100 +150 +200 +250 +300 +Translation Time (ns) +MPI_BYTE +MPI_CHAR +MPI_INT +MPI_FLOAT +MPI_DOUBLE +MPI_LONG +Figure 6. Comparing the datatype translation overhead in +MPIWasm. +84.72ns, 99.78ns, 96.32ns, 103.35ns, and 104.79ns across all +message sizes for the MPI datatypes BYTE, CHAR, INT, FLOAT, +DOUBLE, and LONG respectively. We observe an increase in the +translation overhead for message sizes greater than 256KB. +This can be attributed to an increased latency for acquiring +read locks from the Env structure that maintains the global +state for translations in MPIWasm (§3.7). +5 +Into the Future: Wasm and HPC +In this section, we highlight and discuss the different exten- +sions proposed by the Wasm community to the current Wasm +specification [52] that can be implemented in an embedder +for HPC applications to enhance performance and portability. +Controlled Threading for Wasm modules. The Wasm +Threads proposal [16] lays the foundation for utilizing Wasm +for multithreaded algorithms. It enables Wasm modules to de- +fine shared memories, informing the embedder that the mod- +ule expects the memory to be accessed by multiple threads. +To enable safe multithreaded access to shared memory, the +proposal also defines atomic Wasm instructions that can be +used to implement locks and atomic data structures in the +functions of the module. To enable HPC applications to make +use of the functionality added by the threads proposal, an API +that allows Wasm modules to create additional threads on its +own needs to be added to the embedder. Implementing the +POSIX threads [15] and OpenMP [41] APIs in the embed- +der would enable compatibility with the threading code in +existing HPC applications. +Wasm Extended SIMD. The current Wasm SIMD pro- +posal [52] only specifies 128-bit SIMD instructions, while +modern CPUs support higher-width-SIMD, for instance the +AVX-512 instruction set extensions for x86_64, which speci- +fies 512-bit SIMD instructions. Towards this, the Wasm Flex- +ible Vector proposal [13] aims to provide support for SIMD +instructions that are wider than 128-bit. Moreover, the Wasm +relaxed SIMD instructions [21] aim to make it possible to uti- +lize hardware SIMD instructions that are not well defined, i.e., +they differ in rounding behavior from the Wasm specification. +Implementing these proposals in the embedder would allow +compiled Wasm modules for HPC applications that contain +vectorizable code to make better use of SIMD instructions +available in modern CPU architectures. +20 +22 +24 +26 +28 +210 +2 +4 +6 +Bytes +Iteration Time (usec) +PingPong (time) ≤ 1024 Bytes +MPIWasm +Faasm +212 214 216 218 220 222 +100 +101 +102 +103 +Bytes +Iteration Time (usec) +PingPong (time) > 1024 Bytes +MPIWasm +Faasm +Figure 7. Comparing the performance of MPIWasm and +Faasm [78]. +Wasm PGAS. Partitioned Global Address Space (PGAS) +is a programming model for parallel distributed memory ap- +plications that introduces a memory address space that spans +the local memory of multiple processes. With a memory ad- +dress from this global address space a process can read from +and write to the memory of other processes. Since Wasm +already specifies the concept of defining and importing mem- +ories, the embedder could be extended to provide non-local +memory to the Wasm module. To support this use-case, the +Wasm Multi-Memory proposal [14] needs to be implemented, +which allows a Wasm module to define or import more than +one memory. +Dynamic Linking of Wasm Modules. While there is ex- +isting work on establishing an ABI for dynamic linking be- +tween Wasm modules [12], it has not been standardized yet. +Supporting dynamic linking would significantly decrease the +size of Wasm binaries for more complex applications as they +would no longer need to statically link parts of wasi-libc. +For HPC, it would enable commonly used libraries such as +BLAS to be provided by MPIWasm. Combining dynamic link- +ing with efforts to provide repositories for Wasm modules +such as WAPM [88] could enable automatic dependency man- +agement for Wasm applications. +Compiling Fortran applications to Wasm Currently, the +support for compiling Fortran-based applications to Wasm is +very nascent with only one known attempt based on Dragon- +egg [17]. However, the implementation of the Memory64 +proposal [20] should enable the usage of existing Fortran +LLVM compilers such as F18 for easily compiling Fortran- +based applications to Wasm [18]. +Wasm and Accelerators The module execution hints pro- +posal [23] highlights the changes required in the Wasm speci- +fication to enable the support for executing Wasm modules +on hardware accelerators such as GPUs. Implementing the +proposal in the embedder would enable compatibility with +existing GPU-based HPC applications. +6 +Related Work +Solutions for Packaging and distributing HPC applica- +tions. Recently several HPC-focused tools such as Char- +liecloud [72] and Singularity [58] have been introduced for +distributing HPC applications through containerization. In +contrast, we utilize the universal binary instruction format +Wasm to package and distribute HPC applications. Moreover, + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +while containerization requires building HPC application con- +tainers for different platforms, HPC applications can be com- +piled once to Wasm and executed on any platform using a +supporting Wasm embedder. +MPI and WebAssembly. To the best of our knowledge, +Faasm [78] is the only compute platform that enables the exe- +cution of MPI applications compiled to Wasm. It is based on +a gRPC-based distributed messaging library called Faabric +and contains a Wasm runtime, a workload scheduler, and +a distributed state store. In order to run an application on +Faasm, it needs to be compiled to Wasm and uploaded to the +shared function storage. Following this, the application can +be invoked using events such as HTTP requests. For support- +ing MPI applications, Faasm implements only a subset of +the MPI-1 specification on top of its messaging library and +its own workload scheduler. Moreover, it also does not sup- +port user-defined communicators required by the Intel MPI +benchmarks (§4.1). In contrast, we take an inverted approach, +where MPIWasm builds on top of existing native MPI libraries +and provides a way for Wasm modules to call functions from +them efficiently. Figure 7 shows the performance comparison +between MPIWasm and Faasm for the PingPong benchmark +(§4.2). With MPIWasm, we achieve a GM average speedup +of 4.28x across all message sizes as compared to Faasm. +7 +Conclusion and Future Work +In this paper, we took the first step towards bringing We- +bAssembly to the HPC ecosystem and presented MPIWasm, a +Wasm embedder that enables high performance execution of +MPI applications compiled to Wasm across different proces- +sor architectures. In the future, we plan to extend MPIWasm +to support acceralators such as GPUs found on HPC sys- +tems [23]. +8 +Acknowledgement +We thank our shepherd Milind Chabbi for his help in prepar- +ing the final version of this paper. Furthermore, we thank +the anonymous reviewers for their insightful comments and +valuable feedback that significantly improved the quality of +this paper. This work was supported by the funding of the +German Federal Ministry of Education and Research (BMBF) +in the scope of the Software Campus program. +References +[1] [n.d.]. AWS Graviton 2 Processors. +https://aws.amazon.com/ec2/ +graviton/ +[2] [n.d.]. Blake-3 Hash function. +https://github.com/BLAKE3-team/ +BLAKE3 +[3] [n.d.]. Cranelift Compiler. +https://github.com/bytecodealliance/ +wasmtime/tree/main/cranelift +[4] [n.d.]. Intel Omni-Path Fabric. https://www.intel.com/content/www/ +us/en/high-performance-computing-fabrics/omni-path-fabric- +software-components.html +[5] [n.d.]. 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Slurm: Simple +linux utility for resource management. In Workshop on Job Scheduling +Strategies for Parallel Processing. Springer, 44–60. +A +Artifact Appendix +A.1 +Description +MPIWasm is an embedder for MPI-based HPC applications +based on Wasmer [87]. It enables the high performance execu- +tion of these applications compiled to WebAssembly (Wasm) +and serves two purposes: +1. Delivering close to native application performance, i.e., +when applications are executed directly on a host ma- +chine without using Wasm. +2. Enabling the distribution of MPI-based HPC applica- +tions as Wasm binaries. +Our artifact contains the source code for MPIWasm, toolchain +for compiling C/C++ based MPI applications to Wasm, the +Wasm binaries for the different standardized HPC bench- +marks used in this paper, pre-built versions of our embedder +for different operating systems, and scripts for parsing experi- +ment data and generating plots. The artifact is available at: +https://doi.org/10.5281/zenodo.7468121 +or +https://github.com/kky-fury/MPIWasm +A.2 +Getting Started +For testing our Wasm embedder for executing MPI appli- +cations compiled to WebAssembly, we provide a pre-built +docker image for the linux/amd64 platform with all the re- +quired dependencies. +1 +sudo +docker +run − i t +kkyfury / ppoppae : v2 +/ bin / bash +2 +#Executing the HPCG benchmark compiled to Wasm +3 +mpirun −−allow −run −as − r o o t −np 4 +. / t a r g e t / r e l e a s e / embedder +\ +4 +examples / xhpcg . wasm +5 +#Executing the IntelMPI benchmarks compiled to Wasm +6 +mpirun −−allow −run −as − r o o t −np 4 +. / t a r g e t / r e l e a s e / embedder +\ +7 +examples / imb . wasm +Listing 5. Getting started with MPIWasm. +Towards this, a user can follow the steps described in Listing 5. +Following this, MPIWasm should successfully execute the +HPCG and IntelMPI benchmarks. We provide sample output +for the two benchmarks in the provided artifact. The user can +increase/decrease the number of processes (-np) for executing +the benchmarks. However, depending on the system where + +Exploring the Use of WebAssembly in HPC +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +the container is executing, the user might need to provide the +-oversubscribe flag to mpirun. +A.3 +Running Experiments with MPIWasm +This section describes how to run experiments with our em- +bedder to obtain plots similar to the ones in this paper. +A.3.1 +Running small-scale experiments. To run small-scale +experiments inside the docker container, we provide an end- +to-end script. This script: +1. Executes the HPCG, IS, and IntelMPI benchmarks for +their native execution and when they are executed using +MPIWasm. +2. Parses the obtained results and generates the relevant +plots. +1 +sudo +docker +run − i t +kkyfury / ppoppae : v2 +/ bin / bash +2 +cd +run_experiments +3 +. / runme . sh +Listing 6. Running small-scale experiments with MPIWasm. +For running the experiments, the user can follow the steps de- +scribed in Listing 6. The script can take around 10-15 minutes +to finish execution. After completion, you can see the gener- +ated data in the run_experiments/experiment_data folder. The +generated plots can be found in the run_experiments/Plots +folder. We provide sample plots for the different benchmarks +in our artifact. However, on executing benchmarks inside +the container, the performance difference between the native +execution of the application and MPIWasm can be around +8-12%. +A.3.2 +Running large-scale experiments on an HPC sys- +tem. For running large-scale experiments with our embedder, +a user needs to do the following: +1. Build a version of the embedder for your HPC sys- +tem depending on the particular architecture, operating +system, and the MPI library on the system. MPIWasm +currently supports the OpenMPI library with limited sup- +port for MPICH and MVAPICH. We provide examples for +building MPIWasm for different operating systems and +architectures in our artifact. +2. Execute the MPI applications using the built embedder +on the HPC system. This can be done via submitting +jobs to a RJMS software on an HPC system such as +SLURM [92]. We provide sample job scripts for our HPC +system, i.e., SuperMUC-NG that uses SLURM in our +artifact. +3. After executing the applications, the user can utilize +the different parsers provided in our artifact to parse +the benchmark data. Following this, the results can +be visualized using the plotting helper provided in the +artifact. +A.4 +Compiling C/C++ applications to Wasm +We have setup a docker container with the required depen- +dencies for compiling different MPI applications conformant +to the MPI-2.2 standard to Wasm. The artifact also includes +HPCG, IntelMPI, and IS benchmarks as examples. +1 +sudo +docker +run − i t +kkyfury / w a s i t o o l c h a i n : v1 +/ bin / bash +2 +#Compiling HPCG +3 +cd +/ work / example / hpcg −benchmark +4 +. / wasi −cmake . sh +5 +cd cmake− build −wasi +6 +make +Listing 7. Compiling applications to Wasm. +Listing 7 describes the steps a user can follow for compiling +the HPCG benchmark to Wasm. For steps to compile the other +benchmarks to Wasm, please look at the base Readme.md +file provided with the artifact. All the different applications +compiled to Wasm that we used in this paper are also present +in the artifact. +A.5 +Using MPIWasm +For detailed usage instructions, please look at the base Readme.md +file provided with the artifact. +A.6 +Modifying MPIWasm +For modifying our embedder, we recommend using our pro- +vided docker-compose file in the artifact. This docker-compose +file mounts the volume with the embedder’s source code in- +side the container. As a result, any changes to it’s source code +will be reflected inside it. For our embedder, we currently +support the following operating systems: +1. CentOS-8.2 +2. Opensuse-15-1 +3. Ubuntu-20-04 +4. MacOS-monterey +1 +cd wasi −mpi− r s +2 +docker +compose run +centos −8−2 +3 +cargo +b u i l d −− r e l e a s e +4 +#After the build process , you can see the built embedder in +5 +# +the /target/release/ folder. +Listing 8. Building MPIWasm. +Listing 8 describes the steps for building MPIWasm for +CentOS-8.2 after modifications. The instructions for building +the embedder for other operating systems are provided in the +base Readme.md file inside the artifact. After the build process, +the embedder can be copied to the user’s local filesystem +using the docker-cp command as shown in Listing 9. +1 +docker +cp < c o n t a i n e r −id > : / s / t a r g e t / r e l e a s e / embedder +\ +2 +< d e s t i n a t i o n −path −user − f i l e s y s t e m > +Listing 9. Copying MPIWasm. +We provide provide the base image dockerfiles for the dif- +ferent supported operating systems inside the artifact. These +example dockerfiles can be easily extended to support other +different linux distributions. + +PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada +Mohak Chadha et al. +A.7 +Support for aarch64 +Our embedder also supports execution on linux/arm64 plat- +forms. We provide pre-built versions of our embedder for +arm64 for the different supported operating systems in the +artifact. +A.7.1 +Building images for aarch64. If the user is building +the docker image on an x86_64 system then docker-buildx +is required. Note that, in this case, building the image might +take around 12 hours. +1 +sudo +docker +buildx +c r e a t e +−−name mybuilder −−use −− b o o t s t r a p +2 +cd wasi −mpi− r s / . g i t l a b / c i / images / +3 +sudo +docker +buildx +b u i l d −−push −f +ubuntu −20 −04. D o c k e r f i l e +\ +4 +−− p l at f or m +l i n u x / arm64 +\ +5 +− t +kkyfury / ubuntumodifiedbase : v1 +. +6 +cd +. . / . . / . . / +7 +sudo +docker +buildx +b u i l d −−push −f +D o c k e r f i l e +\ +8 +−− p l at f or m +l i n u x / arm64 +\ +9 +− t +kkyfury / embedderarm : v1 +. +Listing 10. Building MPIWasm for aarch64 on x86_64. +Listing 10 describes the steps required for building MPI- +Wasm for arm systems with docker-buildx. The user should +change the docker image tags according to their docker reg- +istry account, i.e., replace kkyfury with your registry user- +name. On the other hand, if the user is using an aardch64 +system then please follow the instructions described in §A.6. + diff --git a/69E2T4oBgHgl3EQfkweU/content/tmp_files/load_file.txt b/69E2T4oBgHgl3EQfkweU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..381465baac226031a5071e7b32800484f8aea037 --- /dev/null +++ b/69E2T4oBgHgl3EQfkweU/content/tmp_files/load_file.txt @@ -0,0 +1,1472 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf,len=1471 +page_content='Exploring the Use of WebAssembly in HPC Mohak Chadha, Nils Krueger, Jophin John, Anshul Jindal, Michael Gerndt {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='lastname}@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='de Chair of Computer Architecture and Parallel Systems, Technische Universität München, Germany Shajulin Benedict shajulin@iiitkottayam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='in Department of Computer Science and Engg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Indian Institute of Information Technology Kottayam, Kerala Abstract Containerization approaches based on namespaces offered by the Linux kernel have seen an increasing popularity in the HPC community both as a means to isolate applications and as a format to package and distribute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, their adoption and usage in HPC systems faces several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These include difficulties in unprivileged running and build- ing of scientific application container images directly on HPC resources, increasing heterogeneity of HPC architectures, and access to specialized networking libraries available only on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These challenges of container-based HPC appli- cation development closely align with the several advantages that a new universal intermediate binary format called We- bAssembly (Wasm) has to offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These include a lightweight userspace isolation mechanism and portability across oper- ating systems and processor architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In this paper, we explore the usage of Wasm as a distribution format for MPI- based HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, we present MPIWasm, a novel Wasm embedder for MPI-based HPC applications that enables high-performance execution of Wasm code, has low- overhead for MPI calls, and supports high-performance net- working interconnects present on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We evaluate the performance and overhead of MPIWasm on a production HPC system and AWS Graviton2 nodes using standardized HPC benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Results from our experiments demonstrate that MPIWasm delivers competitive native application per- formance across all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, we observe that Wasm binaries are 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5x smaller on average as compared to the statically-linked binaries for the different standardized benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' CCS Concepts: • Software and its engineering → Process management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='nnnnnnn ACM Reference Format: Mohak Chadha, Nils Krueger, Jophin John, Anshul Jindal, Michael Gerndt and Shajulin Benedict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Exploring the Use of We- bAssembly in HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In The 28th ACM SIGPLAN Annual Sympo- sium on Principles and Practice of Parallel Programming (PPoPP ’23), February 25-March 1, 2023, Montreal, QC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' ACM, New York, NY, USA, 16 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' nnnnnnn 1 Introduction Linux containers, due to their portability and high availability, have become the de-facto standard for developing, testing, and deploying a wide range of applications from enterprise to web services in cloud environments [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is because containers enable users to package their application along with its custom software dependencies as a single unit into easy-to-deploy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Motivated by their popularity in the cloud, containers have also seen a growing interest in the HPC community [30, 74, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For HPC systems, containers provide flexibility to users and allow them to define custom software stacks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', user-defined software stack (UDSS) for their large- scale scientific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, they enable easy, reli- able, and verifiable environments that can be reproduced in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, several HPC-focused containerization solutions, such as Charliecloud [72], Shifter [49], Singular- ity [58], Podman [48], and Sarus [33] have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In contrast to previous approaches, this paper investigates us- ing a new novel technology called WebAssembly (Wasm) [52], dubbed as an alternative to Linux containers [81], for packag- ing and distributing HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Despite their increasing popularity, the adoption and us- age of containers in HPC systems is still significantly lim- ited [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This can be attributed to the several challenges commonly faced by users in running and building container images for their applications on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For execut- ing containers, most containerization solutions require root privileges which are not possible for normal HPC users due to shared filesystems and their UNIX permissions in HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While HPC-focused containerization solutions such as Sin- gularity [58] and Podman [48] support rootless-containers through fakeroot [61], their current implementations do not support distributed filesystems such as GPFS commonly found on HPC systems [70, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, as argued by [71], building Open Container Initiative (OCI) [69] compliant con- tainer images on HPC resources by unprivileged (normal) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='03982v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='DC] 10 Jan 2023 PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' users where the applications will eventually run is signifi- cantly hard and requires support from the supercomputing center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is because most container building solutions such as Docker [43] also require root privileges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, most users use their own local systems for building/developing their application container images and then transfer the built image to a login/front-end node of an HPC system for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, this scenario leads to several problems in container- based HPC application development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' First, HPC nodes are becoming more heterogeneous [11] with different processor architectures such as x86_64 or aarch64 and have special- ized accelerators such as GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As application performance is critical in HPC, compiling an application using the specific microarchitectural features of a particular processor is signifi- cantly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While building container images for mul- tiple platforms either by cross-compiling HPC applications or by emulation with QEMU is possible with plugins such as build-x [44], it is not widely supported by HPC application build procedures and requires the presence of specific Linux kernel features (binfmt_misc [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, testing and developing HPC applications offers insights only on the target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In addition, most container images can range from several MiBs to several GiBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, frequent network transfers from the local to the HPC system can be cumber- some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Second, building HPC applications requires access to specialized networking libraries and licenses to compilers that are not available on the local user systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Finally, while different containerization solutions have almost no impact on the performance of the containerized application [72, 75, 84], building high-performant HPC application container images is non-trivial, involves a steep learning curve, and requires knowledge about specific MPI library versions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Open- MPI [10] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='0) and high performance network interconnect hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Intel OmniPath [4]) and libraries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Intel Performance Scaled Messaging [5]) present on the target sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These challenges of container-based HPC application development closely align with the several advantages and core problems that Wasm [52] aims to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm is a low-level, statically typed universal binary in- struction format for memory-safe, sandboxed execution in a virtual machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It offers portability across modern proces- sor architectures and operating systems, fast execution, and a low-level memory model [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Although originally meant for execution in Web browsers, due to its simplicity and gen- erality, Wasm has seen widespread adoption and usage in non-Web domains such as serverless computing [78], edge computing [47, 53], and Internet of Things [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It does not require garbage collection and is designed to be a universal compilation target with mature support for programming lan- guages with an LLVM [59] front-end such as C, C++, C#, and Rust [32, 35, 40, 45, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 1 demonstrates a general workflow for using Wasm in HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Developers can compile their HPC applications to Wasm once on their local systems ahead-of-time (AoT) and HPC Application WebAssembly x86_64 aarch64 WebAssembly embedder Compile Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' An HPC application can be compiled to WebAssem- bly and distributed to multiple platforms where it can be executed efficiently by a supporting WebAssembly embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' distribute it across multiple platforms instead of distribut- ing source code or building application containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Typically, Wasm binaries have a smaller size as compared to native x86_64 binaries [52, 56, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, the resulting binary can be executed on any platform using a standalone Wasm embedder [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm embedder serves two major purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' First, it provides an isolated execution environ- ment for running a Wasm binary on a platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In contrast to container-based approaches that utilize different Linux names- paces [71] for isolation and security, Wasm provides light- weight isolation at the application level based on software fault isolation (SFI) [85] and control flow integrity (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Second, it is responsible for compiling Wasm binaries to native machine code, either by using Just-in-Time (JIT) en- gines at the time of execution, or AoT by using the same JIT engines or AoT compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Note that, Wasm binaries can be executed by normal users and are completely unpriv- ileged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Several open-source standalone embedders such as Wasmer [87], Wasmtime [37], and Wasm3 [79] are currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, none of them support the execution of HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As the first step towards bringing Wasm to the HPC ecosys- tem, we only focus on MPI-based [6] HPC applications in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We chose MPI due to its understanding and in- fluence in the HPC community [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, our key contributions are: We implement and present MPIWasm, a novel Wasm embedder for MPI-based HPC applications based on Wasmer [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPIWasm enables high performance exe- cution of Wasm code, has low-overhead for MPI calls through zero-copy memory operations, and supports high-performance networking interconnects such as Intel OmniPath [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We demonstrate with extensive experiments the low- overhead and performance of MPIWasm using standard- ized HPC benchmarks on a production HPC system and AWS Graviton2 [1] nodes based on the x86_64 and the aarch64 architectures respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We elaborate on the different possible future directions for using Wasm in the HPC ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WA口口Exploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' §2 provides a detailed overview on Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In §3, we describe our embedder MPIWasm in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our experimental results are presented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In §5, we describe the different possible directions for using Wasm in the HPC ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' §6 describes some previous approaches related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In §7, we conclude the paper and present an outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2 A primer on WebAssembly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 WebAssembly Overview WebAssembly (Wasm) was introduced in 2015 as an alter- native to JavaScript for web-browser based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It superseded asm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='js [67], a previous attempt by Mozilla which focused on a subset of Javascript code that can be optimized AoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' When an application is compiled to Wasm, the resulting binary is called a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm modules contain function definitions, declarations of global variables, tables, and a lin- ear memory address space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' All of the application code in Wasm is organized in functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The conceptual machine in Wasm is stack-based and does not contain registers, there- fore all instructions pop their operands from the stack of the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, since application control flow is an explicit part of the module and Wasm operations are typed, it is possible to statically predict the layout of the stack at any point in the program which allows compilers to trans- late the stack semantics to a register-based instruction set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Similar to other higher-level programming languages, Wasm allows the definition of global variables that are not scoped to a specific function or block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Tables in Wasm modules are used for storing references to functions [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm ISA currently supports only four data types for variables: (i) i32, 32-bit integers, (ii) i64, 64-bit integers, (iii) f32, 32-bit IEEE 754 floating point numbers, and (iv) f64 64-bit IEEE 754 floating point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For constructing, complex types a combination of these basic types is commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm provides the capability for data and code to be shared between the module and its embedder using the import/export system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' All of the function definitions that can occur in a Wasm module can be imported from the embedder instead of being defined within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Similarly, function definitions that are present in the module can be exported so that the embedder can utilize them (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 WebAssembly Security and Sandboxing Model Wasm utilizes software fault isolation techniques (SFI) [85] to sandbox the executing Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' By default, a Wasm module cannot interact with the host system or perform I/O operations of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Any system interaction that is to be initiated by the Wasm module’s code must be done through the functions imported from the embedder (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, the embedder can act both as a translation layer and as an arbiter to enforce isolation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a translator, it 1 (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=') (func (param i32) (result i32))) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3 (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=') (func (param i32 i32) (result i32))) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 5 (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=') (func (param i32 i32 i32 i32 i32 i32) 6 (result i32))) 7 (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=') (func (param i32 i32 i32 i32) (result i32 ))) 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 9 (import "wasi_snapshot_preview1" "path_open" 10 (func $__wasi_path_open (type 22))) 11 (import "wasi_snapshot_preview1" "fd_close" 12 (func $__wasi_fd_close (type 1))) 13 (import "wasi_snapshot_preview1" "fd_seek" 14 (func $__wasi_fd_seek (type 23))) 15 (import "wasi_snapshot_preview1" "fd_read" 16 (func $__wasi_fd_read (type 15))) 17 (import "wasi_snapshot_preview1" "proc_exit" 18 (func $__wasi_proc_exit (type 0))) 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 (export "_start" (func $_start )) 21 (export "memory" (memory 0)) Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Example representation of a compiled C++ appli- cation’s Wasm module using the WASI-SDK in WebAssembly text format (WAT) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Ellipses signify sections that are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' is possible for the embedder to provide a common interface to the Wasm module even though the underlying system may have different native interfaces, while as an arbiter it is possi- ble for the embedder to restrict access of the Wasm module to system resources based on an application-level security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For instance, it is possible for the embedder to allow file I/O only to files that reside in a specific directory to iso- late the Wasm module from the rest of the filesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While in principle similar to kernel-level system call filtering tech- niques such as Seccomp-BPF [83] on Linux, performing such filtering on the application level allows to define semantically more meaningful policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In Wasm, all memory access is confined to a module’s lin- ear memory which is separate from the code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Currently, the Wasm specification [52] supports 32-bit addresses to in- dex the memory that a module has access to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While this limits a single module’s memory to 4GiB, it also enables hardware accelerated bound checks of memory accesses at runtime [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' If an embedder is a process with a 64-bit memory address space, it can safely execute an untrusted Wasm module in its memory space without requiring additional isolation by re- serving a continuous range of virtual memory for the module to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Not all pages in this range need to be mapped to phys- ical memory, it is sufficient to only map the required number of pages to fit the amount of memory used by the module at a given point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This ensures that a Wasm module can only operate in its own execution environment and cannot corrupt the memory of the embedder, since any out-of-bounds memory access will result in a page fault which can then be handled by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, since the memory instructions in Wasm’s specification [52] work with offsets, it is not possible to read and write to arbitrary memory locations in Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In the assembly produced by C programs, where a func- tion call is expressed as a jump instruction to the address of the function’s first instruction, a typical exploit is to change this address to take control of the program’s control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 typedef int MPI_Comm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2 typedef int MPI_Datatype;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4 int MPI_Init(int* argc , char *** argv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 5 int MPI_Finalize(void);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 6 int MPI_Send( 7 const void* buf , int count , MPI_Datatype datatype , 8 int dest , int tag , MPI_Comm comm 9 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 10 int MPI_Recv( 11 void* buf , int count , MPI_Datatype datatype , 12 int source , int tag , MPI_Comm comm , MPI_Status* status 13 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Excerpt of the custom MPIWasm mpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='h header file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, such exploits are not possible with Wasm since it features control flow integrity by enforcing structured pro- gram control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is because of two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' First, in Wasm, a function is represented as an index in a table (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1) which adds an additional level of indirection to express the function address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Second, the Wasm specification prevents constructing arbitrary memory addresses [42] and the sepa- ration of the embedder and the module’s memory prevents overwriting function instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3 WebAssembly System Interface Since Wasm was originally designed for web browsers, a sys- tem interface that targets POSIX environments and enables execution of Wasm modules on them was not part of the orig- inal specification [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To overcome this, the WebAssembly System Interface (WASI) specification [89] was designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WASI specifies the interface an embedder needs to implement to execute most POSIX applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Embedders that imple- ment the WASI specification will be able to run any generic application compiled with the WASI-SDK [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The WASI-SDK includes the clang compiler and its own C library based on musl libc that call WASI systemcalls imported from the em- bedder instead of relying on Linux systemcalls [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Note that, due to the ubiquity of glibc [26] on Linux systems, some ap- plications have come to depend on glibc-specific functions or behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Such applications will require modifications before they can be compiled to a WASI-compliant Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 1 shows a compiled Wasm module of a C++ appli- cation using the WASI-SDK in the WebAssembly text format (WAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WAT is a human readable format that enables de- velopers to examine the source code of a Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It can be observed that the module contains several functions with integers as parameter and return types (Lines 1-7) (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1), imports WASI functions (Lines 9-18), and exports its _start (main function) and memory (Lines 20-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Exporting these two definitions allows the embedder that executes this module to call its entrypoint function and to read from and write to the module’s linear memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While the import statements on Lines 9-16 enable the Wasm module to open and read from a file, the function proc_exit is used by the embedder to handle the termination of the application, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', by deallocat- ing the memory reserved for the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the module to 1 (import "env" "MPI_Init" ( 2 func $MPI_Init (param i32 i32) (result i32) 3 )) 4 (import "env" "MPI_Finalize" (func $MPI_Finalize (result i32 ))) 5 (import "env" "MPI_Send" ( 6 func $MPI_Send (param i32 i32 i32 i32 i32 i32) (result i32) 7 )) 8 (import "env" "MPI_Recv" ( 9 func $MPI_Recv (param i32 i32 i32 i32 i32 i32 i32) 10 (result i32) 11 )) Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WAT representation of module imports that corre- spond to the functions shown in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' execute, the imported functions need to be implemented by the embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3 MPIWasm In this section, we describe MPIWasm, our embedder for executing Wasm modules that utilize functions from the MPI standard in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 Overview The purpose of MPIWasm is to support the execution of MPI applications compiled to Wasm on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To facili- tate its adoption and suitability in HPC environments, it (i) supports high-performance execution of MPI-based HPC ap- plications compiled to Wasm (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3), (ii) has low-overhead for MPI calls through zero-copy memory operations (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6), and (iii) supports high-performance interconnects such as Infiniband [64] and Intel OmniPath [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These network inter- connects are utilized by MPI libraries on HPC systems for high-performance inter-rank communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To enable the immediate support for network interconnects present on mod- ern HPC systems, MPIWasm links against the MPI library on the target HPC system at runtime and provides a translation layer between the Wasm module and the host1 MPI library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, the developer doesn’t need to be aware about the particular networking libraries or network interconnects present on the target HPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Depending on the partic- ular host MPI library such as OpenMPI [10] or MPICH [8], MPIWasm needs to be built separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Both of these libraries are currently supported by MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our embedder currently supports the execution of MPI applications written in C/C++ and conforming to the MPI-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 standard [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Integrating the support for MPI-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 [66] is of our interest for the future but is out of scope for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We chose to focus on C/C++ applications due to the stability and maturity of the Wasm backend in the LLVM/Clang [59] project since llvm-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As the base for MPIWasm, we use the open-source Wasm embedder called Wasmer [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasmer supports the execution of Wasm modules on three major plat- forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Linux, Windows, and macOS, and supports both x86_64 and aarch64 instruction set architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, it implements the WASI specification (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3) and provides 1We use the term target and host interchangeably for the system on which the Wasm module is executing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Exploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada ergonomic mechanisms to define additional functions that are provided to the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This dynamic extension of the embedder’s functionality enables the addition of MPI func- tions to the functionality it provides to the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For implementing MPIWasm, we use the Rust programming lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is because of two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' First, it provides high performance comparable to C/C++ with memory-safety [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Second, it has extensive support and documentation for em- bedding Wasmer and using it as a library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 Compiling C/C++ MPI applications to Wasm Most MPI applications expect POSIX functionality to be available in their execution environment, for instance the abil- ity to read from and write to file descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WASI (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3) defines the WebAssembly exports that enable Wasm mod- ules that target it to call most of the functions defined in the C standard libraries shipped on POSIX systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, the WASI-SDK [28] combines the clang compiler and the wasi-libc C library to enable the compilation of C/C++ applications that only make use of POSIX functions and no additional libraries to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The compilation of C/C++ MPI applications is not supported by the stock WASI-SDK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, we implement a custom mpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='h MPI header file and add it to the WASI-SDK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The header file includes the definitions for the different MPI types such as MPI_Op, MPI_Comm, and MPI_Datatype and the definition for the MPI_Status struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, it defines the signatures for the MPI functions according to the MPI-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 [65] standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' An excerpt from the header file is shown in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It is a reduced version of a traditional header file found with MPI libraries with most types defined as integers (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' By combining our header file with the WASI-SDK, a C/C++ MPI application conforming to the MPI-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 standard can be compiled to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, to facilitate the ease-of-use and enable adoption, we imple- ment a custom python-based tool that simplifies the entire compilation process for MPI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 3 shows the different MPI-specific imports present in a Wasm module corresponding to the functions shown in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPIWasm provides definitions for these imports to enable the execution of MPI-based HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In addition, it supports the WASI specification which enables the POSIX functionality for MPI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3 Executing Wasm Code with High Performance There exist several strategies for executing Wasm modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These include using an interpreter [79], Ahead-of-Time (AoT) compilation [37], or Just-in-Time (JIT) compilation [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, for HPC systems the most useful approach is trans- lating the Wasm instructions (Wasm ISA) to the native instruc- tion set of the host machine before the application is executed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', AoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, MPIwasm builds on the code genera- tion infrastructure provided by Wasmer [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasmer currently supports three compiler backends, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', Singlepass [86], Cranelift [3], and LLVM [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The SinglePass compiler Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Comparing compile duration and performance for the different compiler backends supported by Wasmer [87] for the HPCG [73] Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm module was generated using our WASI-SDK (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm module is executed using MPIWasm on an x86_64 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Compiler Compile Duration (ms) Single-Core Performance (GFLOP/s) Singlepass [86] 52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3769 Cranelift [3] 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3240 LLVM [59] 2811 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5426 is designed to emit machine code in linear time and does not perform many code optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Cranelift com- piler is completely based on Rust and is similar to LLVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With Cranelift, the WASM instructions are first translated to the intermediate representation (IR) of Cranelift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', (Cranelift-IR) which are then translated to the native in- struction set of the host machine by taking microarchitecture- specific optimizations into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On the other hand, with LLVM the Wasm ISA is first translated to LLVM-IR followed by the generation of native machine code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Cranelift-IR is similar to LLVM-IR but at a lower level of abstraction which hinders mid-level code optimizations2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' At the end of the com- pilation process, all three compilers produce a shared ob- ject, which can be loaded with a fast dlopen call using the libloading library [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Table 1 shows a comparison of the compile-time and run- time performance of the three different compilers supported by Wasmer for the HPCG benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While LLVM is the slow- est to compile the Wasm module, it also results in the fastest runtime performance for the HPCG application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, we chose LLVM as the compiler backend in MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To offset the longer compilation times required by LLVM as com- pared to the other two compilers, we implement a caching mechanism for the generated machine code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our caching mechanism builds on the FileSystemCache [46] provided by Wasmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In our implementation, we generate a hash for each Wasm module using the Blake-3 hash function [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, we store the generated shared object from LLVM as the generated hash in the local filesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, any changes to the Wasm module lead to the generation of a new hash which triggers the recompilation of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, repeated execution of the same application on a system with MPIWasm will not lead to recompilation overhead for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 Filesystem Isolation with MPIWasm Since in Wasm all system interactions by the application have to be performed by calling functions implemented by the embedder (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2,§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3), it enables the embedder to place addi- tional restrictions on their use and to employ checks on the arguments supplied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In Wasmer, all exported functions 2A more detailed discussion between Cranelift-IR and LLVM-IR can be found here [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Memory address space of MPIWasm with an instan- tiated Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' All memory access instructions to the Wasm module’s linear address space are given offsets relative to the base address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' that handle file I/O perform their own permission handling that is separate from the one employed by the OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This in- process indirection of filesystem accesses allows Wasmer to present a virtual directory tree to the Wasm module that only contains directories that the module is allowed to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In addition, access rights to individual directories can be more restrictive than the permissions granted to the user that is exe- cuting the embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For instance, a user can have read and write access to their home directory and all of its subdirecto- ries, but grant read-only access to one specific subdirectory to a Wasm module executed by the embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPIWasm ex- poses this isolation functionality with its -d flag that grants read-write access to the given directory to the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Note that the full absolute path to the exposed directories is not presented to the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In the virtual direc- tory tree presented to it, all of the subdirectories it has been given access to are direct children of the root directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This approach to mapping directory paths avoids exposure of in- formation contained in the full path to the directories, such as a username in the case of a home directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 Translating from Wasm to Host Memory Address A major part of the Wasm security model is the separation of the host and the module’s linear memory address space (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Since it is the responsibility of the Wasm embedder to uphold capability restrictions, protecting it’s data structures from unintended or malicious access by the modules’ code is significantly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, this separation presents a challenge for supporting MPI applications, because the MPI API is based on the library being able to read and write di- rectly to the memory of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The executing Wasm- based MPI application can only provide memory addresses in its own linear memory address space, while the target MPI library requires addresses in the host memory address space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For executing Wasm modules, MPIWasm reserves a part of its own address space for use by the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, every byte contained in this range can be addressed either with a memory address in the module’s memory space or with a memory address in the embedder’s (host’s) memory space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, while instantiating the module’s linear mem- ory, MPIWasm records its base address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, it is possible to convert an address from the linear address space of the Wasm module to the embedder’s address space and vice- versa by treating the address in the linear address space as an offset relative to the module’s base address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In particular, MPIWasm directly converts 32-bit Wasm pointers that refer to the module’s linear address space to 64-bit pointers that refer to the embedder’s address space and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, MPIWasm directly utilizes the MPI library present on the host system without copying any data from the module’s address space to a different location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', it supports zero-copy memory operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our mechanism for memory address translation does not violate memory-safety because: (i) a malicious Wasm module cannot violate control flow integrity (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2) and (ii) since the size of the linear memory is always known, MPIWasm can perform runtime bound checks for all memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, a module cannot access the memory of the embedder or the memory of the underlying operating system unless explicitly given access to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 Translating MPI Datatypes MPI is implemented as a library with the most common being OpenMPI [10], MPICH [8], and MVAPICH [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Hence, it does not guarantee an Application Binary Interface (ABI) and interoperability between libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This means that chang- ing the MPI implementation requires recompilation of the entire application code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' One of the reasons for ABI incom- patibility is that the MPI standard does not specify explicit types for its datatypes such as MPI_Op and their implemen- tation is completely up to the MPI library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, since Wasm modules are designed to be portable not just between the different MPI libraries but also between different CPU architectures, it becomes necessary to add an abstraction be- tween the datatypes used by the host’s MPI library and the datatypes exposed to the Wasm module by MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' An abstraction is possible since most MPI datatypes such as MPI_Comm, MPI_Datatype and MPI_Op are opaque to the ap- plication and only used as arguments to MPI functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPI- Wasm defines most MPI datatypes as 32-bit integers from the perspective of the Wasm module (Listing 2) and transpar- ently translates these datatypes to the host equivalents (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We use integers as datatypes since MPIWasm internally uses IDs to identify data structures that it creates on behalf of the module in order to communicate with the host MPI library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7 Implementing MPI Functions in MPIWasm Wasm imports are referred to by namespace and name of the definition to import.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' By default, any symbols that are not defined while compiling C/C++ applications to Wasm will be resolved by making them imports of the module in the env namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is also demonstrated in Listing 3 with the function imports related to the MPI standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPI- Wasm provides definitions for all these functions with the same name as the original MPI function and exports them in the env namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For implementing these functions, we 0x0 OxFFFF_FFFFExploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada combine the memory address and MPI datatype translations as described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, we maintain a structure called Env that stores the global state required by these translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This structure includes infor- mation about the memory allocated to the Wasm module, it’s base pointer (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5) and information about the different used datatypes such as MPI_Comm by the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For directly utiliz- ing the host MPI library, we use the project rsmpi [7] in MPI- Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' rsmpi provides MPI bindings for Rust and supports OpenMPI [10] and MPICH [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It utilizes the rust-bindgen project to generate foreign function interfaces tailored to spe- cific MPI libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Each MPI function in MPIWasm directly calls the equivalent function in rsmpi with the appropriate arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While for most functions in the MPI-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 standard MPI- Wasm directly defers the execution to the host MPI library, the implementation of the MPI functions MPI_Alloc_mem and MPI_Free_mem is done differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With these functions, it is possible to allocate memory for use with other MPI functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' When a Wasm module calls MPI_Alloc_mem, it expects a 32- bit memory address in the module’s address space, while call- ing the MPI_Alloc_mem function of the host MPI library re- turns a 64-bit memory address in the embedder’s memory ad- dress space which is not inside the chunk of memory reserved for the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To overcome this, MPIWasm only supports MPI_Alloc_mem and MPI_Free_mem if the Wasm module defines and exports the functions malloc and free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' When MPI_Alloc_mem is called, MPIWasm simply invokes the exported malloc and receives a suitable 32-bit module memory address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This address can then be used as the return value for MPI_Alloc_mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We implement MPI_Free_mem in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='8 Limitations The Wasm specification currently assumes little-endian byte order for multi-byte values [52] in execution environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' By giving direct access to the Wasm module’s memory to the host MPI library, we assume that the byte order of values in the module address space and embedder address space is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, MPIWasm does not support big-endian CPU architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is not a disadvantage since most processor architectures in HPC systems are little-endian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, due to the current linear 32-bit memory space for a Wasm module, HPC applications compiled to Wasm cannot have more than 4GiB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The support for 64-bit memory addresses is an important milestone for the Wasm specification and is highlighted in the Wasm Memory64 proposal [20], but is out of scope for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4 Experimental Results In this section, we present performance results for our embed- der MPIWasm across different processor architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For 1 mpirun -np .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='/ mpiWasm mpi -app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='wasm Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Executing MPI applications compiled to Wasm with MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Comparing the size of native dynamically-linked, statically-linked, and Wasm binaries for the different MPI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The native applications are compiled for the x86_64 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Application Native Size Dynamic (KiB) Native Size Static (MiB) Wasm Size (KiB) Intel MPI Benchmarks [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1087 27 893 HPCG [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 164 26 722 IOR [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 364 16 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='32 IS [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 36 15 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='88 DT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 40 15 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='51 all our experiments, we follow best practices while reporting results [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 System Description For analyzing the performance of our implemented Wasm embedder, we use two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' First, a production HPC clus- ter located at our institute, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', SuperMUC-NG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Second, an AWS EC2 virtual machine (VM) instance with the Gravi- ton2 processor [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our HPC cluster contains eight islands comprising a total of 6480 compute nodes based on the Intel Skylake-SP architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Each compute node has two sockets, comprising two Intel Xeon Platinum 8174 processors, with 24 cores each and a total of 96GiB of main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The nominal operating core frequency for each core is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Hyper-Threading and Turbo Boost are disabled on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The internal interconnect on our system is a fast Intel Omni- Path [4] network with a bandwidth of 100 Gbit/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, our cluster provides a general parallel filesystem based on the Lenovo DSS-G for IBM Spectrum Scale [19] with an aggregate bandwidth of 200 GiB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For our experiments, we use up to 128 nodes of the HPC system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', 6144 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On the other hand, the AWS Graviton2 processor based on the 64-bit ARMv8-A Neoverse-N1 [24] architecture consists of 32 cores each with a nominal frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='50 GHz and a total main memory of 64GiB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We limit our experiments to one node for the Graviton2 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 HPC Benchmarks For our experiments with MPIWasm, we use the Intel MPI Benchmarks [55], two benchmarks from the the NASA Ad- vanced Supercomputing (NAS) Parallel Benchmark (NPB) suite [31], the IOR benchmark [60], and the High Perfor- mance Compute Gradient (HPCG) benchmark [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Intel MPI benchmarks perform a set of MPI perfor- mance measurements for point-to-point and global communi- cation operations for a range of message sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We use them since they characterize the performance of a cluster and are an indication of the efficiency of the used MPI implementa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The NPB suite includes a set of benchmarks that aim to evaluate the overall performance of HPC clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Due to the support for compiling Fortran to Wasm being in the early stages (§5), only the Integer Sort (IS) and Data Transfer (DT) PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 Bytes Iteration Time (usec) PingPong (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 101 102 Bytes Iteration Time (usec) PingPong (time) > 1024 Bytes Native WASM (a) PingPong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 2 4 Bytes Iteration Time (usec) Sendrecv 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 101 102 103 104 Bytes Iteration Time (usec) Sendrecv 6144 Ranks (time) > 1024 Bytes Native WASM (b) SendRecv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0 20 40 Bytes Iteration Time (usec) Bcast 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 100 102 104 Bytes Iteration Time (usec) Bcast 6144 Ranks (time) > 1024 Bytes Native WASM (c) Broadcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 22 24 26 28 210 40 60 Bytes Iteration Time (usec) Allreduce 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 102 104 Bytes Iteration Time (usec) Allreduce 6144 Ranks (time) > 1024 Bytes Native WASM (d) AllReduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0 1 2 104 Bytes Iteration Time (usec) Allgather 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 217 105 106 Bytes Iteration Time (usec) Allgather 6144 Ranks (time) > 1024 Bytes Native WASM (e) AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0 1 2 105 Bytes Iteration Time (usec) Alltoall 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 106 Bytes Iteration Time (usec) Alltoall 6144 Ranks (time) > 1024 Bytes Native WASM (f) Alltoall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 22 24 26 28 210 0 10 20 30 Bytes Iteration Time (usec) Reduce 768 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 100 102 104 Bytes Iteration Time (usec) Reduce 768 Ranks (time) > 1024 Bytes Native WASM 22 24 26 28 210 0 20 40 60 Bytes Iteration Time (usec) Reduce 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 100 102 104 Bytes Iteration Time (usec) Reduce 6144 Ranks (time) > 1024 Bytes Native WASM (g) Reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='210 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Native ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='WASM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='212 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='214 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='216217218219220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Iteration Time (usec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Gather 768 Ranks (time) > 1024 Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Native ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='WASM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='210 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Iteration Time (usec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Gather 6144 Ranks (time) ≤ 1024 Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Native ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='WASM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='212 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='214 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='216 217 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Iteration Time (usec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Gather 6144 Ranks (time) > 1024 Bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='Native ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='WASM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='(h) Gather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0 100 200 300 Bytes Iteration Time (usec) Scatter 768 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216217218219220 101 103 105 Bytes Iteration Time (usec) Scatter 768 Ranks (time) > 1024 Bytes Native WASM 20 22 24 26 28 210 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='000 Bytes Iteration Time (usec) Scatter 6144 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 217 101 103 105 Bytes Iteration Time (usec) Scatter 6144 Ranks (time) > 1024 Bytes Native WASM (i) Scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Performance comparison of the Intel MPI benchmarks for MPIWasm and their native execution on our HPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='8 1 Bytes Iteration Time (usec) PingPong (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 100 101 102 Bytes Iteration Time (usec) PingPong (time) > 1024 Bytes Native WASM (a) PingPong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 Bytes Iteration Time (usec) Sendrecv 32 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 100 101 102 103 Bytes Iteration Time (usec) Sendrecv 32 Ranks (time) > 1024 Bytes Native WASM (b) SendRecv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 22 24 26 28 210 2 4 6 8 Bytes Iteration Time (usec) Allreduce 32 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 101 102 103 104 Bytes Iteration Time (usec) Allreduce 32 Ranks (time) > 1024 Bytes Native WASM (c) AllReduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 0 10 20 Bytes Iteration Time (usec) Allgather 32 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 102 103 104 105 Bytes Iteration Time (usec) Allgather 32 Ranks (time) > 1024 Bytes Native WASM (d) AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 20 40 Bytes Iteration Time (usec) Alltoall 32 Ranks (time) ≤ 1024 Bytes Native WASM 212 214 216 218 220 222 102 104 Bytes Iteration Time (usec) Alltoall 32 Ranks (time) > 1024 Bytes Native WASM (e) Alltoall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 12 4 8 16 32 0 10 20 Ranks GFLOP/s HPCG GFLOPS Native WASM 12 4 8 16 32 0 50 100 150 Ranks GB/s HPCG Bandwidth Native WASM (f) HPCG Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Performance comparison of selected Intel MPI benchmarks and HPCG for MPIWasm against their native execution on the AWS Graviton2 Processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' benchmarks from this suite were used since they are written in pure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The IS benchmark performs bucketed parallel sort- ing of integers across all participating processes, while the DT benchmark tests the communication and the performance of 64-bit floating point operations of a HPC cluster by send- ing data through a topology of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We use the topologies Black-Hole (bh), White-Hole (wh), and Shuffle (sh) for the DT benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For our experiments, we use the classes C and B for the IS and DT benchmarks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The IOR Benchmark measures the filesystem I/O performance avail- able to MPI processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It supports multiple backends that utilize different APIs to perform system I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For our experi- ments with MPIWasm, we use the POSIX API backend since the POSIX filesystem APIs are included in the WASI specifi- cation (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The HPCG benchmark aims to evaluate the real-world performance of HPC systems by solving a sys- tem of linear equations with the conjugate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For our experiments, we use the default available problem Exploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada size for HPCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Note that, in our experiments we use the ver- sions 2019 Update 6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 for the Intel MPI and NAS parallel benchmarks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3 Experiment Setup For all our experiments, we execute the benchmarks in a pure- MPI configuration without shared memory parallelization with OpenMP, as it is currently not supported by MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We use OpenMPI-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='0 as the MPI library since it is available on our HPC system and can be easily installed on the AWS Graviton2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For compiling the native applications on our HPC system (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1), we use the clang-11 compiler, while for the AWS Graviton2 node, we use the gcc7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In both cases, the applications were compiled with the -O3 optimization flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For compiling the different benchmarks to Wasm, we use the clang-11 compiler along with our cus- tomized WASI-SDK with -O3 -msimd128 flags for both test systems (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The -msimd128 flag enables the generation of SIMD instructions in Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We compile the applications to Wasm only once on our local systems and execute them directly with MPIWasm on the different test systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We build MPIWasm on our local system for the different plat- forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', x86_64 and aarch64 with OpenMPI to generate bindings for rsmpi (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, we directly exe- cute the applications compiled to Wasm on the test systems as shown in Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Each MPI rank corresponds to one instance of the embedder with it’s own Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The native applications were executed directly using mpirun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 Comparing Wasm Binary Size Table 2 shows the comparison between the absolute binary sizes for the different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The static versions of the binaries are generated by supplying the -static flag to the clang-11 compiler (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3) and linking the different applica- tions with the static versions of the required libraries such as libmpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='a, libopen-rte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='a, and libz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To this end, we made necessary changes to the Make [50] and CMake [39] files used by the different applications (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While the stack- based instruction set and compact binary format give Wasm the potential to produce smaller binaries for the same appli- cations as compared to the native dynamically-linked bina- ries [52], three out of five applications that we used had a bigger binary size when compiled to WebAssembly in com- parison to the equivalent dynamically-linked native binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While Wasm can benefit from a smaller representation on a function-by-function basis, in practice dynamically-linked native binaries can offset that advantage by being able to rely on commonly used libraries to be present on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For instance, a native binary can dynamically link against glibc, while a Wasm binary must statically include functions from wasi-libc (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, in contrast to containers, Wasm binaries are significantly smaller making them more feasible for application distribution in HPC environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In addi- tion, Wasm binaries are 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5x smaller on average than the statically-linked binaries of the different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This is because the linker, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', lld copies all library routines from the different libraries used by an application into the binary during static linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 Benchmarking MPIWasm Figure 3 and Figure 4 show the iteration times for the different Intel MPI benchmarks for their native execution as compared to their execution with MPIWasm on our HPC system and the AWS Graviton2 processor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For execution with MPIWasm, the iteration times don’t include the time required for compiling the Wasm modules to native machine code (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To avoid repetition, we omit some results for the Graviton2 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Error bars in the graphs represent mini- mum and maximum values for iteration timings as reported by the Intel MPI Benchmarks, while points in the graphs represent the average timings as reported by the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the PingPong benchmark using MPIWasm leads to a geometric mean (GM) average slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='05x for the x86_64 system and a GM average speedup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='01x for the aarch64 system across all message sizes (Figures 3a, 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We calculate this value by dividing the metric t_avg_us reported by the benchmarks for their native execution by the value reported for execution with MPIWasm, followed by a GM of the obtained values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For computing slowdown, we subtract one from the obtained GM value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We observe a maximum bandwidth of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='80 GiB/s and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='98 GiB/s for the native exe- cution of the PingPong benchmark on the HPC and Graviton2 processor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On the other hand, with MPIWasm, we observe a maximum bandwidth of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='44 GiB/s and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='61 GiB/s on the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the SendRecv benchmark, we observe a GM average slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='06x and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='07x with MPIWasm across all message sizes on the x86_64 and aarch64 systems respectively (Figures 3b, 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the native version of the benchmark, we observe a maximum bandwidth of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='24 GiB/s and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='01 GiB/s on the two systems, while with MPIWasm, we observe a maximum bandwidth of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='50 GiB/s and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='83 GiB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the collective communication Broadcast routine, we observe an average GM slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='13x with MPIWasm across all message sizes for 128 nodes as shown in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We observe an average GM slow- down of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='06x and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='10x with MPIWasm across all message sizes for the collective communication AllReduce routine as shown in Figures 3d and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For AllGather with MPI- Wasm, we observe an average GM slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='06x and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='09x across all message sizes for the HPC system and Gravi- ton2 processor respectively (Figure 3e, 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Similarly, for the Alltoall collective communication routine, we observe an average GM slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='10x for the two systems across all message sizes with MPIWasm as shown in Figures 3f and 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For 16 nodes of our HPC system, we observe an average GM slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='12x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='14x, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='05x across message sizes for the routines Reduce, Gather, and Scatter as shown in Figures 3g, 3h, and 3i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On the other hand, for 128 nodes, PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 64128 256 512 1,024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 104 Ranks Mop/s IS Total Operations/s Native WASM bh wh sh 0 500 1,000 1,500 Topology MB/s DT Total Throughput Native WASM w/o SIMD WASM w SIMD (a) NPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 4 8 12 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 104 Block Size (MiB) MiB/s IOR Bandwidth (Read) Native WASM 1 4 8 12 16 3 4 104 Block Size (MiB) MiB/s IOR Bandwidth (Write) Native WASM (b) IOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 48 16 48 96 144 0 20 40 60 80 Ranks GFLOP/s HPCG GFLOPS Native WASM 48 16 48 96 144 0 200 400 600 Ranks GB/s HPCG Bandwidth Native WASM 192 768 1,536 3,072 6,144 0 2,000 4,000 Ranks GFLOP/s HPCG GFLOPS Native WASM 192 768 1,536 3,072 6,144 0 1 2 3 104 Ranks GB/s HPCG Bandwidth Native WASM (c) HPCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Performance comparison of standardized HPC benchmarks for MPIWasm against their native execution on our HPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' we observe an average GM slowdown of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='05x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='10x, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='08x for the three routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The results for testing MPIWasm with the Intel MPI Benchmarks compiled to Wasm demon- strate that neither the mechanism for calling host functions in Wasmer [87] nor the translation layer implemented in MPI- Wasm induce significant overhead for MPI communication (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5,§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6,§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We expand on the translation overhead in MPIWasm in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Overall, our results indicate that MPI- Wasm delivers close to native performance for the different MPI routines on both x86_64 and aarch64 architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The performance of MPIWasm on our HPC system for the IS and DT benchmarks is shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the IS benchmark with MPIWasm, we observe 8260 average mega operations per second across all processes as compared to 8546 average mega operations per second for the native exe- cution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the DT benchmark with different topologies (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2), execution with MPIWasm leads to decreased throughput as compared to the native execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The DT benchmark per- forms a significant number of pairwise comparison opera- tions which benefit greatly from vectorization with SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To demonstrate the effect of SIMD for the DT benchmark, we compile it to Wasm by disabling and enabling the generation of SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm version of the DT benchmark with SIMD leads to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='36x better throughput as compared to the Wasm version without SIMD (Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The difference in performance as compared to the native ver- sion of the DT benchmark can be attributed to the support for only 128-bit SIMD instructions in the Wasm specifica- tion [52] as compared to 512-bit SIMD instructions present in modern Intel processors [38, 57, 76] (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Support for higher-width SIMD in Wasm is an important milestone in its road-map but out of scope for this work (§5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 5b shows total aggregated read and write band- width available to all MPI processes for the IOR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Points in the graph represent the average bandwidth reported by the benchmark, while error bars in the graph represent the maximum and minimum bandwidth observed over all iterations of the benchmark with the same block size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With four nodes of our HPC system, the upper bound for achiev- able bandwidth in our setup (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1) with IOR is 400 GBit/s (≈ 47684 MiB/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With MPIWasm, we observe similar read (29411 MiB/s) and write (40206 MiB/s) bandwidth averaged across all block sizes as compared to the native execution of the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Testing the filesystem I/O performance of the MPIWasm demonstrates that the userspace permission handling and virtual directory tree implemented by Wasmer to provide filesystem isolation (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4) has no significant im- pact on the achievable bandwidth when performing I/O with the POSIX filesystem API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For the HPCG benchmark, we ob- serve similar performance when executed with MPIWasm as compared to it’s native execution on the HPC system and the Graviton2 processor up to 192 MPI processes (Figures 5c and 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On increasing the number of processes, the native exe- cution of the HPCG benchmark outperforms the execution with MPIWasm as shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For 6144 MPI processes, we observe a 14% reduction in GFLOP/s on execution with MPI- Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This behavior can be attributed to the significantly fre- quent amount of communication required by the HPCG bench- mark [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' HPCG repeatedly uses the Allreduce routine to reduce a single variable of size double over all MPI processes to finalize vector-vector dot operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With increasing num- ber of processes, the number of times the Allreduce routine is called also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For instance, executing HPCG with 768 processes results in four times more calls to Allreduce as compared to the execution with 192 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a re- sult, the repeated datatype translations in MPIWasm increase the cost for invoking the collective communication routine leading to performance degradation (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 Analyzing Datatype Translation Overhead To measure the datatype translation overhead in MPIWasm, we implement a custom PingPong application that sends/re- ceives messages of varying sizes between two processes and iterates over the different MPI datatypes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', BYTE, CHAR, INT, FLOAT, DOUBLE, and LONG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, we instru- ment the Send routine in MPIWasm to determine the latency for translating the different datatypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Finally, we execute the application on our HPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 6 shows the trans- lation overhead for different datatypes and message sizes in MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We observe an average overhead of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='44ns, Exploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada 8 64 256 1024 32768 262144 1048576 2097152 4194304 Bytes 0 50 100 150 200 250 300 Translation Time (ns) MPI_BYTE MPI_CHAR MPI_INT MPI_FLOAT MPI_DOUBLE MPI_LONG Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Comparing the datatype translation overhead in MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='72ns, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='78ns, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='32ns, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='35ns, and 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='79ns across all message sizes for the MPI datatypes BYTE, CHAR, INT, FLOAT, DOUBLE, and LONG respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We observe an increase in the translation overhead for message sizes greater than 256KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This can be attributed to an increased latency for acquiring read locks from the Env structure that maintains the global state for translations in MPIWasm (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 5 Into the Future: Wasm and HPC In this section, we highlight and discuss the different exten- sions proposed by the Wasm community to the current Wasm specification [52] that can be implemented in an embedder for HPC applications to enhance performance and portability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Controlled Threading for Wasm modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The Wasm Threads proposal [16] lays the foundation for utilizing Wasm for multithreaded algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It enables Wasm modules to de- fine shared memories, informing the embedder that the mod- ule expects the memory to be accessed by multiple threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To enable safe multithreaded access to shared memory, the proposal also defines atomic Wasm instructions that can be used to implement locks and atomic data structures in the functions of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To enable HPC applications to make use of the functionality added by the threads proposal, an API that allows Wasm modules to create additional threads on its own needs to be added to the embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Implementing the POSIX threads [15] and OpenMP [41] APIs in the embed- der would enable compatibility with the threading code in existing HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm Extended SIMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The current Wasm SIMD pro- posal [52] only specifies 128-bit SIMD instructions, while modern CPUs support higher-width-SIMD, for instance the AVX-512 instruction set extensions for x86_64, which speci- fies 512-bit SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, the Wasm Flex- ible Vector proposal [13] aims to provide support for SIMD instructions that are wider than 128-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, the Wasm relaxed SIMD instructions [21] aim to make it possible to uti- lize hardware SIMD instructions that are not well defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', they differ in rounding behavior from the Wasm specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Implementing these proposals in the embedder would allow compiled Wasm modules for HPC applications that contain vectorizable code to make better use of SIMD instructions available in modern CPU architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 20 22 24 26 28 210 2 4 6 Bytes Iteration Time (usec) PingPong (time) ≤ 1024 Bytes MPIWasm Faasm 212 214 216 218 220 222 100 101 102 103 Bytes Iteration Time (usec) PingPong (time) > 1024 Bytes MPIWasm Faasm Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Comparing the performance of MPIWasm and Faasm [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm PGAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Partitioned Global Address Space (PGAS) is a programming model for parallel distributed memory ap- plications that introduces a memory address space that spans the local memory of multiple processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With a memory ad- dress from this global address space a process can read from and write to the memory of other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Since Wasm already specifies the concept of defining and importing mem- ories, the embedder could be extended to provide non-local memory to the Wasm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To support this use-case, the Wasm Multi-Memory proposal [14] needs to be implemented, which allows a Wasm module to define or import more than one memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Dynamic Linking of Wasm Modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' While there is ex- isting work on establishing an ABI for dynamic linking be- tween Wasm modules [12], it has not been standardized yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Supporting dynamic linking would significantly decrease the size of Wasm binaries for more complex applications as they would no longer need to statically link parts of wasi-libc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For HPC, it would enable commonly used libraries such as BLAS to be provided by MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Combining dynamic link- ing with efforts to provide repositories for Wasm modules such as WAPM [88] could enable automatic dependency man- agement for Wasm applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Compiling Fortran applications to Wasm Currently, the support for compiling Fortran-based applications to Wasm is very nascent with only one known attempt based on Dragon- egg [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, the implementation of the Memory64 proposal [20] should enable the usage of existing Fortran LLVM compilers such as F18 for easily compiling Fortran- based applications to Wasm [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Wasm and Accelerators The module execution hints pro- posal [23] highlights the changes required in the Wasm speci- fication to enable the support for executing Wasm modules on hardware accelerators such as GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Implementing the proposal in the embedder would enable compatibility with existing GPU-based HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 6 Related Work Solutions for Packaging and distributing HPC applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Recently several HPC-focused tools such as Char- liecloud [72] and Singularity [58] have been introduced for distributing HPC applications through containerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In contrast, we utilize the universal binary instruction format Wasm to package and distribute HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' while containerization requires building HPC application con- tainers for different platforms, HPC applications can be com- piled once to Wasm and executed on any platform using a supporting Wasm embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPI and WebAssembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To the best of our knowledge, Faasm [78] is the only compute platform that enables the exe- cution of MPI applications compiled to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It is based on a gRPC-based distributed messaging library called Faabric and contains a Wasm runtime, a workload scheduler, and a distributed state store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In order to run an application on Faasm, it needs to be compiled to Wasm and uploaded to the shared function storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, the application can be invoked using events such as HTTP requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For support- ing MPI applications, Faasm implements only a subset of the MPI-1 specification on top of its messaging library and its own workload scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Moreover, it also does not sup- port user-defined communicators required by the Intel MPI benchmarks (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In contrast, we take an inverted approach, where MPIWasm builds on top of existing native MPI libraries and provides a way for Wasm modules to call functions from them efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Figure 7 shows the performance comparison between MPIWasm and Faasm for the PingPong benchmark (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' With MPIWasm, we achieve a GM average speedup of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='28x across all message sizes as compared to Faasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 7 Conclusion and Future Work In this paper, we took the first step towards bringing We- bAssembly to the HPC ecosystem and presented MPIWasm, a Wasm embedder that enables high performance execution of MPI applications compiled to Wasm across different proces- sor architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In the future, we plan to extend MPIWasm to support acceralators such as GPUs found on HPC sys- tems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 8 Acknowledgement We thank our shepherd Milind Chabbi for his help in prepar- ing the final version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Furthermore, we thank the anonymous reviewers for their insightful comments and valuable feedback that significantly improved the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This work was supported by the funding of the German Federal Ministry of Education and Research (BMBF) in the scope of the Software Campus program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' References [1] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' AWS Graviton 2 Processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://aws.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' wapm is the WebAssembly Package Manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://wapm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='io [89] WebAssembly Community Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' WebAssembly System Inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='com/WebAssembly/WASI [90] Miguel G Xavier, Marcelo V Neves, Fabio D Rossi, Tiago C Ferreto, Timoteo Lange, and Cesar AF De Rose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Performance evalua- tion of container-based virtualization for high performance computing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In 2013 21st Euromicro International Conference on Parallel, Distributed, and Network-Based Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' IEEE, 233–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1109/PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='41 [91] Yutian Yan, Tengfei Tu, Lijian Zhao, Yuchen Zhou, and Weihang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Understanding the Performance of Webassembly Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In Proceedings of the 21st ACM Internet Measurement Conference (Virtual Event) (IMC ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 533–549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1145/3487552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3487827 [92] Andy B Yoo, Morris A Jette, and Mark Grondona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Slurm: Simple linux utility for resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' In Workshop on Job Scheduling Strategies for Parallel Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Springer, 44–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A Artifact Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 Description MPIWasm is an embedder for MPI-based HPC applications based on Wasmer [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' It enables the high performance execu- tion of these applications compiled to WebAssembly (Wasm) and serves two purposes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Delivering close to native application performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', when applications are executed directly on a host ma- chine without using Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Enabling the distribution of MPI-based HPC applica- tions as Wasm binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Our artifact contains the source code for MPIWasm, toolchain for compiling C/C++ based MPI applications to Wasm, the Wasm binaries for the different standardized HPC bench- marks used in this paper, pre-built versions of our embedder for different operating systems, and scripts for parsing experi- ment data and generating plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The artifact is available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7468121 or https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='com/kky-fury/MPIWasm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 Getting Started For testing our Wasm embedder for executing MPI appli- cations compiled to WebAssembly, we provide a pre-built docker image for the linux/amd64 platform with all the re- quired dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 sudo docker run − i t kkyfury / ppoppae : v2 / bin / bash 2 #Executing the HPCG benchmark compiled to Wasm 3 mpirun −−allow −run −as − r o o t −np 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / t a r g e t / r e l e a s e / embedder \\ 4 examples / xhpcg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' wasm 5 #Executing the IntelMPI benchmarks compiled to Wasm 6 mpirun −−allow −run −as − r o o t −np 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / t a r g e t / r e l e a s e / embedder \\ 7 examples / imb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' wasm Listing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Getting started with MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Towards this, a user can follow the steps described in Listing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, MPIWasm should successfully execute the HPCG and IntelMPI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide sample output for the two benchmarks in the provided artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The user can increase/decrease the number of processes (-np) for executing the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, depending on the system where Exploring the Use of WebAssembly in HPC PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada the container is executing, the user might need to provide the oversubscribe flag to mpirun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3 Running Experiments with MPIWasm This section describes how to run experiments with our em- bedder to obtain plots similar to the ones in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 Running small-scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' To run small-scale experiments inside the docker container, we provide an end- to-end script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This script: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Executes the HPCG, IS, and IntelMPI benchmarks for their native execution and when they are executed using MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Parses the obtained results and generates the relevant plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 sudo docker run − i t kkyfury / ppoppae : v2 / bin / bash 2 cd run_experiments 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / runme .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' sh Listing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Running small-scale experiments with MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For running the experiments, the user can follow the steps de- scribed in Listing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The script can take around 10-15 minutes to finish execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' After completion, you can see the gener- ated data in the run_experiments/experiment_data folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The generated plots can be found in the run_experiments/Plots folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide sample plots for the different benchmarks in our artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' However, on executing benchmarks inside the container, the performance difference between the native execution of the application and MPIWasm can be around 8-12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 Running large-scale experiments on an HPC sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For running large-scale experiments with our embedder, a user needs to do the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Build a version of the embedder for your HPC sys- tem depending on the particular architecture, operating system, and the MPI library on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MPIWasm currently supports the OpenMPI library with limited sup- port for MPICH and MVAPICH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide examples for building MPIWasm for different operating systems and architectures in our artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Execute the MPI applications using the built embedder on the HPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This can be done via submitting jobs to a RJMS software on an HPC system such as SLURM [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide sample job scripts for our HPC system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', SuperMUC-NG that uses SLURM in our artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' After executing the applications, the user can utilize the different parsers provided in our artifact to parse the benchmark data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Following this, the results can be visualized using the plotting helper provided in the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='4 Compiling C/C++ applications to Wasm We have setup a docker container with the required depen- dencies for compiling different MPI applications conformant to the MPI-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 standard to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The artifact also includes HPCG, IntelMPI, and IS benchmarks as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 sudo docker run − i t kkyfury / w a s i t o o l c h a i n : v1 / bin / bash 2 #Compiling HPCG 3 cd / work / example / hpcg −benchmark 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / wasi −cmake .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' sh 5 cd cmake− build −wasi 6 make Listing 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Compiling applications to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 7 describes the steps a user can follow for compiling the HPCG benchmark to Wasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For steps to compile the other benchmarks to Wasm, please look at the base Readme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='md file provided with the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' All the different applications compiled to Wasm that we used in this paper are also present in the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='5 Using MPIWasm For detailed usage instructions, please look at the base Readme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='md file provided with the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6 Modifying MPIWasm For modifying our embedder, we recommend using our pro- vided docker-compose file in the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' This docker-compose file mounts the volume with the embedder’s source code in- side the container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' As a result, any changes to it’s source code will be reflected inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' For our embedder, we currently support the following operating systems: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' CentOS-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Opensuse-15-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Ubuntu-20-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' MacOS-monterey 1 cd wasi −mpi− r s 2 docker compose run centos −8−2 3 cargo b u i l d −− r e l e a s e 4 #After the build process , you can see the built embedder in 5 # the /target/release/ folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Building MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 8 describes the steps for building MPIWasm for CentOS-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='2 after modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The instructions for building the embedder for other operating systems are provided in the base Readme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='md file inside the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' After the build process, the embedder can be copied to the user’s local filesystem using the docker-cp command as shown in Listing 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 docker cp < c o n t a i n e r −id > : / s / t a r g e t / r e l e a s e / embedder \\ 2 < d e s t i n a t i o n −path −user − f i l e s y s t e m > Listing 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Copying MPIWasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide provide the base image dockerfiles for the dif- ferent supported operating systems inside the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' These example dockerfiles can be easily extended to support other different linux distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada Mohak Chadha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7 Support for aarch64 Our embedder also supports execution on linux/arm64 plat- forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' We provide pre-built versions of our embedder for arm64 for the different supported operating systems in the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='1 Building images for aarch64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' If the user is building the docker image on an x86_64 system then docker-buildx is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Note that, in this case, building the image might take around 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 1 sudo docker buildx c r e a t e −−name mybuilder −−use −− b o o t s t r a p 2 cd wasi −mpi− r s / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' g i t l a b / c i / images / 3 sudo docker buildx b u i l d −−push −f ubuntu −20 −04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' D o c k e r f i l e \\ 4 −− p l at f or m l i n u x / arm64 \\ 5 − t kkyfury / ubuntumodifiedbase : v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' 6 cd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' / 7 sudo docker buildx b u i l d −−push −f D o c k e r f i l e \\ 8 −− p l at f or m l i n u x / arm64 \\ 9 − t kkyfury / embedderarm : v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Building MPIWasm for aarch64 on x86_64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' Listing 10 describes the steps required for building MPI- Wasm for arm systems with docker-buildx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' The user should change the docker image tags according to their docker reg- istry account, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=', replace kkyfury with your registry user- name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content=' On the other hand, if the user is using an aardch64 system then please follow the instructions described in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfkweU/content/2301.03982v1.pdf'} diff --git a/69E4T4oBgHgl3EQfCAuy/content/tmp_files/2301.04857v1.pdf.txt b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/2301.04857v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96a20109005e0e82cca8c28319582b07591f1f4a --- /dev/null +++ b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/2301.04857v1.pdf.txt @@ -0,0 +1,1309 @@ +Neural Spline Search for Quantile Probabilistic Modeling +Ruoxi Sun1*, Chun-Liang Li1*, Sercan Ö. Arık1, Michael W. Dusenberry2, Chen-Yu Lee1, Tomas +Pfister 1 +1Google Cloud AI 2Google Research, Brain Team +{ruoxis, chunliang, soarik, dusenberrymw, chenyulee, tpfister}@google.com +Abstract +Accurate estimation of output quantiles is crucial in many use +cases, where it is desired to model the range of possibility. +Modeling target distribution at arbitrary quantile levels and at +arbitrary input attribute levels are important to offer a compre- +hensive picture of the data, and requires the quantile function +to be expressive enough. The quantile function describing the +target distribution using quantile levels is critical for quantile +regression. Althought various parametric forms for the distri- +butions (that the quantile function specifies) can be adopted, +an everlasting problem is selecting the most appropriate one +that can properly approximate the data distributions. In this +paper, we propose a non-parametric and data-driven approach, +Neural Spline Search (NSS), to represent the observed data +distribution without parametric assumptions. NSS is flexible +and expressive for modeling data distributions by transform- +ing the inputs with a series of monotonic spline regressions +guided by symbolic operators. We demonstrate that NSS out- +performs previous methods on synthetic, real-world regression +and time-series forecasting tasks. +Introduction +For many machine learning applications, modeling the pre- +diction intervals (e.g. estimating the ranges all individual +predictions observation fall), beyond point estimates, is cru- +cial (Salinas et al. 2020; Wen et al. 2017; Tagasovska and +Lopez-Paz 2019; Gasthaus et al. 2019; Pearce et al. 2018). +The prediction intervals can help with decision making for +retail sales optimization (Simchi-Levi et al. 2008), medi- +cal diagnoses (Begoli, Bhattacharya, and Kusnezov 2019; +Mhaskar, Pereverzyev, and van der Walt 2017; Jiang et al. +2012), information safety (Smith, Dinev, and Xu 2011), fi- +nancial investment management (Engle 1982), robotics and +control (Buckman et al. 2018), autonomous transformation +(Xu et al. 2014) and many others. +To estimate prediction intervals, we would need to estimate +different levels of quantiles for the target distribution using +quantile regression (Koenker and Regression 2005; Wald- +mann 2018). A real-world challenge is to select the paramet- +ric forms of target distributions, which is specified by the +quantile function (also known as the inverse CDF function), +*These authors contributed equally. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +X +2 +1 +0 +1 +2 +Y +X=X0 +P(Y|X) +True +Quantile 25% +Quantile 75% +Figure 1: Modeling multiple quantiles at different +condition-levels with a universal quantile function. The +goal is to model target data distribution y at any arbitrary +quantile level and attribute level X, using one versatile quan- +tile function. Gray dots are observed data points, while green +and blue lines indicate 25% and 75% quantile levels. The +data distribution y varies at different levels of X, say variance +of y increases when X is away from zero. Red dots are data +points at X = X0, p(Y |X0)). +to properly align with observed data distribution. Different +choices for the target distribution (Gaussian, Poisson, Neg- +ative Binomial, Student-t etc.) may yield different quantile +predictions, and misalignment of the assumption with the real +distribution may hinder the performance of the model. There- +fore, such heuristic or empirical hand-picking based paramet- +ric assumptions for the distribution can be sub-optimal. An +approach based on learning from the data in an automated +way, would be highly desirable, from both foundational and +practical perspectives. +For learnable parametric modeling, one challenge is how to +model all quantiles for all input attributes level in a com- +putationally efficient way. First, modeling an any arbitrary +quantile, as opposed to a couple of pre-defined quantile levels, +offers a more comprehensive view on the target distribution, +and provides convenience to use the quantile model (e.g. no +need to re-train the model when quantiles at testing are dif- +ferent from the ones at training). Second, real-world data can +have complex distributions beyond what simple assumptions +can model. Fig. 1 shows different input attribute X levels +arXiv:2301.04857v1 [cs.AI] 12 Jan 2023 + +0 +1 +2 +3 +4 +5 +Y|X +0.000 +0.005 +0.010 +0.015 +0.020 +Probability Density +PDF(Y|x) +0 +1 +2 +3 +4 +5 +Y|X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Probability Density +CDF(Y|x) +Figure 2: An example target distribution with a complex +shape, in PDF and CDF space. Black lines are observed tar- +get distributions, in the form of mixture of the other three dis- +tributions shown with color. Fitting the black line accurately +would be extremely difficult for most of the commonly-used +single parametric splines, motivating for the use of learnable +spline family composed of multiple splines. +have different dependency dynamics with target y level (i.e. +the variance of y increases when X apart from 0). Fig. 2 +shows that the observed distribution cannot trivially fit well +with one single distribution. Therefore, in order to model all +quantiles at all X, we need a quantile function with a com- +plexity that does not increase significantly with number of +input attributes and the number of quantiles. This necessitates +a versatile and highly-expressive quantile function. +There has been many efforts on improving various aspects +of quantile regression. Gasthaus et al. (2019) proposes linear +spline interpolation between knots in the inverse CDF space +to model the target distribution in time-series forecasting +setup. This is proposed to avoid the assumption on paramet- +ric form of the target distribution. Park et al. (2022) and Moon +et al. (2021) focus on learning a valid quantile function with- +out quantile crossing (e.g. quantiles violate monotonically +increasing property), via special design of the neural network +architecture or first-order inequality constraint optimization. +Despite being distribution agnostic, these approaches for de- +scribing the target distribution (specified by quantile function) +are restricted to one function family (e.g. linear spline), which +may limit the expressiveness to represent the target distribu- +tion. In this paper, with the goal of designing an expressive +quantile function for various quantiles and input levels, we +propose a data-driven approach Neural Spline Search (NSS), +which transforms the inputs with a series of monotonic spline +regressions guided by symbolic operators. The contributions +of our paper can be summarized as: +1. We propose an efficient search space and mechanism to +find an expressive quantile function to model the data +distribution, avoiding specifying a parametric form of the +observed distribution as prior. +2. We propose a novel approach to generate an expressive +quantile function using a combination of different distri- +butions and operators guided by symbolic operators. +3. The proposed method can be incorporated into other tasks +(including but not limited to time series forecasting) as +their quantile function. +4. We demonstrate significant accuracy improvements across +numerous regression or time series forecasting tasks. For +example, on UCI benchmarks, we show 3.5%-7.0% im- +provement compared to next best methods. +Related Work +Quantile regression is used to estimate the target distribu- +tion at different quantile levels. The α-quantile estimator +is the solution when minimizing quantile loss at level α +(Koenker and Bassett Jr 1978). Another quantile regression +related loss is continuous ranked probability score (CRPS) +(Gneiting and Raftery 2007), which is the averaging over all +quantile levels, instead of one single quantile. +Neural network quantile forecasting. To model sequential +dependency of time series, several forecasting models pro- +pose a hidden state-emission framework ((Salinas et al. 2020; +Wen et al. 2017; Gasthaus et al. 2019; de Bézenac et al. 2020; +Wang et al. 2019)), where the dynamics of hidden states +are modeled by auto-regressive recurrent neural works (e.g. +LSTM), which takes previous hidden states and current ob- +servations as input and outputs current observation. Different +from modeling the likelihood with parametric distributions +(e.g. Gaussian (Salinas et al. 2020)), emission models for +quantile estimation is to learn the parameters of quantile +function. The overall framework is optimized by employing +a quantile (Wen et al. 2017) or CRPS (Gasthaus et al. 2019) +loss. +Symbolic regression has shown great success in many fields, +including program synthesis (Parisotto et al. 2016), mathe- +matical expressions extraction (Cranmer et al. 2020), physics- +based learning (Li et al. 2019; Petersen et al. 2019). As the +search space is enormous and scaled exponentially with the +length of operators, symbolic regression rule operators are +usually set to be a small number and are learned by Monte +Carlo Tree Search guided evolutionary strategies (Li et al. +2019) or reinforcement learning (Petersen et al. 2019). +Methods +Learning quantile function in quantile regression +Let the input data attributes X and the target variable y are +jointly distributed as p(X, y). The conditional cumulative +distribution function (CDF) is F(Y = y|X) = P(Y ≤ +y|X). The quantile function, which is also called the inverse +CDF function, takes quantile level as inputs and returns a +threshold value Y below which random draws from the given +CDF would fall quantile percent of the time. Specifically, the +α-th quantile function of y|X = x is denoted as: +q(α, x) = F −1 +y|X=x(α) = inf{y : F(y|X = x) ≥ α} +(1) +Here we can think the quantile function is to perform a +transformation on a uniform-distributed random variable +α ∼ U(0, 1) to the target distribution p(y|X). Quantile +function is able to fully specify a distribution. So specifying +the quantile function is describing the target distribution +p(y|X). +Quantile regression estimates different conditional quantile +levels of the target variable given a certain level of input + +P(y|X) +Inverse CDF +alpha +y +P(y|X) +Inverse CDF +alpha +y +P(y|X) +Inverse CDF +alpha +y +P(y|X) +Inverse CDF +alpha +y +Spline Basis +P(y|X) +Inverse CDF +alpha +y +S +C ++ +P(y|X) +Inverse CDF +alpha +y +Spline Basis +P(y|X) +Inverse CDF +alpha +y +.... +P(y|X) +Inverse CDF +alpha +y +P(y|X) +Inverse CDF +alpha +y +... +.... +.... +P(y|X) +Inverse CDF +alpha +y +P(y|X) +Inverse CDF +alpha +y +.... +.... +NSS-sum +Initial distribution +target distribution +Spline Basis +Spline Basis +NSS-chain +P(y|X) +Inverse CDF +alpha +y +Operators ++ +Sum +S +Scale +C +Chaining +.... +.... +Figure 3: Overview of Neural Spline Search (NSS). Modeling the target data distribution can be done by learning the quantile +function (e.g. inverse CDF), which maps a [0, 1]-variable (quantile) to a target value y. Unlike parametric methods which specify +a distribution family and learn the parameters, NSS can generate the target distribution through a set of transformations on the +inverse CDF space (quantile space), where the transformation is guided by a series of operators. Here, the bottom gray box shows +possible operators (denoted as circles), including but not limited to summation (“+”), scale (“S”), and chaining (“C”). The basis +splines are shown with color-shaded squares. The initial distribution is a uniform distribution, as shown in the leftmost panel +(blue shaded), and the target distribution is the rightmost distribution (purple shaded). There is no obvious parametric distribution +to achieve this transformation. Therefore, NSS is used to search for the suitable transformation through simple operators. In +the first row of the middle panel, we show operators for NSS-sum, where the initial uniform distribution is transformed by +the red- and the yellow-shaded splines (e.g. c-spline) through sum (“+”) and scale (“S”) operators. The second row shows the +chaining transformation of the initial distribution, where the orange and cyan splines are used to transform the initial spline. The +parameters of the splines are learned by a neural network. In general, the operators and transformations in NSS are not limited to +two splines (we represent them as the gray splines next to the yellow and cyan shaded splines). +attributes, as opposed to regression, which estimates the con- +ditional mean of the target variable. In quantile regression, +a particular quantile level α of the conditional distribution +of y given X = x, q(α, x) is estimated by minimizing the +pinball loss ρ (or quantile loss), as the the quantile function +q is shown to be the minimizer of the expected pinball loss +(Koenker and Bassett Jr 1978): +ρα(y, q) = (y − q)(α − 1(y < q)), +(2) +q(α, x) = arg min +q +Ey[ρα(y, q)]. +(3) +where 1 is the indicator function. One shortcoming of pinball +loss is only measuring the loss at a single quantile level, +which hinders the estimated q for a global picture of the +distribution (i.e. other α levels). On contrast, the continuous +ranked probability score (CRPS) considers all quantile levels +by integrating the pinball loss over α = [0, 1] (Matheson and +Winkler 1976; Gneiting and Raftery 2007). +CRPS(y, q) = +� 1 +0 +2ρα(y, q)dα +(4) +As a proper scoring rule (Gneiting and Raftery 2007), CRPS +is minimized when the quantile function is q = F. That is, +F −1 +y += arg min +q +Ey[CRPS(y, q)]. +(5) +Please refer (Koenker and Regression 2005) for detailed +proof. +Improving the expressiveness of quantile function +Fig. 2 demonstrate the need of an expressive quantile func- +tion for modeling target distribution. Inspired from neural +architecture search (NAS) (Elsken, Metzen, and Hutter 2019), +we propose an approach to search for the suitable combina- +tion of distributions. The search is over different operations +and basis distributions. We first introduce parametrization of +quantile function, and the two non-parametric spline-based +distributions. +Parameterizing quantile functions +We propose to param- +eterize the quantile function qθ(α, x) using a deep neural +network with parameters θ. The quantile function is aimed +to be accurate for any quantile levels α and input attributes +level X = x. X is high dimensional in real data, not as the +one dimensional in the toy examples in Fig. 1 and Fig. 2. +C-spline distribution +The c-spline (yα = qcsplie +θ +(α, x)) +describes the CDF (Fig. 2, Right Panel) of a probability +distribution Fy|X by setting K anchor points (denoted as +knots) on the CDF curve and performing linear interpolation +to fill in the gap between the knots. Specifically, the knots +split CDF curve into bins and c-spline learns the width wi +and height hi of bins by neural networks NN that depend on +the input attributes level X = x. +{wi, hi}K = NNθ(x) +yα = r({wi, hi}K, α) +∀α ∈ [0 : 1] + +where hi and wi are non-negative delta values imposed by +non-negative activation (i.e. Relu or Sigmoid), and the loca- +tion of each bin (e.g. Y|X) is Li = �i +k=0 wk and quantile +level αi = �i +k=0 hk. The accumulation sum design is to en- +sure that quantile function is monotically increasing and there +is no quantile crossing. r is a function to convert knots to +output of quantile function: for quantile level αi that is on the +knots, we can directly read from li , for quantile levels that +are off the knots, quantile values can be computed through +linear algebra operations on the two nearby knots r(α) = +� +li + (α−αi)(lj−li) +αj−αi +, +if αi ≤ α ≤ αj +0 ≤ i, j ≤ K +lk, +if hk = α +P-spline distribution +The difference between p-spline +from c-spline is having anchor knots in PDF space, instead +of CDF space. Similarly with C-spline, P-spline also per- +form linear interpolation over knots, and the quantile level is +achieved by integration over pdf via polynomial operations. +Neural Spline Search (NSS) +We describe our proposed method, Neural Spline Search +(NSS), which is overviewed in Fig. 3. Similar to symbolic +regression (Parisotto et al. 2016; Li et al. 2019), NSS effec- +tively searches in the space of discrete symbolic operators +and distribution space for a candidate that can better fit the +target data distribution. Specifically, let T(O, S, k) denote the +space of all transformations, via operators O on all distribu- +tion S with a maximum sequence length k. NSS aims to find +the function f(x) selecting operators and distributions in the +space T such that {f(x) ∈ T(O, S, k) : ℓ(f(x), xtrain) ≤ δ +}, where ℓ denotes loss function CRPS, xtrain is training +data and δ is the acceptance threshold. Given the large search +space composed of combinations of numerous splines and +operators, we restrict to use spline-based distribution as the +basis distribution, and limit the operator search space to sum- +mation and chaining operations upon the transformation basis +spline regressions. Note that this work can be easily extend +to other operations and distributions, which we leave to fu- +ture work. We describe the following NSS transformations as +they are observed to work well consistently across different +datasets: NSS with summation (NSS-sum) and NSS with +chaining (NSS-chain). Algorithm 1 and Fig. 4(b) +NSS-sum +NSS-sum performs transformations using the scale and sum- +mation operators. We represent this scenario with two splines: +Spline 1: c-spline and Spline 2: p-spline, and two operators: +scale O1 : O(a) = λa and summation O2 : O(a, b) : a + b; +therefore, the overall transformation is (Spline 1-Operator 1) - +(Spline 2-Operator 2), which yields: f = c-spline + λ p-spline. +Essentially, NSS-sum performs weighted sum of different +splines. The motivation behind is that c-spline with fewer +parameters can be more robust against overfitting, whereas +p-spline increases the expressiveness of the splines. +NSS-chain +Another proposed NSS design is NSS-chain. We focus on +the chaining operator due to its expressiveness. This design +is inspired by the success of normalizing flow (Rezende and +Mohamed 2015), where a sequence of bijector transforms is +utilized to transform distributions. Different from normaliz- +ing flow which has practical applicability challenges, NSS- +chain only requires the forward pass of the transformation, +not the inverse as normalizing flow does. This significantly +reduces the computational complexity and broadens the fea- +sibility of transformations. As mentioned, quantile function +takes input attributes level (X) to predict the target value (y) +at quantile level (α). +y = qθ(X, α), +(6) +where X ∈ Rm and α ∈ [0, 1]. We present two designs to +chain different transformations (see Fig. 4 (a)). We note that +chaining of transformation is not limited to the two designs. +Algorithm 1: Neural Spline Search +Operators = {+, ×, Scale, Chain, ...} +Splines = {c-spline, p-spline, Gaussian, Cauchy ...} +Data: Quantile level α ∈ [0, 1], N data points +{X ∈ Rd, y ∈ R1}N, d ≥ 1, with chain depth k. +Transform indicates the transformation using the +input spline Sθ and operator O. +Result: p(y|X) and F −1 +y|X(α) +k ← 1; +while k ≤ K do +Select O = {Oi}no ∈ Operators ; +Select S = {Sj}ns ∈ Splines ; +θ ← MLP(X) ; +ypred ← Transform(Sθ, O, α); +if α NSS-chain then +Normalize ypred to [0, 1] as y′ +pred ; +α ← y′ +pred; +else +X ← Y +▷ if X-NSS-chain ; +end +k ← k + 1; +end +• α-chaining +The α-chaining is when we consider the condition level +(X) unchanged during the chain of transformation, and +the output of each transformation is a scaled version of +quantile level for the next transformation. In particular, +after each transformation, we normalize the output y to +be in the range [0, 1], and then the normalized output is +re-input as the new α to the next transformation. This is +repeated until the maximum depth is reached. This design +is more similar with normalizing flow methods. +y = qθK(X, ...fn(qθ2(X, fn(qθ1(X, α))))) +(7) +θk for k=1,2,..K are parameters for different splines in +K-length chain. fn is the normalization function. +• X-chaining +X-chaining is when we consider quantile level α level is +unchanged during chaining, as each transformation learns + +alpha +P(y|X) +y_alpha +Inverse CDF +alpha +y + MLP +X +Spline's parameters +alpha +P(y|X) +y_alpha +Inverse CDF +alpha +y + MLP +X +Spline's parameters +x-chaining +alpha +P(y|X) +y_alpha +Inverse CDF +alpha +y + MLP +X +Spline's parameters +alpha-chaining +NNS Chain +Figure 4: (a) Illustration of NSS-chain methods. The dia- +gram demonstrates chaining for NSS-chain. Left: α-chaining. +The output y of the spline, after re-scaling to [0, 1], is re- +inputted to the quantile spline at quantile level α. Right: X- +chaining. The output y is instead re-inputted to the quantile +spline as X. Both rely on input attributes X. +a suitable condition level (or feature) for next iteration. +Similarly with α-chaining in the iterative manner, except +that the output y of each transformation is projected to +generate X for the next iteration of Eq. 6. +y = qθK(...qθ2(qθ1(X, α), α), α) +(8) +The advantage of this approach, compared tp α-chaining, +is that we keep quantile levels α unchanged, and re- +normalizing output is not needed. +Remarks on NSS: . (1) why a simple spline-based algo- +rithm, e.g. C-spline, is not enough? Although in theory +spline-based algorithms can represent any arbitrary distribu- +tions with sufficiently high number of knots K, in practice, +we find a large K often lead to unstable training, as also +studied in (Park et al. 2022). In contrast, we find the combina- +tion (combined or chained) over a relatively restricted splines +are more robust in capturing the overall of the target distribu- +tion (2) Include both spline-based distribution and classic +parametric distribution In addition to spline-based distribu- +tion, we also encourage incorporating parametric distribution +(e.g. Gaussian) as basis distribution for NSS, especially when +prior knowledge (say Gaussian noise) is available. Because, it +is challenging for spline based methods to reconstruct Gaus- +sian distribution even with infinite number of knots; and , the +benefits of combining the two are the parametric distribution +offers advantage of classic statistics and robust to noise, and +the non-parametric spline offers flexibility. +Training +Once we select the operators and splines, the parameters of +the splines are trained in an end-to-end way by optimizing +CRPS (Eq. 4). Specifically, during training, we fit parameters +by optimizing over with the empirical mean of CRPS over N +data points: +θ∗ = arg min +θ +1/N +N +� +i=1 +Ey[CRPS(y, qθ(Xi, α))]. +(9) +Algorithm 2 overviews the training of NSS for spline parame- +ter selection. Because of the form of the transformations, the +analytical solution of CRPS integration is intractable. Thus, +we use a Monte Carlo estimation for the CRPS loss. In par- +ticular, we sample m number of α values from the range of +[0, 1] and average them for the corresponding pinball loss. +Algorithm 2: Training with CRPS +Data: N data points {Xi ∈ Rd, yi ∈ R1}N +i=1, m +quantile levels, T transformation, which +takes selected splines Sselect and selected +operators Oselect from NSS. lr is learning +rate. +Result: Neural network weights θ +e ← 1; +while e ≤ Nepoch do +f = Transform(Sselect, Oselect) ℓ ← 0 ; +for α in [0, 1 +m, 2 +m, ..1] do +ypred +α += fθ(X, α) ; +ℓ ← ℓ + pinball_loss (ypred +α +, y, α) +end +CRPS = ℓ/m ; +θ ← θ − lr · ∇θ CRPS ; +e ← e + 1; +end +Experiments +Comparison methods +QD (Pearce et al. 2018) generates prediction intervals (PIs) +for estimating uncertainty for regression tasks with the as- +sumption that high-quality PIs should be as narrow as possi- +ble. Deep Quantile Aggregation (Kim et al. 2021) proposes +weighted ensembling strategies where aggregation weights +vary over both individual models and feature values plus +(pairs of) quantile levels. The monotonization layer in the +network is applied to avoid crossing of quantile estimates. +RQspline (Durkan et al. 2019) proposes a fully-differentiable +module based on monotonic rational-quadratic splines, which +enhances the flexibility of coupling and autoregressive trans- +forms while retaining analytic invertibility. Global-Coarse +(Ratcliff 1979) provides an analysis of distribution statis- +tics of group reaction time distributions. MLE (NB) and +Mix. MLE are Negative Binomial and mixture likelihood +based methods (Awasthi et al. 2021). C-spline is proposed in +(Gasthaus et al. 2019), where C-spline is used as the quantile +function in time-series forecasting. +Metrics +For point predictions, we focus on the following metrics: +Mean absolute error (MAE): 1 +n +�n +t=1 |Tt − Pt| where Tt and +Pt are true and predicted value; Mean Absolute Percentage +Error (MAPE): 1 +n +�n +t=1 | Tt−Pt +Tt +|. Weighted Average Percent- +age Error (WAPE): +�n +t=1 |Tt−Pt| +�n +t=1 |Tt| +; and Root Mean Square +Error (RMSE): +� �N +t (Tt−Pt)2 +n +. For quantile predictions, we +use the Pinball Loss (Eq. 2), with 50%-th, Q50; 90%-th, Q90; +and 10%-th Q10 quantiles. + +Methods +Boston +Concrete +kin8nm +Power +Protein +Wine +Gaussian +0.0754 +0.0564 +0.048 +0.0449 +0.2116 +0.0978 +QD +0.5003 +0.4150 +0.3945 +0.3688 +0.6689 +0.4456 +RQspline +0.0917 +0.0622 +0.0479 +0.0485 +0.2153 +0.0912 +p-sline +0.0778 +0.0570 +0.0444 +0.0453 +— +0.0966 +c-spline +0.0806 +0.0543 +0.0430 +0.0447 +0.2002 +0.0947 +NSS-X-chain +0.0787 +0.0588 +0.0430 +0.0448 +0.2052 +0.0962 +NSS-α-chain +0.0846 +0.0568 +0.0417 +0.0448 +0.2067 +0.0976 +NSS-sum +0.0709 +0.0512 +0.0414 +0.0442 +0.1949 +0.0957 +Gain percentage +12.0% +17.7% +3.7% +1.1% +2.6% +- +Table 1: Mean Absolute Error (MAE) on UCI benchmarks. Test performance of the proposed method (NSS) and existing +methods on UCI benchmarks. We use the 50th quantile estimator as our estimates. The dash indicates unavailability. The shaded +area is the proposed methods. Bold is the top one. Lower is better. Gaussian: Gaussian kernel; QD is quantity-driven methods +proposed in (Pearce et al. 2018); RQ spline proposed in (Durkan et al. 2019); c-spline proposed in (Gasthaus et al. 2019). Boston, +Concrete, Power is short for Boston Housing, Concrete Strength, Power Plant. Gain percentage is computed as (best nss - best +baseline)/best baseline. +Training +For simplicity, the proposed NSS methods use depth- +2 +splines, +which +contain +{(c-spline, +p-spline), +(c- +spline, +p-spline), +(c-spline, +c-spline), +(p-spline, +p- +spline)}. NSS-sum is tuned with λ in the range of +[0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. +NSS-chain +nor- +malizing of y in α chaining can be achieved by applying +sigmoid layer or scaling by max value. As splines are +monotonically-increasing functions, the spline value y with +α = 0 is the minimum value of y and α = 1 yields the +maximum value of y. Scale is yscale = +y−ymin +ymax−ymin . We use +a batch size=128 and a learning rate of 0.005 for 100 epochs. +Results +To demonstrate the effectiveness of proposed methods, we +conduct experiments on synthetic, real-world tabular regres- +sion, and time series forecasting datasets. +Synthetic data +Dataset. We generate 2000 data points (X +∈ R1 and +y ∈ R1), where X is in the range of [−2, 2] and y has Gaus- +sian distribution y ∼ N(0.3 sin(3x), 0.2x2), where sin is the +sinusodial function. We construct the validation and test sets +to come from the same distribution. Unlike real-world data, +the synthetic data would have known quantile levels, that can +be used for evaluating the accuracy of quantile estimates. We +make the task more challenging by setting a data-dependent +variance for the Gaussian noise to evaluate the ability of learn- +ing condition-specific quantile values. Fig. 5 shows that the +proposed NSS-chain and NSS-sum can capture the true under- +lying quantiles, whereas QD (Pearce et al. 2018) struggles on +the varying variance locations (e.g. around x = 0). The upper +and lower black lines are the predicted 2.5%-th and 97.5%-th +quantiles for the observed data (e.g. red dots), shown along +with the ground truth quantiles (e.g. shaded red area). The +results indicate that more expressive NSS transformations +are superior in more challenging scenarios, where true data +points are distributed differently (e.g., distributions depend +on the value of the inputs"). Fig. 6 shows the calibration plot +of the predicted vs. true distributions at different quantile +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +x +2 +1 +0 +1 +2 +y +QD +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +x +2 +1 +0 +1 +2 +y +NSS-SUM. +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +x +2 +1 +0 +1 +2 +y +NSS-CHAIN. +Figure 5: NSS on Synthetic data. We compare the per- +formance of proposed NSS against existing methods QD +(Pearce et al. 2018). The red dots are observed data points, +shaded red area is the ground truth 2.5% and 97.5% quantile +levels, and the dark black lines are the predicted 2.5% and +97.5% quantile levels. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True percentile +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted percentile +QD +NSS-SUM +NSS-CHAIN +2 +0 +2 +X +y +X=0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True percentile +Calibration plot +QD +NSS-SUM +NSS-CHAIN +2 +0 +2 +X +y +X=1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True percentile +QD +NSS-SUM +NSS-CHAIN +2 +0 +2 +X +y +X=1.5 +Figure 6: Calibration plots. Predicted vs. ground truth per- +centiles at condition levels: X=0.5, 1.0 and 1.5. The perfect +calibration would correspond to the diagonal (red dotted) +line. +levels. Here, we show the true percentile p as the fraction of +data in the dataset such that the p percentile of the predictive +distribution is larger than the ground truth data. The perfect +prediction would be the diagonal line. Fig. 6 indicates that +the proposed methods NSS-sum and NSS-chain can capture +the proposed true distribution at various levels by close to the +red line, whereas QD does not fit as well. +Real-world tabular regression +We use UCI benchmarks (Asuncion and Newman 2007) that +contain tabular data from diverse domains (e.g. real estate +and physics). Following (Salem, Langseth, and Ramampiaro +2020), the datasets are normalized with z-score standardiza- + +Methods +Boston +Concrete +kin8nm +Power +Protein +Wine +Gaussian +0.0276 +0.0203 +0.0171 +0.0158 +0.0725 +0.0357 +Global-Coarse∗ +0.0745 +0.0596 +0.0681 +0.0473 +0.1321 +— +Deep Quantile Aggregation∗ +0.0754 +0.0541 +0.0684 +0.0441 +0.1253 +— +QD +0.1212 +0.1076 +0.1004 +0.0972 +0.1547 +0.1164 +RQspline +0.0458 +0.0418 +0.0203 +0.0189 +0.0863 +0.0424 +p-sline +0.0308 +0.0211 +0.016 +0.0160 +— +0.0358 +c-spline +0.0312 +0.0198 +0.0157 +0.0159 +0.0688 +0.0351 +NSS-X-chain +0.0311 +0.0216 +0.0165 +0.0162 +0.0707 +0.0358 +NSS-α-chain +0.0322 +0.0208 +0.0151 +0.0159 +0.0726 +0.0363 +NSS-sum +0.0265 +0.0191 +0.0152 +0.0157 +0.0674 +0.0357 +Gain percentage +4.0% +3.5% +3.8% +0.6% +7.0% +- +Table 2: Average pinball loss on UCI benchmarks. The test pinball loss (the lower, the better) is over 99 quantile levels, +α = {0.01, 0.02, ...0.99}. The compared methods are Global-Coarse proposed in (Ratcliff 1979); QD (Pearce et al. 2018); Deep +Quantile Aggregation (DQA) (Kim et al. 2021); RQspline (Durkan et al. 2019); ∗ indicates entries are from (Kim et al. 2021) +(under the same experiment setup). +Methods +MAPE +WAPE +RMSE +Q50 +Q90 +Q10 +MLE (NB) +0.44434 +0.27240 +7.70958 +0.27240 +0.10907 +0.15275 +Mix MLE +0.44839 +0.26838 +7.22556 +0.26838 +0.10293 +0.14508 +c-spline +0.44672 +0.26635 +7.06332 +0.26635 +0.10238 +0.14241 +p-spline +0.44912 +0.26834 +7.14643 +0.26834 +0.10343 +0.14333 +NSS-sum +0.44501 +0.26545 +6.96697 +0.26545 +0.10238 +0.14266 +NSS-chain +0.44883 +0.26420 +6.91726 +0.26420 +0.10243 +0.14149 +Table 3: Performance comparisons for time series forecasting on M5. Different evaluation metrics are included in this table +for M5. Detailed descriptions of the metrics are in Sec . Qk indicates the pinball loss of k-th quantile. e.g. Q50 is the pinball loss +of 50th quantile. Lower is better. +tion. +We evaluate the accuracy for both point predictions and quan- +tiles. As the point predictions, we use the 50th quantile es- +timator as our estimates. Table 1 shows that the proposed +NSS methods outperform the other existing methods on most +datasets in mean absolute error (MAE). In mean square error +(MSE), the results are provided in Appendix Table 4. We +observe that the NSS-sum performs better than NSS-chain. +For quantile metrics, we use the pinball loss (Eq. 2) over +100 quantile levels α = {0.01, 0.02, ...0.99} in Table 2. The +results indicate that NSS consistently outperforms other al- +ternatives across different UCI benchmarks. In pinball loss, +NSS-sum performs better than NSS-chain. We attribute the +superiority of NSS-sum for regression to make balance be- +tween different transformation, which is helpful in explaining +the variance in the data. +Retail demand forecasting +For time series forecasting, we focus on the M5 dataset, +which contains time-varying sales data for retail goods, along +with other relevant covariates like price, promotions, day +of the week, special events etc. It represents an important +real-world scenario, where the accurate estimation of the +output distribution is crucial, as retailers use them to optimize +prices or promotions. +The time series forecasting experiments are conducted by +performing one-step ahead prediction, yielding predictions +in an autoregressive way. Table 3 shows the results of +our method compared to other alternatives. We observe +consistent outperformance of NSS in various forecasting +evaluation metrics. Different from regression tasks, we +observe that NSS-chain is better than NSS-sum, indicating +its benefit in capturing time-dependent relationship. +Remarks on NSS-sum vs NSS-chain. The results show that +NSS-sum is superior on regression, while NSS-chain has +advantages on time series forecasting. The observations may +indicate NSS-sum is suitable for more constrained tasks (e.g. +regression, one time step time series-forecasting), where be- +ing moderately expressive would suffice. NSS-sum is also +more robust and easier to train. On the other hand, NSS-chain +may be more expressive, which is beneficial to fit tasks re- +quires more complex distributions at different time steps of +the time series, but for individual step NSS-chain is not as +accurate as NSS-sum in fitting the distribution. +Conclusion +We propose a novel approach for modeling uncertainty. 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Motion +planning under uncertainty for on-road autonomous driving. +In ICRA. + +Appendix +Mean square error for UCI dataset + +Methods +Bost House +Concr Stren +kin8nm +Power plant +Protein +Wine +Gaussian +0.0105 +0.0054 +0.0042 +0.0032 +0.0648 +0.0164 +(Salem, Langseth, and Ramampiaro 2020)∗ +0.1120 +0.0560 +0.0600 +0.0420 +0.3100 +0.5970 +QD +0.2705 +0.1839 +0.1613 +0.1393 +0.5277 +0.2164 +RQspline +0.0255 +0.0070 +0.0040 +0.0037 +0.0809 +0.0195 +p-sline +0.0136 +0.0058 +0.0032 +0.0032 +— +0.0162 +c-spline +0.0162 +0.0050 +0.0031 +0.0032 +0.0757 +0.0159 +NSS-X-chain +0.0128 +0.0056 +0.0031 +0.0032 +0.0751 +0.0164 +NSS-α-chain +0.0184 +0.0058 +0.0029 +0.0032 +0.0760 +0.0169 +NSS-sum +0.0112 +0.0046 +0.0029 +0.0032 +0.0711 +0.0160 +Table 4: Mean Square Error of UCI datasets + diff --git a/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ceb9cb33b64f01c26f6505e455a41490784885c --- /dev/null +++ b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt @@ -0,0 +1,1090 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf,len=1089 +page_content='Neural Spline Search for Quantile Probabilistic Modeling Ruoxi Sun1*, Chun-Liang Li1*, Sercan Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Arık1, Michael W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Dusenberry2, Chen-Yu Lee1, Tomas Pfister 1 1Google Cloud AI 2Google Research, Brain Team {ruoxis, chunliang, soarik, dusenberrymw, chenyulee, tpfister}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='com Abstract Accurate estimation of output quantiles is crucial in many use cases, where it is desired to model the range of possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Modeling target distribution at arbitrary quantile levels and at arbitrary input attribute levels are important to offer a compre- hensive picture of the data, and requires the quantile function to be expressive enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function describing the target distribution using quantile levels is critical for quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Althought various parametric forms for the distri- butions (that the quantile function specifies) can be adopted, an everlasting problem is selecting the most appropriate one that can properly approximate the data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In this paper, we propose a non-parametric and data-driven approach, Neural Spline Search (NSS), to represent the observed data distribution without parametric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS is flexible and expressive for modeling data distributions by transform- ing the inputs with a series of monotonic spline regressions guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate that NSS out- performs previous methods on synthetic, real-world regression and time-series forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Introduction For many machine learning applications, modeling the pre- diction intervals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' estimating the ranges all individual predictions observation fall), beyond point estimates, is cru- cial (Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Tagasovska and Lopez-Paz 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The prediction intervals can help with decision making for retail sales optimization (Simchi-Levi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2008), medi- cal diagnoses (Begoli, Bhattacharya, and Kusnezov 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Mhaskar, Pereverzyev, and van der Walt 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2012), information safety (Smith, Dinev, and Xu 2011), fi- nancial investment management (Engle 1982), robotics and control (Buckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018), autonomous transformation (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2014) and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' To estimate prediction intervals, we would need to estimate different levels of quantiles for the target distribution using quantile regression (Koenker and Regression 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wald- mann 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' A real-world challenge is to select the paramet- ric forms of target distributions, which is specified by the quantile function (also known as the inverse CDF function), These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 X 2 1 0 1 2 Y X=X0 P(Y|X) True Quantile 25% Quantile 75% Figure 1: Modeling multiple quantiles at different condition-levels with a universal quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The goal is to model target data distribution y at any arbitrary quantile level and attribute level X, using one versatile quan- tile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gray dots are observed data points, while green and blue lines indicate 25% and 75% quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The data distribution y varies at different levels of X, say variance of y increases when X is away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Red dots are data points at X = X0, p(Y |X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' to properly align with observed data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different choices for the target distribution (Gaussian, Poisson, Neg- ative Binomial, Student-t etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=') may yield different quantile predictions, and misalignment of the assumption with the real distribution may hinder the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There- fore, such heuristic or empirical hand-picking based paramet- ric assumptions for the distribution can be sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' An approach based on learning from the data in an automated way, would be highly desirable, from both foundational and practical perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For learnable parametric modeling, one challenge is how to model all quantiles for all input attributes level in a com- putationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' First, modeling an any arbitrary quantile, as opposed to a couple of pre-defined quantile levels, offers a more comprehensive view on the target distribution, and provides convenience to use the quantile model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' no need to re-train the model when quantiles at testing are dif- ferent from the ones at training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Second, real-world data can have complex distributions beyond what simple assumptions can model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 1 shows different input attribute X levels arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='04857v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='AI] 12 Jan 2023 0 1 2 3 4 5 Y|X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='020 Probability Density PDF(Y|x) 0 1 2 3 4 5 Y|X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 Probability Density CDF(Y|x) Figure 2: An example target distribution with a complex shape, in PDF and CDF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Black lines are observed tar- get distributions, in the form of mixture of the other three dis- tributions shown with color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fitting the black line accurately would be extremely difficult for most of the commonly-used single parametric splines, motivating for the use of learnable spline family composed of multiple splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' have different dependency dynamics with target y level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' the variance of y increases when X apart from 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2 shows that the observed distribution cannot trivially fit well with one single distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Therefore, in order to model all quantiles at all X, we need a quantile function with a com- plexity that does not increase significantly with number of input attributes and the number of quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This necessitates a versatile and highly-expressive quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There has been many efforts on improving various aspects of quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2019) proposes linear spline interpolation between knots in the inverse CDF space to model the target distribution in time-series forecasting setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This is proposed to avoid the assumption on paramet- ric form of the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2022) and Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2021) focus on learning a valid quantile function with- out quantile crossing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' quantiles violate monotonically increasing property), via special design of the neural network architecture or first-order inequality constraint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Despite being distribution agnostic, these approaches for de- scribing the target distribution (specified by quantile function) are restricted to one function family (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' linear spline), which may limit the expressiveness to represent the target distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In this paper, with the goal of designing an expressive quantile function for various quantiles and input levels, we propose a data-driven approach Neural Spline Search (NSS), which transforms the inputs with a series of monotonic spline regressions guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The contributions of our paper can be summarized as: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We propose an efficient search space and mechanism to find an expressive quantile function to model the data distribution, avoiding specifying a parametric form of the observed distribution as prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We propose a novel approach to generate an expressive quantile function using a combination of different distri- butions and operators guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The proposed method can be incorporated into other tasks (including but not limited to time series forecasting) as their quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate significant accuracy improvements across numerous regression or time series forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For example, on UCI benchmarks, we show 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% im- provement compared to next best methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Related Work Quantile regression is used to estimate the target distribu- tion at different quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The α-quantile estimator is the solution when minimizing quantile loss at level α (Koenker and Bassett Jr 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Another quantile regression related loss is continuous ranked probability score (CRPS) (Gneiting and Raftery 2007), which is the averaging over all quantile levels, instead of one single quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Neural network quantile forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' To model sequential dependency of time series, several forecasting models pro- pose a hidden state-emission framework ((Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' de Bézenac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019)), where the dynamics of hidden states are modeled by auto-regressive recurrent neural works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' LSTM), which takes previous hidden states and current ob- servations as input and outputs current observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from modeling the likelihood with parametric distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian (Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020)), emission models for quantile estimation is to learn the parameters of quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The overall framework is optimized by employing a quantile (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017) or CRPS (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Symbolic regression has shown great success in many fields, including program synthesis (Parisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2016), mathe- matical expressions extraction (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020), physics- based learning (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As the search space is enormous and scaled exponentially with the length of operators, symbolic regression rule operators are usually set to be a small number and are learned by Monte Carlo Tree Search guided evolutionary strategies (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) or reinforcement learning (Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods Learning quantile function in quantile regression Let the input data attributes X and the target variable y are jointly distributed as p(X, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The conditional cumulative distribution function (CDF) is F(Y = y|X) = P(Y ≤ y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function, which is also called the inverse CDF function, takes quantile level as inputs and returns a threshold value Y below which random draws from the given CDF would fall quantile percent of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, the α-th quantile function of y|X = x is denoted as: q(α, x) = F −1 y|X=x(α) = inf{y : F(y|X = x) ≥ α} (1) Here we can think the quantile function is to perform a transformation on a uniform-distributed random variable α ∼ U(0, 1) to the target distribution p(y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Quantile function is able to fully specify a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' So specifying the quantile function is describing the target distribution p(y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Quantile regression estimates different conditional quantile levels of the target variable given a certain level of input P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y Spline Basis P(y|X) Inverse CDF alpha y S C + P(y|X) Inverse CDF alpha y Spline Basis P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. NSS-sum Initial distribution target distribution Spline Basis Spline Basis NSS-chain P(y|X) Inverse CDF alpha y Operators + Sum S Scale C Chaining .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. Figure 3: Overview of Neural Spline Search (NSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Modeling the target data distribution can be done by learning the quantile function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' inverse CDF), which maps a [0, 1]-variable (quantile) to a target value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Unlike parametric methods which specify a distribution family and learn the parameters, NSS can generate the target distribution through a set of transformations on the inverse CDF space (quantile space), where the transformation is guided by a series of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Here, the bottom gray box shows possible operators (denoted as circles), including but not limited to summation (“+”), scale (“S”), and chaining (“C”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The basis splines are shown with color-shaded squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The initial distribution is a uniform distribution, as shown in the leftmost panel (blue shaded), and the target distribution is the rightmost distribution (purple shaded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There is no obvious parametric distribution to achieve this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Therefore, NSS is used to search for the suitable transformation through simple operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In the first row of the middle panel, we show operators for NSS-sum, where the initial uniform distribution is transformed by the red- and the yellow-shaded splines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' c-spline) through sum (“+”) and scale (“S”) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The second row shows the chaining transformation of the initial distribution, where the orange and cyan splines are used to transform the initial spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The parameters of the splines are learned by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In general, the operators and transformations in NSS are not limited to two splines (we represent them as the gray splines next to the yellow and cyan shaded splines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' attributes, as opposed to regression, which estimates the con- ditional mean of the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In quantile regression, a particular quantile level α of the conditional distribution of y given X = x, q(α, x) is estimated by minimizing the pinball loss ρ (or quantile loss), as the the quantile function q is shown to be the minimizer of the expected pinball loss (Koenker and Bassett Jr 1978): ρα(y, q) = (y − q)(α − 1(y < q)), (2) q(α, x) = arg min q Ey[ρα(y, q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (3) where 1 is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' One shortcoming of pinball loss is only measuring the loss at a single quantile level, which hinders the estimated q for a global picture of the distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' other α levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' On contrast, the continuous ranked probability score (CRPS) considers all quantile levels by integrating the pinball loss over α = [0, 1] (Matheson and Winkler 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gneiting and Raftery 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' CRPS(y, q) = � 1 0 2ρα(y, q)dα (4) As a proper scoring rule (Gneiting and Raftery 2007), CRPS is minimized when the quantile function is q = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' That is, F −1 y = arg min q Ey[CRPS(y, q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (5) Please refer (Koenker and Regression 2005) for detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Improving the expressiveness of quantile function Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2 demonstrate the need of an expressive quantile func- tion for modeling target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Inspired from neural architecture search (NAS) (Elsken, Metzen, and Hutter 2019), we propose an approach to search for the suitable combina- tion of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The search is over different operations and basis distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We first introduce parametrization of quantile function, and the two non-parametric spline-based distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Parameterizing quantile functions We propose to param- eterize the quantile function qθ(α, x) using a deep neural network with parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function is aimed to be accurate for any quantile levels α and input attributes level X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' X is high dimensional in real data, not as the one dimensional in the toy examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline distribution The c-spline (yα = qcsplie θ (α, x)) describes the CDF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2, Right Panel) of a probability distribution Fy|X by setting K anchor points (denoted as knots) on the CDF curve and performing linear interpolation to fill in the gap between the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, the knots split CDF curve into bins and c-spline learns the width wi and height hi of bins by neural networks NN that depend on the input attributes level X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' {wi, hi}K = NNθ(x) yα = r({wi, hi}K, α) ∀α ∈ [0 : 1] where hi and wi are non-negative delta values imposed by non-negative activation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Relu or Sigmoid), and the loca- tion of each bin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Y|X) is Li = �i k=0 wk and quantile level αi = �i k=0 hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The accumulation sum design is to en- sure that quantile function is monotically increasing and there is no quantile crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' r is a function to convert knots to output of quantile function: for quantile level αi that is on the knots, we can directly read from li , for quantile levels that are off the knots, quantile values can be computed through linear algebra operations on the two nearby knots r(α) = � li + (α−αi)(lj−li) αj−αi , if αi ≤ α ≤ αj 0 ≤ i, j ≤ K lk, if hk = α P-spline distribution The difference between p-spline from c-spline is having anchor knots in PDF space, instead of CDF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similarly with C-spline, P-spline also per- form linear interpolation over knots, and the quantile level is achieved by integration over pdf via polynomial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Neural Spline Search (NSS) We describe our proposed method, Neural Spline Search (NSS), which is overviewed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similar to symbolic regression (Parisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019), NSS effec- tively searches in the space of discrete symbolic operators and distribution space for a candidate that can better fit the target data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, let T(O, S, k) denote the space of all transformations, via operators O on all distribu- tion S with a maximum sequence length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS aims to find the function f(x) selecting operators and distributions in the space T such that {f(x) ∈ T(O, S, k) : ℓ(f(x), xtrain) ≤ δ }, where ℓ denotes loss function CRPS, xtrain is training data and δ is the acceptance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Given the large search space composed of combinations of numerous splines and operators, we restrict to use spline-based distribution as the basis distribution, and limit the operator search space to sum- mation and chaining operations upon the transformation basis spline regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Note that this work can be easily extend to other operations and distributions, which we leave to fu- ture work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We describe the following NSS transformations as they are observed to work well consistently across different datasets: NSS with summation (NSS-sum) and NSS with chaining (NSS-chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4(b) NSS-sum NSS-sum performs transformations using the scale and sum- mation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We represent this scenario with two splines: Spline 1: c-spline and Spline 2: p-spline, and two operators: scale O1 : O(a) = λa and summation O2 : O(a, b) : a + b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' therefore, the overall transformation is (Spline 1-Operator 1) - (Spline 2-Operator 2), which yields: f = c-spline + λ p-spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Essentially, NSS-sum performs weighted sum of different splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The motivation behind is that c-spline with fewer parameters can be more robust against overfitting, whereas p-spline increases the expressiveness of the splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-chain Another proposed NSS design is NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We focus on the chaining operator due to its expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This design is inspired by the success of normalizing flow (Rezende and Mohamed 2015), where a sequence of bijector transforms is utilized to transform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from normaliz- ing flow which has practical applicability challenges, NSS- chain only requires the forward pass of the transformation, not the inverse as normalizing flow does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This significantly reduces the computational complexity and broadens the fea- sibility of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As mentioned, quantile function takes input attributes level (X) to predict the target value (y) at quantile level (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθ(X, α), (6) where X ∈ Rm and α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We present two designs to chain different transformations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We note that chaining of transformation is not limited to the two designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 1: Neural Spline Search Operators = {+, ×, Scale, Chain, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='} Splines = {c-spline, p-spline, Gaussian, Cauchy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='} Data: Quantile level α ∈ [0, 1], N data points {X ∈ Rd, y ∈ R1}N, d ≥ 1, with chain depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Transform indicates the transformation using the input spline Sθ and operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Result: p(y|X) and F −1 y|X(α) k ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' while k ≤ K do Select O = {Oi}no ∈ Operators ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Select S = {Sj}ns ∈ Splines ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' θ ← MLP(X) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ypred ← Transform(Sθ, O, α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' if α NSS-chain then Normalize ypred to [0, 1] as y′ pred ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' α ← y′ pred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' else X ← Y ▷ if X-NSS-chain ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end α-chaining The α-chaining is when we consider the condition level (X) unchanged during the chain of transformation, and the output of each transformation is a scaled version of quantile level for the next transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In particular, after each transformation, we normalize the output y to be in the range [0, 1], and then the normalized output is re-input as the new α to the next transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This is repeated until the maximum depth is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This design is more similar with normalizing flow methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθK(X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='fn(qθ2(X, fn(qθ1(X, α))))) (7) θk for k=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='.K are parameters for different splines in K-length chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' fn is the normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=" X-chaining X-chaining is when we consider quantile level α level is unchanged during chaining, as each transformation learns alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters x-chaining alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters alpha-chaining NNS Chain Figure 4: (a) Illustration of NSS-chain methods." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The dia- gram demonstrates chaining for NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Left: α-chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The output y of the spline, after re-scaling to [0, 1], is re- inputted to the quantile spline at quantile level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Right: X- chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The output y is instead re-inputted to the quantile spline as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Both rely on input attributes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' a suitable condition level (or feature) for next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similarly with α-chaining in the iterative manner, except that the output y of each transformation is projected to generate X for the next iteration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='qθ2(qθ1(X, α), α), α) (8) The advantage of this approach, compared tp α-chaining, is that we keep quantile levels α unchanged, and re- normalizing output is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Remarks on NSS: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (1) why a simple spline-based algo- rithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline, is not enough?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Although in theory spline-based algorithms can represent any arbitrary distribu- tions with sufficiently high number of knots K, in practice, we find a large K often lead to unstable training, as also studied in (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In contrast, we find the combina- tion (combined or chained) over a relatively restricted splines are more robust in capturing the overall of the target distribu- tion (2) Include both spline-based distribution and classic parametric distribution In addition to spline-based distribu- tion, we also encourage incorporating parametric distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian) as basis distribution for NSS, especially when prior knowledge (say Gaussian noise) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Because, it is challenging for spline based methods to reconstruct Gaus- sian distribution even with infinite number of knots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and , the benefits of combining the two are the parametric distribution offers advantage of classic statistics and robust to noise, and the non-parametric spline offers flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Training Once we select the operators and splines, the parameters of the splines are trained in an end-to-end way by optimizing CRPS (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, during training, we fit parameters by optimizing over with the empirical mean of CRPS over N data points: θ∗ = arg min θ 1/N N � i=1 Ey[CRPS(y, qθ(Xi, α))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (9) Algorithm 2 overviews the training of NSS for spline parame- ter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Because of the form of the transformations, the analytical solution of CRPS integration is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Thus, we use a Monte Carlo estimation for the CRPS loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In par- ticular, we sample m number of α values from the range of [0, 1] and average them for the corresponding pinball loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 2: Training with CRPS Data: N data points {Xi ∈ Rd, yi ∈ R1}N i=1, m quantile levels, T transformation, which takes selected splines Sselect and selected operators Oselect from NSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' lr is learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Result: Neural network weights θ e ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' while e ≤ Nepoch do f = Transform(Sselect, Oselect) ℓ ← 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' for α in [0, 1 m, 2 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='.1] do ypred α = fθ(X, α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ℓ ← ℓ + pinball_loss (ypred α , y, α) end CRPS = ℓ/m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' θ ← θ − lr · ∇θ CRPS ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' e ← e + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end Experiments Comparison methods QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018) generates prediction intervals (PIs) for estimating uncertainty for regression tasks with the as- sumption that high-quality PIs should be as narrow as possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Deep Quantile Aggregation (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021) proposes weighted ensembling strategies where aggregation weights vary over both individual models and feature values plus (pairs of) quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The monotonization layer in the network is applied to avoid crossing of quantile estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQspline (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) proposes a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of coupling and autoregressive trans- forms while retaining analytic invertibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Global-Coarse (Ratcliff 1979) provides an analysis of distribution statis- tics of group reaction time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' MLE (NB) and Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' MLE are Negative Binomial and mixture likelihood based methods (Awasthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline is proposed in (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019), where C-spline is used as the quantile function in time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Metrics For point predictions, we focus on the following metrics: Mean absolute error (MAE): 1 n �n t=1 |Tt − Pt| where Tt and Pt are true and predicted value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Mean Absolute Percentage Error (MAPE): 1 n �n t=1 | Tt−Pt Tt |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Weighted Average Percent- age Error (WAPE): �n t=1 |Tt−Pt| �n t=1 |Tt| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and Root Mean Square Error (RMSE): � �N t (Tt−Pt)2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For quantile predictions, we use the Pinball Loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2), with 50%-th, Q50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 90%-th, Q90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and 10%-th Q10 quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods Boston Concrete kin8nm Power Protein Wine Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0978 QD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4456 RQspline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} 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+page_content='2002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0947 NSS-X-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0588 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0962 NSS-α-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0976 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0957 Gain percentage 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6% Table 1: Mean Absolute Error (MAE) on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Test performance of the proposed method (NSS) and existing methods on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We use the 50th quantile estimator as our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The dash indicates unavailability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The shaded area is the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Bold is the top one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian: Gaussian kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' QD is quantity-driven methods proposed in (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQ spline proposed in (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' c-spline proposed in (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Boston, Concrete, Power is short for Boston Housing, Concrete Strength, Power Plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gain percentage is computed as (best nss - best baseline)/best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Training For simplicity, the proposed NSS methods use depth- 2 splines, which contain {(c-spline, p-spline), (c- spline, p-spline), (c-spline, c-spline), (p-spline, p- spline)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-sum is tuned with λ in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-chain nor- malizing of y in α chaining can be achieved by applying sigmoid layer or scaling by max value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As splines are monotonically-increasing functions, the spline value y with α = 0 is the minimum value of y and α = 1 yields the maximum value of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Scale is yscale = y−ymin ymax−ymin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We use a batch size=128 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='005 for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Results To demonstrate the effectiveness of proposed methods, we conduct experiments on synthetic, real-world tabular regres- sion, and time series forecasting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Synthetic data Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We generate 2000 data points (X ∈ R1 and y ∈ R1), where X is in the range of [−2, 2] and y has Gaus- sian distribution y ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3 sin(3x), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2x2), where sin is the sinusodial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We construct the validation and test sets to come from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Unlike real-world data, the synthetic data would have known quantile levels, that can be used for evaluating the accuracy of quantile estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We make the task more challenging by setting a data-dependent variance for the Gaussian noise to evaluate the ability of learn- ing condition-specific quantile values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 5 shows that the proposed NSS-chain and NSS-sum can capture the true under- lying quantiles, whereas QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018) struggles on the varying variance locations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' around x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The upper and lower black lines are the predicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-th quantiles for the observed data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' red dots), shown along with the ground truth quantiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' shaded red area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results indicate that more expressive NSS transformations are superior in more challenging scenarios, where true data points are distributed differently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=', distributions depend on the value of the inputs").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6 shows the calibration plot of the predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' true distributions at different quantile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y QD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y NSS-SUM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y NSS-CHAIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Figure 5: NSS on Synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We compare the per- formance of proposed NSS against existing methods QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The red dots are observed data points, shaded red area is the ground truth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% quantile levels, and the dark black lines are the predicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 Predicted percentile QD NSS-SUM NSS-CHAIN 2 0 2 X y X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile Calibration plot QD NSS-SUM NSS-CHAIN 2 0 2 X y X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile QD NSS-SUM NSS-CHAIN 2 0 2 X y X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 Figure 6: Calibration plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ground truth per- centiles at condition levels: X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The perfect calibration would correspond to the diagonal (red dotted) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Here, we show the true percentile p as the fraction of data in the dataset such that the p percentile of the predictive distribution is larger than the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The perfect prediction would be the diagonal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6 indicates that the proposed methods NSS-sum and NSS-chain can capture the proposed true distribution at various levels by close to the red line, whereas QD does not fit as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Real-world tabular regression We use UCI benchmarks (Asuncion and Newman 2007) that contain tabular data from diverse domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' real estate and physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Following (Salem, Langseth, and Ramampiaro 2020), the datasets are normalized with z-score standardiza- Methods Boston Concrete kin8nm Power Protein Wine Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0357 Global-Coarse∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1321 — Deep Quantile Aggregation∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1253 — QD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1164 RQspline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0424 p-sline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0160 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0358 c-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0351 NSS-X-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0358 NSS-α-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0363 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0357 Gain percentage 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% Table 2: Average pinball loss on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The test pinball loss (the lower, the better) is over 99 quantile levels, α = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='02, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='99}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The compared methods are Global-Coarse proposed in (Ratcliff 1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Deep Quantile Aggregation (DQA) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQspline (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ∗ indicates entries are from (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021) (under the same experiment setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods MAPE WAPE RMSE Q50 Q90 Q10 MLE (NB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='27240 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='70958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='27240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='15275 Mix MLE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26838 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='22556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14508 c-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26635 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='06332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14241 p-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26834 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14333 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26545 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='96697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14266 NSS-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26420 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='91726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14149 Table 3: Performance comparisons for time series forecasting on M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different evaluation metrics are included in this table for M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Detailed descriptions of the metrics are in Sec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Qk indicates the pinball loss of k-th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Q50 is the pinball loss of 50th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We evaluate the accuracy for both point predictions and quan- tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As the point predictions, we use the 50th quantile es- timator as our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Table 1 shows that the proposed NSS methods outperform the other existing methods on most datasets in mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In mean square error (MSE), the results are provided in Appendix Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We observe that the NSS-sum performs better than NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For quantile metrics, we use the pinball loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2) over 100 quantile levels α = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='02, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='99} in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results indicate that NSS consistently outperforms other al- ternatives across different UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In pinball loss, NSS-sum performs better than NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We attribute the superiority of NSS-sum for regression to make balance be- tween different transformation, which is helpful in explaining the variance in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Retail demand forecasting For time series forecasting, we focus on the M5 dataset, which contains time-varying sales data for retail goods, along with other relevant covariates like price, promotions, day of the week, special events etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' It represents an important real-world scenario, where the accurate estimation of the output distribution is crucial, as retailers use them to optimize prices or promotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The time series forecasting experiments are conducted by performing one-step ahead prediction, yielding predictions in an autoregressive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Table 3 shows the results of our method compared to other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We observe consistent outperformance of NSS in various forecasting evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from regression tasks, we observe that NSS-chain is better than NSS-sum, indicating its benefit in capturing time-dependent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Remarks on NSS-sum vs NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results show that NSS-sum is superior on regression, while NSS-chain has advantages on time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The observations may indicate NSS-sum is suitable for more constrained tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' regression, one time step time series-forecasting), where be- ing moderately expressive would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-sum is also more robust and easier to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' On the other hand, NSS-chain may be more expressive, which is beneficial to fit tasks re- quires more complex distributions at different time steps of the time series, but for individual step NSS-chain is not as accurate as NSS-sum in fitting the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Conclusion We propose a novel approach for modeling uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The proposed Neural Spline Search (NSS) method employs a se- ries of monotonic spline regression transformations, guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate the effectiveness of NSS for superior modeling of output distributions, on both synthetic and real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We leave the extensions to different operators and splines, including parametric distribu- tion transformations to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' References Asuncion, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Motion planning under uncertainty for on-road autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In ICRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Appendix Mean square error for UCI dataset Methods Bost House Concr Stren kin8nm Power plant Protein Wine Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0054 0.' 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+size 1236612 diff --git a/89A0T4oBgHgl3EQfOv-8/content/tmp_files/2301.02166v1.pdf.txt b/89A0T4oBgHgl3EQfOv-8/content/tmp_files/2301.02166v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..381ed83ae55f800a793bbba034741e10b74764a1 --- /dev/null +++ b/89A0T4oBgHgl3EQfOv-8/content/tmp_files/2301.02166v1.pdf.txt @@ -0,0 +1,681 @@ +Identification of lung nodules CT scan using YOLOv5 based on +convolution neural network +Haytham Al Ewaidat +ID 1,*, Youness El Brag +ID 2 +1Jordan University of Science and Technology, Faculty of Applied Medical Sciences, Department of Allied Medical +Sciences-Radiologic Technology, Irbid, Jordan, 22110 +2Abdelmalek Essaˆadi University of Science and Technology, Faculty of Multi-Disciplinary Larache, Department of +Computer Sciences, ksar el kebir , Morocco, 92150 +Correspondence author: Dr Haytham Al Ewaidat, Department of Allied Medical Sciences-Radiologic Technology, +Faculty of Applied Medical Sciences, Jordan University of Science and Technology. PO Box 3030, Irbid 22110, +Jordan Tel: (+962)27201000-26939; Fax: (+962)27201087; E-mail: haewaidat@just.edu.jo +arXiv:2301.02166v1 [eess.IV] 31 Dec 2022 + +Abstract +Purpose: The lung nodules localization in CT scan images is the most difficult task due to the complexity of the +arbitrariness of shape, size, and texture of lung nodules. This is a challenge to be faced when coming to developing +different solutions to improve detection systems. the deep learning approach showed promising results by using +convolutional neural network (CNN), especially for image recognition and it’s one of the most used algorithm in +computer vision. +Approach: we use (CNN) building blocks based on YOLOv5 (you only look once) to learn the features representations +for nodule detection labels, in this paper, we introduce a method for detecting lung cancer localization. Chest X-rays +and low-dose computed tomography are also possible screening methods, When it comes to recognizing nodules +in radiography, computer-aided diagnostic (CAD) system based on (CNN) have demonstrated their worth. One- +stage detector YOLOv5 trained on 280 annotated CT SCAN from a public dataset LIDC-IDRI based on segmented +pulmonary nodules. +Results: we analyze the predictions performance of the lung nodule locations, and demarcates the relevant CT scan +regions. In lung nodule localization the accuracy is measured as mean average precision (mAP). the mAP takes into +account how well the bounding boxes are fitting the labels as well as how accurate the predicted classes for those +bounding boxes, the accuracy we got 92.27% . +Conclusion: this study was to identify the nodule that were developing in the lungs of the participants. It was difficult +to find information on lung nodules in medical literature, +Keywords: computer-aided diagnostic, deep learning, Convolutional Neural Networks ,Lung Nodule. +*Address all correspondence to Haytham Al Ewaidat , haewaidat@just.edu.jo +1 +introduction +As far as noninvasive therapy and clinical assessment are concerned, medical image analysis offers +a tremendous advantage. X-rays, CTs, MRIs, and ultrasounds are utilized to make precise diag- +noses based on the obtained restorative images. By using attractive fields, CT can capture pictures +on film in medical imaging. One-of-a-kind lung cancer is responsible for 1.61 million fatalities per +year. Most of the cases of lung cancer in Indonesia are observed in the MIoT centers. If the tumor +is identified early, the survival percentage is better then. It’s not an easy task to find lung cancer +in its early stages. Approximately 80% of cancer patients are diagnosed at the core or accelerated +phase of the disease. Lung cancer is the second most common cancer among men and the tenth +most common among women worldwide. After breast and colorectal cancer, lung cancer is the +thirdly most common cancer among women. Features extraction in image processing is one of the +simplest and most efficient dimensionality reduction approaches. The non-invasive nature of CT +imaging is one of its most notable characteristics. It’s surprising to see angles increasing when +compared to other imaging modalities. +Computed tomography imaging is the best technique for examining lung disorders. CT scans, on +the other hand, have a high probability of false-positive results and are associated with cancer- +causing radiation exposure. When compared to standard-dose CT, low-dose CT utilizes a lot less +radiation contact power. The findings reveal that the detection sensitivity of low-dose and standard- +dose CT images is not significantly different. A well know database the National Lung Screening +Trial database shows that cancer-related fatalities were considerably decreased in the group that +was subjected to low-dose CT scans rather than chest radiography. The sensitivity of lung nodule +1 + +identification may be improved by the use of more detailed anatomical information, and better +image registration methods. As a result, the datasets have grown enormously. Up to 500 seg- +ments/slice may be generated from a single scan, depending on how thick the slice is. A single +slice is examined by a competent radiologist in 2–3.5 minutes. A radiologist’s workload keeps +rising while screening a Ct for the presence of a suspicious nodule. The detection sensitivity of +nodules is influenced by a variety of factors, including the size, location, form, nearby structures, +edges, and density, in addition to the CT slice section thickness. +Only 68 percent of lung cancer nodules are properly identified when only one radiologist doctor +views the scan, and up to 82% of the time when two radiologists check the scan, according to +the study results. Early diagnosis of malignant lung nodules by radiologists is a tough, time- +consuming, and laborious process in and of itself. The radiologist needs a lot of time to carefully +screen a large number of images, but this method is prone to mistake when looking for microscopic +nodules. +An aid for radiologists is required in this case to speed up readings, catch any missing nodules, +and enable improved localization. A primary goal of computer-aided detection systems was to +minimize radiologists’ labor and boost the detection rate of nodules. Newer CAD systems, on +the other hand, can distinguish between benign, and malignant nodules, which is helpful in the +screening process. CAD systems frequently beat professional radiologists in nodule identification +and localization tasks because of recent breakthroughs in deep learning models, particularly in +image processing. CAD systems, on the other hand, have an FP rate of 1–8.2 per scan and a +detection range of 38–100%, according to different studies. As a result of their likeness to one +other, benign, and malignant nodule remain a difficult challenge to solve. +During the screening process, a variety of mistakes might occur. For example, if a scan fails to +capture or recognize a specific region of the lesion or fails to distinguish between benign, and +malignant lesions in a patient’s body, the patient may be at risk of misdiagnosis. Most people +die as a result of misdiagnoses and delays in treatment because of these mistakes. In radiology, +over 4% of reports include diagnostic mistakes on a daily, and about 30% of aberrant radiological +diagnoses are ignored. Early-stage lung nodules may be detected and classified more accurately +using different methodologies such as deep learning. +Lung nodule identification using deep learning with a specific methodology is presented in this +research. lung CT images, physiological symptoms, and clinical indicators, the suggested ap- +proach reduces false-positive findings and eventually prevents invasive procedures. YOLOv5 is +used which has convolutional networks were built to identify and classify nodules. For nodule +identification. Nodule identification and classification using the publicly accessible data set LIDC- +IDRI surpasses state-of-the-art deep learning techniques. Using a variety of techniques, we were +able to reduce the number of false positives in the learning algorithm. +Lung nodule computer-aided detection (CAD) systems were originally developed in the late 1980s, +but the processing resources required for sophisticated image analysis methods at the time made +these efforts unattractive. For image analysis, and decision support systems based on computers, +the graphics processing unit and convolutional neural networks revolutionized their performance. +Some of the most important lung nodule identification and classification approaches have been +suggested by researchers in deep learning based medical images analysis models. For lung nodule +2 + +classification, Yutong Xie et al1 . proposed a method that utilizes Texture, Shape, and Deep Model +learned Data at the choice level. +Nodule heterogeneity may be shown with the use of this algorithm’s GLCM-based surface de- +scriptor, Fourier-shape descriptor, and a DCNN. Based on CNNs Chougrad et al.2 studied the +classification of breast cancer using a CAD framework. Transfer learning, on the other hand, takes +just a small number of medical images to train a system. With the use of the transfer learning +approach, the CNNs were taught to their fullest potential. In terms of accuracy, CNN came out on +top with a score of 98.94 percent. Using the wavelet transform and principal component analysis, +Heba Mohsen et al3 developed a DNN classifier for brain tumor classification. A technique of reg- +ularized linear discriminant analysis was developed in 2015 by Sharma et al,4 and it used a regular- +ization parameter to perform a standard cross-validation methodology. An appropriate collection +of characteristics is needed to evaluate medical data for illness prediction. Several evolutionary +algorithms have been used to find the best possible traits. Gravitational search and Elephant Herd +optimizations have recently been used to choose the best features.5 Another deep learning-based +model created by Kuruvilla, and Gunavathi, K. in 2014, an ANN-based cancer classification for +CT scans. Development of the statistical model used to classify the data was completed. Compared +to feed-forward networks feed-forward backpropagation networks are more accurate, according to +research. Classifier accuracy may be improved even more by using the skewness feature.6 +Lung cancer detection categorization is becoming more and more popular due to the rapid advance- +ment of pattern recognition and image processing methods. Textural evaluation of thin-section CT +images has been used in the literature to help distinguish various obstructive lung disorders. At- +tenuation distribution statistics, acquisition-length parameter, and co-occurrence descriptor are all +included in 13-dimensional vectors of local textures information developed by Chabat et al.7 A +Bayesian classifier is used for feature segmentation. These five scalar metrics, max, entropy, en- +ergy, contrast, and homogeneity were recovered per each co-occurrence matrix to minimize the +feature vector’s dimensionality. The textural characteristics of Solitary Pulmonary Nodules dis- +covered by CT have been described and assessed by Yanjie Zhu et al.8 It took 300 generations for +67 characteristics to be retrieved, however, only 25 features were picked. SVM-based classifiers +are used for classification. For Interstitial Lung Disease, Sang Cheol Park and colleagues9 used a +genetic algorithm to identify the best picture attributes (ILD). Hiram et al10 used the frequency do- +main, and SVM with RBF to classify lung nodule classifications. Solitary pulmonary nodules may +be automatically detected using an algorithm provided by Hong et al.11 True nodules are identified +and labeled on original images using an SVM classifier. The LIDC-IDRI images database was +used by Antonio et al,12 to classify lung nodules. Ecological taxonomic diversity and taxonomic +distinctness indexes are used for classification using SVM.13 Results show a 98.11% accuracy rate. +The mesh grid region growth approach was used in CT to select and analyze just the pixels that +were most likely to be relevant to the diagnosis. The ILD status of all unselected pixels was deter- +mined to be negative. To recognize lung cancer cells, Zhi-Hua et al14 presented Neural Ensemble- +based Detection (NED), which makes use of an artificial neural network ensemble. Using this +technology, it is possible to accurately identify cancer cells. An algorithm developed by Hui Chen +et al,15 uses a Neural Networks Ensemble to construct the categorization of a lung nodule on a thin +section CT image (NNE). A model suggested by Aggarwal, Furquan, and Kalra16 is characterized +by normal lung architecture by which segmentation is done using the best possible thresholds. Ge- +3 + +ometric, statistical, and grey level properties are used to extract features. Classification is done +using LDA. The accuracy is 84%, the sensitivity is 97%, and the specificity is 53%. An inference- +based approach to identify lung cancer nodules has been developed by Roy, Sirohi, and Patle.17 To +improve contrast, this technique employs grey transformations using an active contour model, the +image is segmented. Training the classifier is done by extracting features such as area, mean, major +axis, and minor axis length. Overall, the system’s accuracy rate is 94.12%. This approach has a +disadvantage in that it does not distinguish between benign cancers and those that are malignant. +Authors have used wavelet feature descriptors to classify lung nodules.18 One and two-level de- +compositions of wavelet transformations are used in this example. A total of 19 characteristics are +derived from each wavelet sub-band. SVM is used to distinguish between CT images that include +malignant nodules and those that do not. +2 +Material and Methods +In this section, we introduce our methods for Lung Nodules localization We use a one-stage- +method based on YOLOv5 detection , the methodology has been split into the following Subsec- +tions to explain the whole process of our method . +2.1 +Dataset +For this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image collection, +thoracic CT scans with marked-up annotated lesions are included. For the development, training, +and assessment of computer-assisted diagnostic (CAD) approaches for the detection and diagnosis +of lung cancer is a worldwide web-accessible resource One example of a public-private partnership +founded on consensus-based decision-making is this collaboration between the National Cancer +Institute, the Foundation for the National Institute of Health, and Food and Drug Administration +(FDA), which was spearheaded by NCI and supported by the FDA. This data collection, which +includes 239 Ct images for training and 41 images for validation. is a subset of the original dataset. +Some of the samples are given below in the following Fig.1. +Fig. 1: Samples from dataset LIDC-IDRI Lung Cancer +2.2 +Pre-Processing Data +Real-world data tends to be fragmentary, noisy, and inconclusive. This may lead to low-quality data +collection, which in turn can lead to low-quality models. Data Preprocessing offers procedures that +4 + +LIDC-IDRI-0001 +LightSpeed Plus +1-January-2000 +ST:2.50SL +ST +LittleEndianExplicit +Images:1/1 +400mA120.00kV +Series: +3000566 +WL: +-600WW:1600LIDC-IDRI-0003 +LightSpeed16 +1-January-2008 +ST:2.50S +T +LittleEndianExplicit +Images:1/1 +300mA120.00kV +Series: +3000611 +WL: +-600WW:1600may properly organize the data for better comprehension in the deep learning process to solve these +challenges. Data Preprocessing steps that have been used in this research study are given in the +following Fig.2. +Fig. 2: Preprocessing steps for images +2.3 +Model architecture +As discussed in the introduction in this research YOLOv5 model is used for feature extraction and +detection of lung nodules in CT scans. Let have a brief discussion about Yolov5 and its architecture. +2.3.1 +YOLOv5 for lung nodules localization +the whole structure of Yolov4 Optimal speed and accuracy of object detection19 is shown in Fig.3 +and YOLOv5 illustration representation shown in Fig.4. the YOLO family of models consists of +three main components to every single-stage object detector, and YOLOv5 has its own three main +modules +Fig. 3: Overview of YOLOv5 building blocks model architecture +(1) Backbone Figure 3:it’s mostly used to extract the elements of the most significant feature +from the images that have been provided. Cross Stage Partial Networks(CSP) is the back- +bone of YOLOv5’s feature extraction, which uses them to extract an image’s most informa- +tive details +5 + +input image +output +image +Gray +Noise +Edge +Filter +scale +Removal +DetectionBackbone: CSPDarknet +Neck: PANet +Head: Yolo Layer +BottleNeckCSP +Concat +BottleNeckCSP +Convlx1 +input image +UpSample +Conv3×3 S2 +Conv1×1 +Concat +BottleNeckCSP +Final +BottleNeckCSP +Concat +BottleNeckCSP +Output +Convl×1 +UpSample +Conv3x3 S2 +Convl×1 +Concat +SPP +BottleNeckCSP +BottleNeckCSP +Convl×1 +CSP +Cross Stage Partial Netword +Conv + Convolutional Layer + SPP + Spatial pyramid pooling +Concat + Concatenate Function(2) Neck Figure 3: it used to create feature pyramids. Feature pyramids aid models in generaliz- +ing successfully when it comes to object scaling. It aids in the identification of the same item +at various scales and dimensions. Feature pyramids are quite valuable and can help models +perform effectively on data that has never been examined. It’s not only FPN, BiFPN, and +PANet that are used in feature pyramid models +(3) Head Figure 3: it has layers that generate predictions from anchor boxes on features and +generated final output vectors with probabilities, object classes scores, and bounding boxes., +YOLOv5 uses the following choices for training20 +Fig. 4: Model detection can be considered a regression problem. The image is divided into S * S grids in +which bounding boxes are predicted for each grid cell, along with their confidence value +2.3.2 +Training Model +During the training and validation process, a total of 270 CT Scan images are used of which 239 +CT Scans are used for training and 41 are used for validations. For training, the Google Colab is +used which is an online platform for training models. Which provides 16GB GPU free for training. +The batch size was kept to 16 and the number of epochs was kept to 100. Splitting of data can be +seen in Fig.5 . +6 + +Head +Bounding Boxes + confidence Score +Backbone +Neck +images +Extraction of +Elaboration in +informative +Featuer +Labels +features +Pyramids +S x S Grid on input +image +Bounding Boxes + confidence Score + Localization of Lung Nodule +Class Probability +MapFig. 5: Dataset Splitting Diagram CT Scan images +3 +Results +the model had initial leverage to train faster and predict the location of lung nodules and demarcates +the relevant CT scan regions. before diving into the analysis of the results is necessary to explain +the statistical machine learning knowledge behind those results, the explanations have been split +into the following Subsections to explain the whole analysis of the method we use. +3.1 +Evaluation Metrics +In this section, we describe Charts of evaluation metrics that got from our experiment. It is known +to us that, in the computer Aide system, the main part is detecting the object inside the image. +Common metrics for measuring the performance of classification algorithms such as YOLOv5 +that are based on CNN include, Recall, precision, F-score, mAP, PR curve, F1 curve , IOU,21 +overlapping error, and boundary-based evaluation, the evaluation metrics we used is the mean +Average Precision (mAP),22 the precision, and F1-Curve. We will briefly explain them in the +following part. According to the theory of the statistical machine learning , precision is a two- +category statistical indicator whose formula is . +Precision : measures how accurate is our predictions was. the percentage of our predictions are +correct as shown in Fig.8,and following equation1. +Precision = +TP +TP + FP +(1) +Recall: measures how much of the true bbox were correctly predicted as shown in the following +equation.2. +Recall = +TP +TP + FN +(2) +7 + +Total Data +Distrubtion CT Scan +270Samples +Traning +Validation +images 239 Samples +images 41 Samplesmoreover, it is necessary to know TP, FP, and FN in the localization Nodules task. +(1) True positive (TP): IoU>[formula] (in this work, [formula] takes 0.2) the number of Local- +ization frames (the same Ground Truth is only calculated once23) +(2) False positive (FP): the number of check boxes for IoU<=[formula] or the number of re- +dundant check boxes that detect the same Ground Truth +(3) False negative (FN): the number of Ground Truths not detected +the IoU is a measures of the degree of overlap between two boundaries. We use that to measure +how much our predicted frame overlaps with the ground truth (the actual ground frame) ,the IOU +is shown with Fig.7 as follows, and the formula is as following Fig.6: +Fig. 6: Graphical representation of the Intersection over Union (IoU=0.2) calculation on a narrow-band +imaging. The light blue rectangle represents the ground truth bounding box, while the red rectangle repre- +sents the model prediction. The IoU is calculated by dividing the overlap area by the total area of union +after getting familiar with these definitions of statistical learning formulas, we introduce the mAP +(mean Average Precision). The mAP compares the ground-truth bounding box to the detected box +and returns a score. The higher the score, the more accurate the model is in its detections. +F1-score is defined as the harmonic average of precision and recall as shown in figure 10a: +F1 Score = 2 ∗ Precision ∗ Recall +Precision + Recall +(3) +8 + +LIDC-IDRI-0003 +LightSpeed16 +1-January-20e6 +Ground truth +intersection +0 +area of overlap +Prediction +Iou = +area of union +Ground truth +Ground truth +Prediction +ST:2.58 +Prediction +ittleEndianExplicit +ges: 1/1 +300mA120.00kl +WL:-600hW:1606(a) the Predicted location of a +Single Nodule +(b) the Predicted location of +tow Nodules in region +Fig. 7: Example of output Results +3.2 +Experiment’s Setting +the set Hyper-parameters of our fine tuning model are shown in Table 1, Our experiment uses +Pytorch framework deep learning on GPU Tesla K80 by Google open Platform Colab-research . +Parameters +Value +Batch size +16 +Image size +416 +Epoch +145 +Learning rate +0.01 +Optimizer +SGD +Table 1: Parameters and their value. +3.3 +Experiment’s Result and Analysis +To check the model’s predictions, and generalizations a few evaluation parameters must be tracked +during training and validation. There are several criteria to keep in mind while evaluating a box +loss, Precision, and recall values. The variable box benefits from objectivity and categorization. +Fig.9 shows all of the graphs that were used for this work. And Figure 10a shows the F1 indicator +training process for a single category that we want to be detected. The F1 score tends to be 0 with +increasing confidence . Training and validation box losses are reduced Fig.9, suggesting that the +model is sound good. the mAP is the abbreviation of median accuracy performances. The high +number indicates that this parameter is correct 92.27% as shown blow in Fig.8. +9 + +THORAXW/OCONTRAST +LightSpeed1e +1-January-2000 9:01:09 +Nodules 0.76 +ST:2.502 +ST +LittleEndianExplicit +Images:1/1 +265mA120.00kV +Series: +WL: +-600WW:1600LIDC-IDRI-0011 +LightSpeed16 +1-January-200e +Nodules 0.33 +Nodules 0.78 +ST:2.50SL: +ST +LittleEndianExplicit +Images:1/1 +265mA120.00kV +Series: +3000559 +WL: +-600wW:1600(a) mean Average Precision Evaluation +(b) Precision Evaluation +Fig. 8: the important Evaluation Metrics +(a) the training confidence of object pres- +ence loss +(b) the validation confidence of object pres- +ence loss +(c) the training bounding box regression +loss +(d) the validation bounding box regression +loss +Fig. 9: Results of feature extraction training and validation +Precision is needed to determine how accurate the model forecasts are 92.82% following the figure +8. Only excellent results may be achieved by using the recall method. the model performance +showed a good benefit of using Hyper-parameter tuning to make better Learning from data samples +and generalize good knowledge from distribution can be seen the following Fig.9 . Due to the +importance of both precision and recall, there is a precision-recall curve the shows the tradeoff +between the precision and recall values for different thresholds. This curve helps to select the best +threshold to maximize both metrics, tin the following Fig.10b +10 + +metrics/mAP 0.5 +0.8 +0.6 +0.4 +0.2 +Epoch +0 +0 +20 +40 +60 +80 +100 +120 +140metrics/precision +0.8 +0.6 +0.4 +0.2 +Epoch +0 +0 +20 +40 +60 +80 +100 +120 +140train/obj_loss +0.012 +0.01 +0.008 +0.006 +0.004 +0.002 +Epoch +0 +0 +20 +40 +60 +80 +100 +120 +140val/obj_loss +0.012 +0.01 +0.008 +0.006 +0.004 +0.002 +Epoch +0 +20 +40 +60 +80 +100 +120 +0 +140train/box_loss +0.12 +0.1 +0.08 +0.06 +0.04 +0.02 +Epoch +0 +0 +20 +40 +60 +80 +100 +120 +140val/box_loss +0.1 +0.08 +0.06 +0.04 +0.02 +Epoch +0 +0 +20 +40 +60 +80 +100 +120 +140(a) F1 indicator training process for single +category +(b) Precision — Recall Curve of the valida- +tion data +Fig. 10: the important Evaluation Metrics +4 +Discussion Ana Conclusion +This research examined at how an AI model can help readers detect viewable lung cancer in Ct +images. Residents identified more viewable lung cancer when AI was being used as a second +reader.In this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image col- +lection, thoracic CT scans with marked-up annotated lesions are included. Yolov5 model is used +for feature extraction and detection of lung nodules in CT scans.During the training and validation +process, a total of 270 CT Scan images are used of which 239 CT Scans are used for training and +41 are used for validations. In this study, the model’s performance was assessed using accuracy, +precision, and recall. The accuracy metric indicates how well the model recognised both positive +and negative instances. The precision metric measures how well the model predicts both negative +and positive cases. The model’s high accuracy, precision, and recall imply that it has a small error +possibility. Our findings imply that the AI technique assists low experienced individuals in terms +of recall while benefiting more-experienced audience in terms of precision. Previous research has +revealed that inexperienced readers are more likely to overlook lung malignancies, particularly le- +sions with a limited visibility score. In this research LIDC-IDRI dataset is used which have lung +nodules in it. The purpose of this study was to identify the nodule that were developing in the +lungs of the participants. It was difficult to find information on lung nodules in medical literature. +Research in the medical field often use deep learning. Deep learning will be utilised to develop +an algorithm with the support of previous medical imaging research, according to the findings of +a literature review. Using over 270 CT images , we were able to classify and identify nodules +using a deep learning algorithm. Using medical images analysis based on deep neural networks, +this study found that as much as 92.27% of cancer could be detected. Nodules on radiographs are +easier to see with its help. Using this technology in the future will help treat illnesses including +brain tumours and breast cancer. +5 +Disclosures +The authors declare that they have no conflict of interest +6 +Acknowledgments +We would like to thank our respectful research assistant Moath Alawaqla, for his distinguished role +of data collection. +11 + +1.0 +Nodules +all classes 0.91 at 0.437 +0.8 +0.6 +0.4 +0.2 +0.0 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence1.0 +Nodules 0.923 +all classes 0.923 mAP@0.5 +0.8 +0.6 + Precision +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall7 +Funding +This work supported by Jordan University of Science and Technology, Irbid-Jordan, +References +1 Y. Xie, J. Zhang, Y. 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Sun, et al., “Intelligent solutions in chest abnormality detection based +on yolov5 and resnet50,” Journal of Healthcare Engineering 2021 (2021). +13 + diff --git a/89A0T4oBgHgl3EQfOv-8/content/tmp_files/load_file.txt b/89A0T4oBgHgl3EQfOv-8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31b8f6db28c3a554eae6b50fe91d460f81d25fa9 --- /dev/null +++ b/89A0T4oBgHgl3EQfOv-8/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf,len=508 +page_content='Identification of lung nodules CT scan using YOLOv5 based on convolution neural network Haytham Al Ewaidat ID 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Youness El Brag ID 2 1Jordan University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Faculty of Applied Medical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Department of Allied Medical Sciences-Radiologic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Irbid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 22110 2Abdelmalek Essaˆadi University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Faculty of Multi-Disciplinary Larache,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Department of Computer Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' ksar el kebir ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Morocco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 92150 Correspondence author: Dr Haytham Al Ewaidat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Department of Allied Medical Sciences-Radiologic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Faculty of Applied Medical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Jordan University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' PO Box 3030, Irbid 22110, Jordan Tel: (+962)27201000-26939;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Fax: (+962)27201087;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' E-mail: haewaidat@just.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='jo arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='02166v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='IV] 31 Dec 2022 Abstract Purpose: The lung nodules localization in CT scan images is the most difficult task due to the complexity of the arbitrariness of shape, size, and texture of lung nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' This is a challenge to be faced when coming to developing different solutions to improve detection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the deep learning approach showed promising results by using convolutional neural network (CNN), especially for image recognition and it’s one of the most used algorithm in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Approach: we use (CNN) building blocks based on YOLOv5 (you only look once) to learn the features representations for nodule detection labels, in this paper, we introduce a method for detecting lung cancer localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Chest X-rays and low-dose computed tomography are also possible screening methods, When it comes to recognizing nodules in radiography, computer-aided diagnostic (CAD) system based on (CNN) have demonstrated their worth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' One- stage detector YOLOv5 trained on 280 annotated CT SCAN from a public dataset LIDC-IDRI based on segmented pulmonary nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Results: we analyze the predictions performance of the lung nodule locations, and demarcates the relevant CT scan regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In lung nodule localization the accuracy is measured as mean average precision (mAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the mAP takes into account how well the bounding boxes are fitting the labels as well as how accurate the predicted classes for those bounding boxes, the accuracy we got 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='27% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Conclusion: this study was to identify the nodule that were developing in the lungs of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It was difficult to find information on lung nodules in medical literature, Keywords: computer-aided diagnostic, deep learning, Convolutional Neural Networks ,Lung Nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Address all correspondence to Haytham Al Ewaidat , haewaidat@just.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='jo 1 introduction As far as noninvasive therapy and clinical assessment are concerned, medical image analysis offers a tremendous advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' X-rays, CTs, MRIs, and ultrasounds are utilized to make precise diag- noses based on the obtained restorative images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' By using attractive fields, CT can capture pictures on film in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' One-of-a-kind lung cancer is responsible for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='61 million fatalities per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Most of the cases of lung cancer in Indonesia are observed in the MIoT centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' If the tumor is identified early, the survival percentage is better then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It’s not an easy task to find lung cancer in its early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Approximately 80% of cancer patients are diagnosed at the core or accelerated phase of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Lung cancer is the second most common cancer among men and the tenth most common among women worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' After breast and colorectal cancer, lung cancer is the thirdly most common cancer among women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Features extraction in image processing is one of the simplest and most efficient dimensionality reduction approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The non-invasive nature of CT imaging is one of its most notable characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It’s surprising to see angles increasing when compared to other imaging modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Computed tomography imaging is the best technique for examining lung disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' CT scans, on the other hand, have a high probability of false-positive results and are associated with cancer- causing radiation exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' When compared to standard-dose CT, low-dose CT utilizes a lot less radiation contact power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The findings reveal that the detection sensitivity of low-dose and standard- dose CT images is not significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A well know database the National Lung Screening Trial database shows that cancer-related fatalities were considerably decreased in the group that was subjected to low-dose CT scans rather than chest radiography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The sensitivity of lung nodule 1 identification may be improved by the use of more detailed anatomical information, and better image registration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' As a result, the datasets have grown enormously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Up to 500 seg- ments/slice may be generated from a single scan, depending on how thick the slice is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A single slice is examined by a competent radiologist in 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A radiologist’s workload keeps rising while screening a Ct for the presence of a suspicious nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The detection sensitivity of nodules is influenced by a variety of factors, including the size, location, form, nearby structures, edges, and density, in addition to the CT slice section thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Only 68 percent of lung cancer nodules are properly identified when only one radiologist doctor views the scan, and up to 82% of the time when two radiologists check the scan, according to the study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Early diagnosis of malignant lung nodules by radiologists is a tough, time- consuming, and laborious process in and of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The radiologist needs a lot of time to carefully screen a large number of images, but this method is prone to mistake when looking for microscopic nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' An aid for radiologists is required in this case to speed up readings, catch any missing nodules, and enable improved localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A primary goal of computer-aided detection systems was to minimize radiologists’ labor and boost the detection rate of nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Newer CAD systems, on the other hand, can distinguish between benign, and malignant nodules, which is helpful in the screening process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' CAD systems frequently beat professional radiologists in nodule identification and localization tasks because of recent breakthroughs in deep learning models, particularly in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' CAD systems, on the other hand, have an FP rate of 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2 per scan and a detection range of 38–100%, according to different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' As a result of their likeness to one other, benign, and malignant nodule remain a difficult challenge to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' During the screening process, a variety of mistakes might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For example, if a scan fails to capture or recognize a specific region of the lesion or fails to distinguish between benign, and malignant lesions in a patient’s body, the patient may be at risk of misdiagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Most people die as a result of misdiagnoses and delays in treatment because of these mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In radiology, over 4% of reports include diagnostic mistakes on a daily, and about 30% of aberrant radiological diagnoses are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Early-stage lung nodules may be detected and classified more accurately using different methodologies such as deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Lung nodule identification using deep learning with a specific methodology is presented in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' lung CT images, physiological symptoms, and clinical indicators, the suggested ap- proach reduces false-positive findings and eventually prevents invasive procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' YOLOv5 is used which has convolutional networks were built to identify and classify nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For nodule identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Nodule identification and classification using the publicly accessible data set LIDC- IDRI surpasses state-of-the-art deep learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using a variety of techniques, we were able to reduce the number of false positives in the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Lung nodule computer-aided detection (CAD) systems were originally developed in the late 1980s, but the processing resources required for sophisticated image analysis methods at the time made these efforts unattractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For image analysis, and decision support systems based on computers, the graphics processing unit and convolutional neural networks revolutionized their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Some of the most important lung nodule identification and classification approaches have been suggested by researchers in deep learning based medical images analysis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For lung nodule 2 classification, Yutong Xie et al1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' proposed a method that utilizes Texture, Shape, and Deep Model learned Data at the choice level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Nodule heterogeneity may be shown with the use of this algorithm’s GLCM-based surface de- scriptor, Fourier-shape descriptor, and a DCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Based on CNNs Chougrad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2 studied the classification of breast cancer using a CAD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Transfer learning, on the other hand, takes just a small number of medical images to train a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' With the use of the transfer learning approach, the CNNs were taught to their fullest potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In terms of accuracy, CNN came out on top with a score of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='94 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using the wavelet transform and principal component analysis, Heba Mohsen et al3 developed a DNN classifier for brain tumor classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A technique of reg- ularized linear discriminant analysis was developed in 2015 by Sharma et al,4 and it used a regular- ization parameter to perform a standard cross-validation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' An appropriate collection of characteristics is needed to evaluate medical data for illness prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Several evolutionary algorithms have been used to find the best possible traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Gravitational search and Elephant Herd optimizations have recently been used to choose the best features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='5 Another deep learning-based model created by Kuruvilla, and Gunavathi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' in 2014, an ANN-based cancer classification for CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Development of the statistical model used to classify the data was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Compared to feed-forward networks feed-forward backpropagation networks are more accurate, according to research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Classifier accuracy may be improved even more by using the skewness feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='6 Lung cancer detection categorization is becoming more and more popular due to the rapid advance- ment of pattern recognition and image processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Textural evaluation of thin-section CT images has been used in the literature to help distinguish various obstructive lung disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' At- tenuation distribution statistics, acquisition-length parameter, and co-occurrence descriptor are all included in 13-dimensional vectors of local textures information developed by Chabat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='7 A Bayesian classifier is used for feature segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' These five scalar metrics, max, entropy, en- ergy, contrast, and homogeneity were recovered per each co-occurrence matrix to minimize the feature vector’s dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The textural characteristics of Solitary Pulmonary Nodules dis- covered by CT have been described and assessed by Yanjie Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='8 It took 300 generations for 67 characteristics to be retrieved, however, only 25 features were picked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' SVM-based classifiers are used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For Interstitial Lung Disease, Sang Cheol Park and colleagues9 used a genetic algorithm to identify the best picture attributes (ILD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Hiram et al10 used the frequency do- main, and SVM with RBF to classify lung nodule classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Solitary pulmonary nodules may be automatically detected using an algorithm provided by Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='11 True nodules are identified and labeled on original images using an SVM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The LIDC-IDRI images database was used by Antonio et al,12 to classify lung nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Ecological taxonomic diversity and taxonomic distinctness indexes are used for classification using SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='13 Results show a 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='11% accuracy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The mesh grid region growth approach was used in CT to select and analyze just the pixels that were most likely to be relevant to the diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The ILD status of all unselected pixels was deter- mined to be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' To recognize lung cancer cells, Zhi-Hua et al14 presented Neural Ensemble- based Detection (NED), which makes use of an artificial neural network ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using this technology, it is possible to accurately identify cancer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' An algorithm developed by Hui Chen et al,15 uses a Neural Networks Ensemble to construct the categorization of a lung nodule on a thin section CT image (NNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A model suggested by Aggarwal, Furquan, and Kalra16 is characterized by normal lung architecture by which segmentation is done using the best possible thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Ge- 3 ometric, statistical, and grey level properties are used to extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Classification is done using LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The accuracy is 84%, the sensitivity is 97%, and the specificity is 53%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' An inference- based approach to identify lung cancer nodules has been developed by Roy, Sirohi, and Patle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='17 To improve contrast, this technique employs grey transformations using an active contour model, the image is segmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Training the classifier is done by extracting features such as area, mean, major axis, and minor axis length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Overall, the system’s accuracy rate is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' This approach has a disadvantage in that it does not distinguish between benign cancers and those that are malignant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Authors have used wavelet feature descriptors to classify lung nodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='18 One and two-level de- compositions of wavelet transformations are used in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' A total of 19 characteristics are derived from each wavelet sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' SVM is used to distinguish between CT images that include malignant nodules and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 2 Material and Methods In this section, we introduce our methods for Lung Nodules localization We use a one-stage- method based on YOLOv5 detection , the methodology has been split into the following Subsec- tions to explain the whole process of our method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1 Dataset For this research, the dataset has been collected from LIDC-IDRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In LIDC-IDRI image collection, thoracic CT scans with marked-up annotated lesions are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For the development, training, and assessment of computer-assisted diagnostic (CAD) approaches for the detection and diagnosis of lung cancer is a worldwide web-accessible resource One example of a public-private partnership founded on consensus-based decision-making is this collaboration between the National Cancer Institute, the Foundation for the National Institute of Health, and Food and Drug Administration (FDA), which was spearheaded by NCI and supported by the FDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' This data collection, which includes 239 Ct images for training and 41 images for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' is a subset of the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Some of the samples are given below in the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 1: Samples from dataset LIDC-IDRI Lung Cancer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2 Pre-Processing Data Real-world data tends to be fragmentary, noisy, and inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' This may lead to low-quality data collection, which in turn can lead to low-quality models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Data Preprocessing offers procedures that 4 LIDC-IDRI-0001 LightSpeed Plus 1-January-2000 ST:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='50SL ST LittleEndianExplicit Images:1/1 400mA120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='00kV Series: 3000566 WL: 600WW:1600LIDC-IDRI-0003 LightSpeed16 1-January-2008 ST:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='50S T LittleEndianExplicit Images:1/1 300mA120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='00kV Series: 3000611 WL: 600WW:1600may properly organize the data for better comprehension in the deep learning process to solve these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Data Preprocessing steps that have been used in this research study are given in the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 2: Preprocessing steps for images 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='3 Model architecture As discussed in the introduction in this research YOLOv5 model is used for feature extraction and detection of lung nodules in CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Let have a brief discussion about Yolov5 and its architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1 YOLOv5 for lung nodules localization the whole structure of Yolov4 Optimal speed and accuracy of object detection19 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='3 and YOLOv5 illustration representation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the YOLO family of models consists of three main components to every single-stage object detector, and YOLOv5 has its own three main modules Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 3: Overview of YOLOv5 building blocks model architecture (1) Backbone Figure 3:it’s mostly used to extract the elements of the most significant feature from the images that have been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Cross Stage Partial Networks(CSP) is the back- bone of YOLOv5’s feature extraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' which uses them to extract an image’s most informa- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='tive details ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='input image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Gray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Removal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='DetectionBackbone: CSPDarknet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Neck: PANet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Head: Yolo Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Convlx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='input image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='UpSample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Conv3×3 S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Conv1×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Convl×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='UpSample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Conv3x3 S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Convl×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='SPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='BottleNeckCSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Convl×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='CSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Cross Stage Partial Netword ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Convolutional Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='SPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Spatial pyramid pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='Concatenate Function(2) Neck Figure 3: it used to create feature pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Feature pyramids aid models in generaliz- ing successfully when it comes to object scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It aids in the identification of the same item at various scales and dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Feature pyramids are quite valuable and can help models perform effectively on data that has never been examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It’s not only FPN, BiFPN, and PANet that are used in feature pyramid models (3) Head Figure 3: it has layers that generate predictions from anchor boxes on features and generated final output vectors with probabilities, object classes scores, and bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=', YOLOv5 uses the following choices for training20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 4: Model detection can be considered a regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The image is divided into S * S grids in which bounding boxes are predicted for each grid cell, along with their confidence value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2 Training Model During the training and validation process, a total of 270 CT Scan images are used of which 239 CT Scans are used for training and 41 are used for validations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' For training, the Google Colab is used which is an online platform for training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Which provides 16GB GPU free for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The batch size was kept to 16 and the number of epochs was kept to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Splitting of data can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 6 Head Bounding Boxes + confidence Score Backbone Neck images Extraction of Elaboration in informative Featuer Labels features Pyramids S x S Grid on input image Bounding Boxes + confidence Score Localization of Lung Nodule Class Probability MapFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 5: Dataset Splitting Diagram CT Scan images 3 Results the model had initial leverage to train faster and predict the location of lung nodules and demarcates the relevant CT scan regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' before diving into the analysis of the results is necessary to explain the statistical machine learning knowledge behind those results, the explanations have been split into the following Subsections to explain the whole analysis of the method we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1 Evaluation Metrics In this section, we describe Charts of evaluation metrics that got from our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It is known to us that, in the computer Aide system, the main part is detecting the object inside the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Common metrics for measuring the performance of classification algorithms such as YOLOv5 that are based on CNN include, Recall, precision, F-score, mAP, PR curve, F1 curve , IOU,21 overlapping error, and boundary-based evaluation, the evaluation metrics we used is the mean Average Precision (mAP),22 the precision, and F1-Curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' We will briefly explain them in the following part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' According to the theory of the statistical machine learning , precision is a two- category statistical indicator whose formula is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Precision : measures how accurate is our predictions was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the percentage of our predictions are correct as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='8,and following equation1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Precision = TP TP + FP (1) Recall: measures how much of the true bbox were correctly predicted as shown in the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Recall = TP TP + FN (2) 7 Total Data Distrubtion CT Scan 270Samples Traning Validation images 239 Samples images 41 Samplesmoreover, it is necessary to know TP, FP, and FN in the localization Nodules task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' (1) True positive (TP): IoU>[formula] (in this work, [formula] takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2) the number of Local- ization frames (the same Ground Truth is only calculated once23) (2) False positive (FP): the number of check boxes for IoU<=[formula] or the number of re- dundant check boxes that detect the same Ground Truth (3) False negative (FN): the number of Ground Truths not detected the IoU is a measures of the degree of overlap between two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' We use that to measure how much our predicted frame overlaps with the ground truth (the actual ground frame) ,the IOU is shown with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='7 as follows, and the formula is as following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='6: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 6: Graphical representation of the Intersection over Union (IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2) calculation on a narrow-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The light blue rectangle represents the ground truth bounding box, while the red rectangle repre- sents the model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The IoU is calculated by dividing the overlap area by the total area of union after getting familiar with these definitions of statistical learning formulas, we introduce the mAP (mean Average Precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The mAP compares the ground-truth bounding box to the detected box and returns a score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The higher the score, the more accurate the model is in its detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' F1-score is defined as the harmonic average of precision and recall as shown in figure 10a: F1 Score = 2 ∗ Precision ∗ Recall Precision + Recall (3) 8 LIDC-IDRI-0003 LightSpeed16 1-January-20e6 Ground truth intersection 0 area of overlap Prediction Iou = area of union Ground truth Ground truth Prediction ST:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='58 Prediction ittleEndianExplicit ges: 1/1 300mA120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='00kl WL:-600hW:1606(a) the Predicted location of a Single Nodule (b) the Predicted location of tow Nodules in region Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 7: Example of output Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='2 Experiment’s Setting the set Hyper-parameters of our fine tuning model are shown in Table 1, Our experiment uses Pytorch framework deep learning on GPU Tesla K80 by Google open Platform Colab-research .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Parameters Value Batch size 16 Image size 416 Epoch 145 Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='01 Optimizer SGD Table 1: Parameters and their value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='3 Experiment’s Result and Analysis To check the model’s predictions, and generalizations a few evaluation parameters must be tracked during training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' There are several criteria to keep in mind while evaluating a box loss, Precision, and recall values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The variable box benefits from objectivity and categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='9 shows all of the graphs that were used for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' And Figure 10a shows the F1 indicator training process for a single category that we want to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The F1 score tends to be 0 with increasing confidence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Training and validation box losses are reduced Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='9, suggesting that the model is sound good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the mAP is the abbreviation of median accuracy performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The high number indicates that this parameter is correct 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='27% as shown blow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 9 THORAXW/OCONTRAST LightSpeed1e 1-January-2000 9:01:09 Nodules 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='76 ST:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='502 ST LittleEndianExplicit Images:1/1 265mA120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='00kV Series: WL: 600WW:1600LIDC-IDRI-0011 LightSpeed16 1-January-200e Nodules 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='33 Nodules 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='78 ST:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='50SL: ST LittleEndianExplicit Images:1/1 265mA120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='00kV Series: 3000559 WL: 600wW:1600(a) mean Average Precision Evaluation (b) Precision Evaluation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 8: the important Evaluation Metrics (a) the training confidence of object pres- ence loss (b) the validation confidence of object pres- ence loss (c) the training bounding box regression loss (d) the validation bounding box regression loss Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 9: Results of feature extraction training and validation Precision is needed to determine how accurate the model forecasts are 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='82% following the figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Only excellent results may be achieved by using the recall method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' the model performance showed a good benefit of using Hyper-parameter tuning to make better Learning from data samples and generalize good knowledge from distribution can be seen the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Due to the importance of both precision and recall, there is a precision-recall curve the shows the tradeoff between the precision and recall values for different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' This curve helps to select the best threshold to maximize both metrics, tin the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='10b 10 metrics/mAP 0.' metadata={'source': 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+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='002 Epoch 0 0 20 40 60 80 100 120 140val/obj_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='002 Epoch 0 20 40 60 80 100 120 0 140train/box_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='02 Epoch 0 0 20 40 60 80 100 120 140val/box_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='02 Epoch 0 0 20 40 60 80 100 120 140(a) F1 indicator training process for single category (b) Precision — Recall Curve of the valida- tion data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 10: the important Evaluation Metrics 4 Discussion Ana Conclusion This research examined at how an AI model can help readers detect viewable lung cancer in Ct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Residents identified more viewable lung cancer when AI was being used as a second reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='In this research, the dataset has been collected from LIDC-IDRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In LIDC-IDRI image col- lection, thoracic CT scans with marked-up annotated lesions are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Yolov5 model is used for feature extraction and detection of lung nodules in CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='During the training and validation process, a total of 270 CT Scan images are used of which 239 CT Scans are used for training and 41 are used for validations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In this study, the model’s performance was assessed using accuracy, precision, and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The accuracy metric indicates how well the model recognised both positive and negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The precision metric measures how well the model predicts both negative and positive cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The model’s high accuracy, precision, and recall imply that it has a small error possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Our findings imply that the AI technique assists low experienced individuals in terms of recall while benefiting more-experienced audience in terms of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Previous research has revealed that inexperienced readers are more likely to overlook lung malignancies, particularly le- sions with a limited visibility score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' In this research LIDC-IDRI dataset is used which have lung nodules in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' The purpose of this study was to identify the nodule that were developing in the lungs of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' It was difficult to find information on lung nodules in medical literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Research in the medical field often use deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Deep learning will be utilised to develop an algorithm with the support of previous medical imaging research, according to the findings of a literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using over 270 CT images , we were able to classify and identify nodules using a deep learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using medical images analysis based on deep neural networks, this study found that as much as 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='27% of cancer could be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Nodules on radiographs are easier to see with its help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' Using this technology in the future will help treat illnesses including brain tumours and breast cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 5 Disclosures The authors declare that they have no conflict of interest 6 Acknowledgments We would like to thank our respectful research assistant Moath Alawaqla, for his distinguished role of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89A0T4oBgHgl3EQfOv-8/content/2301.02166v1.pdf'} +page_content='0 Nodules all classes 0.' metadata={'source': 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sha256:5656bd6aa85c1de53f713845c8d3e2ff2c45987de0745f2223ed99f532cd27ac +size 96482 diff --git a/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/2301.00315v1.pdf.txt b/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/2301.00315v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c960807eb626edfad4b103cf50cb814861e2a1f --- /dev/null +++ b/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/2301.00315v1.pdf.txt @@ -0,0 +1,3344 @@ +Parametric “Non-nested” Discriminants +for Multiplicities of Univariate Polynomials +Hoon Hong +Department of Mathematics, North Carolina State University +Box 8205, Raleigh, NC 27695, USA +hong@ncsu.edu +Jing Yang∗ +SMS–HCIC–School of Mathematics and Physics, +Center for Applied Mathematics of Guangxi, +Guangxi Minzu University, Nanning 530006, China +yangjing0930@gmail.com +Abstract +We consider the problem of complex root classification, i.e., finding the conditions on the coefficients +of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known +that such conditions can be written as conjunctions of several polynomial equations and one inequation +in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It +is well known that discriminants can be obtained by using repeated parametric gcd’s. The resulting +discriminants are usually nested determinants, that is, determinants of matrices whose entries are deter- +minants, and so son. In this paper, we give a new type of discriminants which are not based on repeated +gcd’s. The new discriminants are simpler in that they are non-nested determinants and have smaller +maximum degrees. +1 +Introduction +In this paper, we consider the problem of complex root classification, i.e., finding the conditions on the +coefficients of a polynomial over the complex field C for every potential multiplicity structure its complex +roots may have. For example, consider a quintic polynomial F = a5x5 +a4x4 +a3x3 +a2x2 +a1x+a0 where +ai’s take values over C. We would like to find conditions C0, C1, . . . , C6 on a = (a0, . . . , a5) such that +multiplicity structure of F = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(1, 1, 1, 1, 1) +if +C0 (a) holds +(2, 1, 1, 1) +if +C1 (a) holds +(2, 2, 1) +if +C2 (a) holds +(3, 1, 1) +if +C3 (a) holds +(3, 2) +if +C4 (a) holds +(4, 1) +if +C5 (a) holds +(5) +if +C6 (a) holds +In general, the problem is stated as follows: +Problem: For every µ = (µ1, . . . , µm) such that µ1 ≥ . . . ≥ µm > 0 and µ1 +· · ·+µm = n, find a condition +on the coefficients of a polynomial Fover C of degree n such that the multiplicity structure of F is µ. +∗Corresponding author. +1 +arXiv:2301.00315v1 [cs.SC] 1 Jan 2023 + +The problem is important because many tasks in mathematics, science and engineering can be reduced to +the problem. Due to its importance, the problem and several related problems have been already carefully +studied [4, 6, 7, 8, 9, 11]. +The problem can be viewed as a generalization of a well known problem of finding a condition on +coefficients such that the polynomial has a given number of distinct roots. +This subproblem has been +extensively studied. For instance, the subdiscriminant theory provides a complete solution to the subproblem: +a univariate polynomial of degree n has m distinct roots if and only if its 0-th, . . ., (n − m − 1)-th psd’s +(i.e., principal subdiscriminant coefficient) vanish and the (n−m)-th psd does not. For details, see standard +textbooks on computational algebra (e.g., [1]). +In [11], Yang, Hou and Zeng gave an algorithm to generate conditions for discriminating different mul- +tiplicity structures of a univariate polynomial (referred as YHZ’s condition hereinafter) by making use of +repeated gcd computation for parametric polynomials [2, 3, 10]. It is based on a similar idea adopted by +Gonzalez-Vega et al. [4] for solving the real root classification and quantifier elimination problems by using +Sturm-Habicht sequences. The conditions produced by these methods are conjunctions of several polynomial +equations and one inequation on the coefficients. Those polynomials in he coefficients are called discrimi- +nants for multiplicities. The maximum degree of the discriminants grow exponentially in the degree of F. +Furthermore, each discriminant is a “nested” determinant, that is, it is a determinant of a matrix whose +entries are again determinants and so on. +In [6], the authors developed a new type of multiplicity discriminants to distinguish different multiplicities +when the number of distinct roots is fixed. The main idea is to convert the multiplicity condition expressed +as a permanent inequation in roots into a sum of determinants in coefficients. In order to generate conditions +for all the possible multiplicity structures of a univariate polynomial, one may first use subdiscriminants in +classical resultant theory to decide the number of distinct complex roots and then add one more inequation +to discriminate different multiplicity structures with the same number of distinct roots. In the new condition, +the maximum degree of the discriminants grow linearly in the degree of F, which makes the size of discrim- +inants significantly smaller. However, the form of resulting discriminants is a sum of many determinants, +which makes the further analysis (reasoning) difficult. +The main contribution in this paper is to provide a new type of discriminants, which are non-nested +determinants and whose max degrees are smaller than those in the previous methods. +The method is +based on a significantly different theory and techniques from the previous methods (which are essentially +based on repeated parametric gcd or subdiscriminant theory). The new condition is given by a newly devised +multiplicity discriminant in coefficients for every potential multiplicity vector of a given degree, which can be +viewed as a generalization of subdiscriminant theory to higher order derivatives. To build up the connection +between the new discriminants and multiple roots, we first convert it into the ratio of two determinants +in terms of generic roots (without considering the multiplicities). Then by making use of the connection +between divided difference with multiple nodes and the derivatives of higher orders at the nodes, we integrate +the multiplicity information into the expression and convert it into an expression in terms of multiple roots. +After careful manipulation, it is shown that the new discriminant can capture the multiplicity information. +The paper is structured as follows. In Section 2, we first present the problem to be solved in a formal +way. In Section 3, we give a precise statement of the main result of the paper (Theorem 9). Then a proof +of Theorem 9 is provided in Section 4. The proof is long thus we divide the proof into three subsections +which are interesting on their own. +In Section 5, we compare the form and size of polynomials in the +multiplicity-discriminant condition in Theorem 9 and those given by previous works. +2 +Problem +Definition 1 (Multiplicity vector). Let F ∈ C [x] with m distinct complex roots, say r1, . . . , rm, with mul- +tiplicities µ1, . . . , µm respectively. Without losing generality, we assume that µ1 ≥ · · · ≥ µm > 0. Then the +multiplicity vector of F, written as mult (F), is defined by +mult (F) = (µ1, . . . , µm) +2 + +Example 2. Let F = x5 − 5x4 + 7x3 + x2 − 8x + 4. Then mult (F) = (2, 2, 1), since it can be verified that +F = (x − 1)2 (x + 1)1 (x − 2)2. Note that the multiplicity vector is a partition of 5, which is the degree of F. +Definition 3 (Potential multiplicity vectors). Let n be a positive integer. Let M(n) stand for the set of all +the potential multiplicity vectors of polynomials of degree n, equivalently, the set of all partitions of n, that +is, +M(n) = {(µ1, . . . , µm) : µ1 + · · · + µm = n, µ1 ≥ · · · ≥ µm > 0} +Example 4. M (5) = { (1, 1, 1, 1, 1) , +(2, 1, 1, 1) , (2, 2, 1) , (3, 1, 1) , (3, 2) , (4, 1) , (5) }. +Problem 5 (Parametric multiplicity problem). The parametric multiplicity problem is stated as: +In : n, a positive integer standing for the polynomial of degree n with parametric coefficients a, that is, +F = +n +� +i=0 +aixi where an ̸= 0 +Out: For each µ ∈ M(n), find a condition Cµ on a such that mult (F) = µ. +3 +Main Result +Definition 6 (Determinant polynomial). Consider a vector of univariate polynomials +P = +� +�� +P0 +... +Pk +� +�� ∈ C[x]k+1 +where deg Pi ≤ k and Pi = � +0≤j≤k aijxj. The coefficient matrix of P, written as C (P) , is defined by +C (P) = coef (P) = +� +�� +coef (P0) +... +coef (Pk) +� +�� = +� +�� +a0k +· · · +a00 +... +... +akk +· · · +ak0 +� +�� +The determinant polynomial of P, written as dp (P) , is defined by +dp (P) = |C (P) | +Definition 7 (Multiplicity Discriminant). Let F = �n +i=0 aixi where an ̸= 0. Let γ = (γ1, . . . , γs) ∈ M (n). +The the γ-discriminant of F, written as D (γ) , is defined by +D (γ) = 1 +an +dp +� +�������������������� +F (0)xγ0−1 +... +F (0)x0 +F (1)xγ1−1 +... +F (1)x0 +... +F (s)xγs−1 +... +F (s)x0 +� +�������������������� +where γ0 is the smallest so that the above matrix is square. It is straightforward to show that γ0 = γ1 − 1. +3 + +Example 8. Let n = 5 and F = �n +i=0 aixi and an ̸= 0. Then +D (5) += +dp +� +������������� +F (0)x3 +F (0)x2 +F (0)x1 +F (0)x0 +F (1)x4 +F (1)x3 +F (1)x2 +F (1)x1 +F (1)x0 +� +������������� += +1 +a5 +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +������������������ +D (4, 1) += +dp +� +����������� +F (0)x2 +F (0)x1 +F (0)x0 +F (1)x3 +F (1)x2 +F (1)x1 +F (1)x0 +F (2)x0 +� +����������� += +1 +a5 +���������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +���������������� +D (3, 2) += +dp +� +��������� +F (0)x1 +F (0)x0 +F (1)x2 +F (1)x1 +F (1)x0 +F (2)x1 +F (2)x0 +� +��������� += +1 +a5 +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +�������������� +D (3, 1, 1) += +dp +� +��������� +F (0)x1 +F (0)x0 +F (1)x2 +F (1)x1 +F (1)x0 +F (2)x0 +F (3)x0 +� +��������� += +1 +a5 +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4 · 3a5 +4 · 3 · 2a4 +3 · 2 · 1a3 +�������������� +D (2, 2, 1) += +dp +� +������� +F (0)x0 +F (1)x1 +F (1)x0 +F (2)x1 +F (2)x0 +F (3)x0 +� +������� += +1 +a5 +������������ +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4 · 3a5 +4 · 3 · 2a4 +3 · 2 · 1a3 +������������ +D (2, 1, 1, 1) += +dp +� +������� +F (0)x0 +F (1)x1 +F (1)x0 +F (2)x0 +F (3)x0 +F (4)x0 +� +������� += +1 +a5 +������������ +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +1a1 +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4 · 3a5 +4 · 3 · 2a4 +3 · 2 · 1a3 +5 · 4 · 3 · 2a5 +4 · 3 · 2 · 1a4 +������������ +D (1, 1, 1, 1, 1) += +dp +� +����� +F (1)x0 +F (2)x0 +F (3)x0 +F (4)x0 +F (5)x0 +� +����� += +1 +a5 +���������� +5a5 +4a4 +3a3 +2a2 +1a1 +5 · 4a5 +4 · 3a4 +3 · 2a3 +2 · 1a2 +5 · 4 · 3a5 +4 · 3 · 2a4 +3 · 2 · 1a3 +5 · 4 · 3 · 2a5 +4 · 3 · 2 · 1a4 +5 · 4 · 3 · 2 · 1a5 +���������� +4 + +Note that the last one D (1, 1, 1, 1, 1) = 5544332211a4 +5. Since a5 ̸= 0, we see that D (1, 1, 1, 1, 1) ̸= 0. +Theorem 9 (Main Result). Let F = �n +i=0 aixi +where an ̸= 0. Let M(n) = +� +µ0, µ1, . . . , µp +� +where the +entries are ordered in the lexicographically increasing order, that is, µ0 ≺lex µ1 ≺lex · · · ≺lex µp. Then we +have the following conditions for the multiplicity vectors. +mult(F) = +� +� +� +� +� +� +� +� +� +µ0 +if +D +� +µp +� +̸= 0 +µ1 +else if +D +� +µp−1 +� +̸= 0 +... +... +... +̸= 0 +µp +else if +D (µ0) +̸= 0 +Equivalently, +mult(F) = µi +⇐⇒ +D +� +µp +� += · · · = D +� +µp−i−1 +� += 0 ∧ D +� +µp−i +� +̸= 0 +Example 10. We have the following condition for each multiplicity vector for degree 5. +mult(F) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(1, 1, 1, 1, 1) +if +D (5) +̸= 0 +(2, 1, 1, 1) +else if +D (4, 1) +̸= 0 +(2, 2, 1) +else if +D (3, 2) +̸= 0 +(3, 1, 1) +else if +D (3, 1, 1) +̸= 0 +(3, 2) +else if +D (2, 2, 1) +̸= 0 +(4, 1) +else if +D (2, 1, 1, 1) +̸= 0 +(5) +else if +D (1, 1, 1, 1, 1) +̸= 0 +Equivalently, for instance, +mult(F) = (2, 2, 1) +⇐⇒ +D (5) = D (4, 1) = 0 ∧ D (3, 2) ̸= 0 +Remark 11. +1. Note that µ0 = (1, . . . , 1) and +D (µ0) = 1 +an +��������� +nan +· · · +1a1 +n (n − 1) an +· · · +2 · 1a2 +... +... +n (n − 1) · · · 1an +��������� += +n +� +i=1 +ii · an−1 +n +̸= 0 +Hence the last condition is always satisfied and there is no need to check the condition. +2. Note that µi and µp−i are conjugates of each other. +4 +Proof of the Main Theorem +Here is a high level view of the proof. +We start with converting D (µ) into the equivalent symmetric +polynomials in generic roots (though displayed as a ratio of two determinants) which is easier to embed the +multiplicity information. Then by making use of the connection between divided difference with multiple +nodes and the derivatives of higher orders at the nodes, we convert the expression in generic roots to that +in distinct roots with multiplicity information integrated. The theorem will be proved by eliminating the +entries in the determinantal expression obtained from the second stage which may vanish under the given +multiplicity structure. +5 + +4.1 +Multiplicity discriminant in terms of roots +We first understand what the multiplicity discriminants look like in terms of roots. . +Notation 12. V (α1, . . . , αn) := +������� +αn−1 +1 +· · · +αn−1 +n +... +... +α0 +1 +· · · +α0 +n +������� +Lemma 13 (Multiplicity discriminant in generic roots). Let F = an(x−α1) · · · (x−αn) and γ = (γ1, . . . , γs) ∈ +M(n). Then +D(γ) = +aγ1−2 +n +· +������������������ +F (1)(α1)αγ1−1 +1 +· · · +F (1)(αn)αγ1−1 +n +... +... +F (1)(α1)α0 +1 +· · · +F (1)(αn)α0 +n +... +... +F (s)(α1)αγs−1 +1 +· · · +F (s)(αn)αγs−1 +n +... +... +F (s)(α1)α0 +1 +· · · +F (s)(αn)α0 +n +������������������ +V (α1, . . . , αn) +(1) +Proof. +1. Since γ1 ≥ · · · ≥ γs and γ0 = γ1 − 1, we have +deg(F (0)xn−2) > · · · > deg(F (0)xγ1−1) > max(F (0)xγ0−1, F (1)xγ1−1, . . . , F (s)xγs−1) +Thus +D(γ) = 1 +an +dp +� +�������������������� +F (0)xγ1−2 +... +F (0)x0 +F (1)xγ1−1 +... +F (1)x0 +... +F (s)xγs−1 +... +F (s)x0 +� +�������������������� += 1 +an +· aγ1−n +n +dp +� +��������������������������� +F (0)xn−2 +... +F (0)xγ1−1 +F (0)xγ1−2 +... +F (0)x0 +F (1)xγ1−1 +... +F (1)x0 +... +F (s)xγs−1 +... +F (s)x0 +� +��������������������������� += aγ1−n−1 +n +dp +� +�������������������� +F (0)xn−2 +... +F (0)x0 +F (1)xγ1−1 +... +F (1)x0 +... +F (s)xγs−1 +... +F (s)x0 +� +�������������������� +2. Now we recall the following result from [6] which is the key for proving the lemma. Let G1, . . . , Gn ∈ +C [x]2n−2 where C [x]2n−2 consists of all the polynomials in x with degree no greater than 2n−2. Then +dp +� +��������� +F (0)xn−2 +... +F (0)x0 +G1 +... +Gn +� +��������� += +an−1 +n +· +������� +G1(α1) +· · · +G1(αn) +... +... +Gn(α1) +· · · +Gn (αn) +������� +V (α1, . . . , αn) +(2) +6 + +3. After specializing G1, . . . , Gn in (2) with F (1)xγ1−1, . . . , F (1)x0, . . . , F (s)xγs−1, . . . , F (s)x0, respectively, +we have +D(γ) = aγ1−n−1 +n +· +an−1 +n +· +����������������� +� +F (1)xγ1−1� +(α1) +· · · +� +F (1)xγ1−1� +(αn) +... +... +� +F (1)x0� +(α1) +· · · +� +F (1)x0� +(αn) +... +... +� +F (s)xγs−1� +(α1) +· · · +� +F (s)xγs−1� +(αn) +... +... +� +F (s)x0� +(α1) +· · · +� +F (s)x0� +(αn) +����������������� +V (α1, . . . , αn) +which can be easily simplified into (1). +Remark 14. It is very important to note that the right hand side is a polynomial function in α1, . . . , αn, +even though written as a rational function, since the numerator is exactly divisible by the denominator. +Hence the above definition should be read as follows: +1. Treating α1, . . . , αn as distinct indeterminates, carry out the exact division obtaining a polynomial. +2. Treating α1, . . . , αn as numbers, evaluate the resulting polynomial. +Lemma 15 (Multiplicity discriminant in multiple roots). Let F be of degree n with m distinct roots +r1, . . . , rm, of multiplicities µ1, . . . , µm, that is µ1 + · · · + µm = n. Let γ = (γ1, . . . , γs) ∈ Γ(n). Then +we have +D(γ) = +c · +����������������� +(F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm) +... +... +... +... +(F (1)x0)(0)(r1) +· · · (F (1)x0)(µ1−1)(r1) +· · · · · · (F (1)x0)(0)(rm) +· · · (F (1)x0)(µm−1)(rm) +... +... +... +... +(F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm) +... +... +... +... +(F (s)x0)(0)(r1) +· · · (F (s)x0)(µ1−1)(r1) +· · · · · · (F (s)x0)(0)(rm) +· · · (F (s)x0)(µm−1)(rm) +����������������� +� +1≤i0 +(αi − αj) +� +αi,αj /∈S1 +j−i>0 +(αi − αj) +� +αi∈S1 +αj /∈S1 +(αi − αj) += ±aγ1−2 +n +· +�� F [α1] +F [α1, α2] +· · · +F [αµ1−1, αµ1] +F [αµ1+1] +· · · +F [αn] +�� +� +αi,αj∈S1 +j−i>1 +(αi − αj) +� +αi,αj /∈S1 +j−i>0 +(αi − αj) +� +αi∈S1 +αj /∈S1 +(αi − αj) += ±aγ1−2 +n +· +�� F [α1] +F [α1, α2] +F [α1, α2, α3] +· · · +F [αµ1−2,µ1−1, αµ1] +F [αµ1+1] +· · · +F [αn] +�� +� +αi,αj∈S1 +j−i>2 +(αi − αj) +� +αi,αj /∈S1 +j−i>0 +(αi − αj) +� +αi∈S1 +αj /∈S1 +(αi − αj) +... += ±aγ1−2 +n +· +�� F [α1] +F [α1, α2] +· · · +F [α1, . . . , αµ1] +F (αµ1+1) +· · · +F (αn) +�� +� +αi,αj /∈S1 +j−i>0 +(αi − αj) +� +αi∈S1 +αj /∈S1 +(αi − αj) +8 + +6. Repeating the procedure for αj’s in each Si for i = 2, . . . , m successively, we get +D(γ) = ±aγ1−2 +n +· +�� F [α1] +· · · +F [α1, . . . , αµ1] +· · · +· · · +F [αµ1+···+µm−1+1] +· · · +F [αµ1+···+µm−1+1, . . . , αn] +�� +� +1≤i 0, the first column of T is all zeros. Hence |T| = 0 and in turn D(λ) = 0 +Arbitrary case. Now we generalize the above ideas to arbitrary cases. +1. Let µ = (µ1, . . . , µm). +Assume that r1, . . . , rm are the m distinct roots of F with multiplicities +µ1, . . . , µm respectively. In other words, F = an(x − r1)µ1 · · · (x − rm)µm. +2. Let γ = ¯µ = (γ1, . . . , γs). By the definition of conjugate, γi = #{µj : µj ≥ i}. Note that s = µ1 since +µ1 ≥ · · · ≥ µm. +3. Consider λ = (λ1, . . . , λt) ∈ M(n) such that γ ≺lex λ. By Lemma 15, we have +D(λ) = +c · +����������������� +(F (1)xλ1−1)(0)(r1) · · · (F (1)xλ1−1)(µ1−1)(r1) · · · · · · (F (1)xλ1−1)(0)(rm) · · · (F (1)xλ1−1)(µm−1)(rm) +... +... +... +... +(F (1)x0)(0)(r1) +· · · (F (1)x0)(µ1−1)(r1) +· · · · · · (F (1)x0)(0)(rm) +· · · (F (1)x0)(µm−1)(rm) +... +... +... +... +(F (t)xλt−1)(0)(r1) · · · (F (t)xλt−1)(µ1−1)(r1) · · · · · · (F (t)xλt−1)(0)(rm) · · · (F (t)xλt−1)(µm−1)(rm) +... +... +... +... +(F (t)x0)(0)(r1) +· · · (F (t)x0)(µ1−1)(r1) +· · · · · · (F (t)x0)(0)(rm) +· · · (F (t)x0)(µm−1)(rm) +����������������� +� +i 0, the first column of T is all zeros. Hence |T| = 0, which implies that D(λ) = 0. +16 + +4.3 +Proof of Theorem 9 +Now we are ready to prove Theorem 9. +Proof of Theorem 9. +The result of Theorem 9 is equivalent to the following claim: let +δ = +max +γ∈M(n) +D(γ)̸=0 +γ +where max is with respect to the lexicographic ordering ≺lex. Then mult(F) = δ. +Next we will show the correctness of the claim. +1. Assume that mult(F) = µ. We will show µ = δ by disproving µ ≺lex δ and δ ≺lex µ. +2. If µ ≺lex δ, then δ ≺lex µ. By the condition for determining δ, we immediately have D(µ) = 0, leading +to a contradiction with Lemma 16. +3. If δ ≺lex µ, then µ +≺lex δ. By Lemma 17, D(δ) = 0. However, it contradicts the condition for +determining δ. +4. Therefore, the only possibility is µ = µ′. +5 +Comparison +In this section, we compare the multiplicity discriminant condition given by Theorem 9 (mentioned as HY22 +hereinafter) and that given by a complex root version of YHZ’s condition [11] as well as the one given by +the authors in [6, Theorem 6] (mentioned as HY21 hereinafter). In particular, we will make comparison on +the forms and the maximum degrees of discriminants appearing in the conditions. +5.1 +Form of discriminants +We will illustrate the forms of conditions generated by the three methods for a fixed µ. For example, we +consider the polynomial F = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 and µ = (2, 2, 1). The condition for F +having the multiplicity structure µ is given as follows: +1. YHZ’s condition: P1 = 0 ∧ P2 = 0 ∧ P3 ̸= 0 where +P1 = +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +������������������ +P2 = +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +�������������� +17 + +P3 = +���������������������������������� +���������� +a5 +a4 +a3 +a2 +a1 +a5 +a4 +a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +���������� +���������� +a5 +a4 +a3 +a2 +a0 +a5 +a4 +a3 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +a1 +5a5 +4a4 +2a2 +���������� +���������� +a5 +a4 +a3 +a2 +a5 +a4 +a3 +a0 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +5a5 +4a4 +a1 +���������� +2 +���������� +a5 +a4 +a3 +a2 +a1 +a5 +a4 +a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +���������� +���������� +a5 +a4 +a3 +a2 +a0 +a5 +a4 +a3 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +a1 +5a5 +4a4 +2a2 +���������� +2 +���������� +a5 +a4 +a3 +a2 +a1 +a5 +a4 +a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +���������� +���������� +a5 +a4 +a3 +a2 +a0 +a5 +a4 +a3 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +a1 +5a5 +4a4 +2a2 +���������� +���������������������������������� +2. HY21’s condition: Q1 = 0 ∧ Q2 = 0 ∧ Q3 ̸= 0 ∧ Q4 ̸= 0 where +Q1 = +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +������������������ +Q2 = +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +�������������� +Q3 = +���������� +a5 +a4 +a3 +a2 +a1 +a5 +a4 +a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +5a5 +4a4 +3a3 +���������� +Q4 = +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +������������������ ++ +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +10a5 +6a4 +3a3 +a2 +������������������ +18 + +�������������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +�������������������� ++ +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +10a5 +6a4 +3a3 +a2 +5a5 +4a4 +3a3 +2a2 +a1 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +������������������ +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +10a5 +6a4 +3a3 +a2 +������������������ +3. HY22’s condition: R1 = 0 ∧ R2 = 0 ∧ R3 ̸= 0 where +R1 = 1 +a5 +������������������ +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +������������������ +R2 = 1 +a5 +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +20a5 +12a4 +6a3 +2a2 +�������������� +R3 = 1 +a5 +�������������� +a5 +a4 +a3 +a2 +a1 +a0 +a5 +a4 +a3 +a2 +a1 +a0 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +5a5 +4a4 +3a3 +2a2 +a1 +20a5 +12a4 +6a3 +2a2 +20a5 +12a4 +6a3 +2a2 +�������������� +From the above conditions, we make the following observations which are also true in general. +1. YHZ’s discriminant involves a nested determinant; +2. HY21’s discriminant involves a sum of several determinants; +3. HY22’s discriminant involves a non-nested determinant. +19 + +5.2 +Maximum degree of discriminants +For the sake of simplicity, we use the following short-hands: +• dYHZ : the maximum of the degrees of the polynomials appearing in YHZ’s conditions ([11]) +• dHY21 : the maximum of the degrees of the polynomials appearing in HY21’s conditions ([6] ) +• dHY22 : the maximum of the degrees of the polynomials appearing in the new conditions (Theorem 9). +Lemma 18. Let dYHZ(µ),dHY21(µ) and dHY22(µ) denote the maximum degrees of the polynomials appear- +ing in YHZ’s condition, HY21’s condition and HY22’s condition for a given µ = (µ1, . . . , µm) ∈ M(n), +respectively. Then we have: +1. Under some minor and reasonable assumption (see [5, Assumption 2]), +dYHZ(µ) = +� +� +� +� +� +� +� +µ2−1 +� +j=0 +(2 mj − 1) +� +� +� +1 +if +µ1 = µ2 +1 + +2 +2mµ2−1−1 +if +µ1 = µ2 + 1 +(2 (µ1 − µ2) − 1) +if +µ1 > µ2 + 1 +≥ +2n + 3µ2 − 4µ2, +for m > 1 +2n − 1, +for m = 1 +where mi is the largest k such that µk > i; +2. dHY21(µ) = 2n − 1; +3. dHY22(µ) = 2n − 2. +Proof. +1. When m = 1, µ = (n). In this case, the condition for the polynomial having multiplicity structure µ +is given by the 0-th,. . . ,(n − 1)-th subdiscriminants. Thus the maximum degree dYHZ(µ) is 2n − 1, +achieved at the 0-th subdiscriminant. +When m > 1, see [5, Appendix] for a detailed proof. +2. Recall that HY21’s condition consists of two parts: (i) the 0-th,. . . ,(n − m)-th subdiscriminants whose +highest degree is 2n − 1; (ii) the multiplicity discriminant given by +� +σ∈Sp +dp +� +��������� +xn−µm−1F +... +x0F +xn−1F(σ1)/σ1! +... +x0F (σn)/σn! +� +��������� +where p = (µ1, . . . , µ1 +� +�� +� +µ1 +, . . . , µm, . . . , µm +� +�� +� +µm +) and Sp is the set of all permutations of p. It is easy to see +that the degree of the multiplicity discriminant is 2n − µm. Hence the maximum degree of the above +discriminants is 2n − 1. +20 + +3. HY22’s condition only consists of the multiplicity discriminants given by +D (γ) = 1 +an +dp +� +�������������������� +F (0)xγ0−1 +... +F (0)x0 +F (1)xγ1−1 +... +F (1)x0 +... +F (s)xγs−1 +... +F (s)x0 +� +�������������������� +where γ = (γ1, . . . , γs) ranges over (n) ≻lex · · · ≻lex µ. Note that the highest degree is achieved when +γ = (n) and γ0 = γ1 − 1. In this case, the degree of D (γ) is 2n − 2. +Remark 19. It is noted that in HY21’s condition, the multiplicity discriminant is always divisible by the +leading coefficient an and thus with this division carried out, the degree can be made smaller by 1. +By Lemma 18, the maximum degree in YHZ’s condition grows exponentially with respect to n while the +maximum degrees in HY21 and HY22’s conditions grow linearly. Below we show a comparison with examples +where n < 10. +n +dYHZ +dHY21 +dHY22 +3 +5 +5 +4 +4 +9 +7 +6 +5 +15 +9 +8 +6 +27 +11 +10 +7 +45 +13 +12 +8 +81 +15 +14 +9 +135 +17 +16 +0 +20 +40 +60 +80 +100 +120 +140 +160 +3 +4 +5 +6 +7 +8 +9 +YHZ +HY21 +HY22 +max deg +n +Acknowledgements. The second author’s work was supported by National Natural Science Foundation of +China (Grant Nos.: 12261010 and 11801101). +References +[1] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry. Springer-Verlag, Berlin- +Heidelberg, 2006. +[2] W. Brown and J. Traub. +On Euclid’s algorithm and the theory of subresultants. +Journal of the +Association for Computing Machinery, 18:505–514, 1971. +[3] G. Collins. Subresultants and Reduced Polynomial Remainder Sequences. Journal of the Association +for Computing Machinery, 14:128–142, 1967. +21 + +[4] L. Gonz´alez-Vega, T. Recio, H. Lombardi, and M.-F. Roy. Sturm-Habicht Sequences, Determinants and +Real Roots of Univariate Polynomials. In Quantifier Elimination and Cylindrical Algebraic Decomposi- +tion. Texts and Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic +Computation, Johannes-Kepler-University, Linz, Austria), pages 300–316. Springer, 1998. +[5] H. Hong and J. Yang. +A condition for multiplicity structure of univariate polynomials. +CoRR, +abs/2001.02388, 2020. +[6] H. Hong and J. Yang. A condition for multiplicity structure of univariate polynomials. Journal of +Symbolic Computation, 104:523–538, 2021. +[7] S. Liang and D. J. Jeffrey. An algorithm for computing the complete root classification of a para- +metric polynomial. In J. Calmet, T. Ida, and D. Wang, editors, Artificial Intelligence and Symbolic +Computation, pages 116–130, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. +[8] S. Liang, D. J. Jeffrey, and M. M. Maza. The complete root classification of a parametric polynomial +on an interval. In International Symposium on Symbolic and Algebraic Computation, 2008. +[9] S. Liang and J. Zhang. A complete discrimination system for polynomials with complex coefficients and +its automatic generation. Science in China Series E: Technological Sciences, 42:113–128, 1999. +[10] R. Loos. Generalized Polynomial Remainder Sequences, pages 115–137. Springer Vienna, Vienna, 1983. +[11] L. Yang, X. Hou, and Z. Zeng. A complete discrimination system for polynomials. Science in China +(Series E), 39(6):628–646, 1996. +22 + diff --git a/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/load_file.txt b/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d50e6908c313751eb0aac2f68b06870d9d4bd44 --- /dev/null +++ b/BdAyT4oBgHgl3EQfd_jz/content/tmp_files/load_file.txt @@ -0,0 +1,2005 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf,len=2004 +page_content='Parametric “Non-nested” Discriminants for Multiplicities of Univariate Polynomials Hoon Hong Department of Mathematics, North Carolina State University Box 8205, Raleigh, NC 27695, USA hong@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='edu Jing Yang∗ SMS–HCIC–School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning 530006, China yangjing0930@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='com Abstract We consider the problem of complex root classification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=', finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Those polynomials in the coefficients are called discriminants for multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' It is well known that discriminants can be obtained by using repeated parametric gcd’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are deter- minants, and so son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In this paper, we give a new type of discriminants which are not based on repeated gcd’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The new discriminants are simpler in that they are non-nested determinants and have smaller maximum degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1 Introduction In this paper, we consider the problem of complex root classification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=', finding the conditions on the coefficients of a polynomial over the complex field C for every potential multiplicity structure its complex roots may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' For example, consider a quintic polynomial F = a5x5 +a4x4 +a3x3 +a2x2 +a1x+a0 where ai’s take values over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' We would like to find conditions C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , C6 on a = (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , a5) such that multiplicity structure of F = � � � � � � � � � � � � � � � � � � � (1, 1, 1, 1, 1) if C0 (a) holds (2, 1, 1, 1) if C1 (a) holds (2, 2, 1) if C2 (a) holds (3, 1, 1) if C3 (a) holds (3, 2) if C4 (a) holds (4, 1) if C5 (a) holds (5) if C6 (a) holds In general, the problem is stated as follows: Problem: For every µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µm) such that µ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' ≥ µm > 0 and µ1 +· · ·+µm = n, find a condition on the coefficients of a polynomial Fover C of degree n such that the multiplicity structure of F is µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='00315v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='SC] 1 Jan 2023 The problem is important because many tasks in mathematics, science and engineering can be reduced to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Due to its importance, the problem and several related problems have been already carefully studied [4, 6, 7, 8, 9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The problem can be viewed as a generalization of a well known problem of finding a condition on coefficients such that the polynomial has a given number of distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' This subproblem has been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' For instance, the subdiscriminant theory provides a complete solution to the subproblem: a univariate polynomial of degree n has m distinct roots if and only if its 0-th, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=', (n − m − 1)-th psd’s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=', principal subdiscriminant coefficient) vanish and the (n−m)-th psd does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' For details, see standard textbooks on computational algebra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=', [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In [11], Yang, Hou and Zeng gave an algorithm to generate conditions for discriminating different mul- tiplicity structures of a univariate polynomial (referred as YHZ’s condition hereinafter) by making use of repeated gcd computation for parametric polynomials [2, 3, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' It is based on a similar idea adopted by Gonzalez-Vega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' [4] for solving the real root classification and quantifier elimination problems by using Sturm-Habicht sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The conditions produced by these methods are conjunctions of several polynomial equations and one inequation on the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Those polynomials in he coefficients are called discrimi- nants for multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The maximum degree of the discriminants grow exponentially in the degree of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Furthermore, each discriminant is a “nested” determinant, that is, it is a determinant of a matrix whose entries are again determinants and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In [6], the authors developed a new type of multiplicity discriminants to distinguish different multiplicities when the number of distinct roots is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The main idea is to convert the multiplicity condition expressed as a permanent inequation in roots into a sum of determinants in coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In order to generate conditions for all the possible multiplicity structures of a univariate polynomial, one may first use subdiscriminants in classical resultant theory to decide the number of distinct complex roots and then add one more inequation to discriminate different multiplicity structures with the same number of distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In the new condition, the maximum degree of the discriminants grow linearly in the degree of F, which makes the size of discrim- inants significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' However, the form of resulting discriminants is a sum of many determinants, which makes the further analysis (reasoning) difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The main contribution in this paper is to provide a new type of discriminants, which are non-nested determinants and whose max degrees are smaller than those in the previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The method is based on a significantly different theory and techniques from the previous methods (which are essentially based on repeated parametric gcd or subdiscriminant theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The new condition is given by a newly devised multiplicity discriminant in coefficients for every potential multiplicity vector of a given degree, which can be viewed as a generalization of subdiscriminant theory to higher order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' To build up the connection between the new discriminants and multiple roots, we first convert it into the ratio of two determinants in terms of generic roots (without considering the multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then by making use of the connection between divided difference with multiple nodes and the derivatives of higher orders at the nodes, we integrate the multiplicity information into the expression and convert it into an expression in terms of multiple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' After careful manipulation, it is shown that the new discriminant can capture the multiplicity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In Section 2, we first present the problem to be solved in a formal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In Section 3, we give a precise statement of the main result of the paper (Theorem 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then a proof of Theorem 9 is provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The proof is long thus we divide the proof into three subsections which are interesting on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' In Section 5, we compare the form and size of polynomials in the multiplicity-discriminant condition in Theorem 9 and those given by previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 2 Problem Definition 1 (Multiplicity vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F ∈ C [x] with m distinct complex roots, say r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , rm, with mul- tiplicities µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Without losing generality, we assume that µ1 ≥ · · · ≥ µm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then the multiplicity vector of F, written as mult (F), is defined by mult (F) = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µm) 2 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F = x5 − 5x4 + 7x3 + x2 − 8x + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then mult (F) = (2, 2, 1), since it can be verified that F = (x − 1)2 (x + 1)1 (x − 2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Note that the multiplicity vector is a partition of 5, which is the degree of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Definition 3 (Potential multiplicity vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let M(n) stand for the set of all the potential multiplicity vectors of polynomials of degree n, equivalently, the set of all partitions of n, that is, M(n) = {(µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µm) : µ1 + · · · + µm = n, µ1 ≥ · · · ≥ µm > 0} Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' M (5) = { (1, 1, 1, 1, 1) , (2, 1, 1, 1) , (2, 2, 1) , (3, 1, 1) , (3, 2) , (4, 1) , (5) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Problem 5 (Parametric multiplicity problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The parametric multiplicity problem is stated as: In : n, a positive integer standing for the polynomial of degree n with parametric coefficients a, that is, F = n � i=0 aixi where an ̸= 0 Out: For each µ ∈ M(n), find a condition Cµ on a such that mult (F) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 3 Main Result Definition 6 (Determinant polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Consider a vector of univariate polynomials P = � �� P0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Pk � �� ∈ C[x]k+1 where deg Pi ≤ k and Pi = � 0≤j≤k aijxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The coefficient matrix of P, written as C (P) , is defined by C (P) = coef (P) = � �� coef (P0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' coef (Pk) � �� = � �� a0k · · a00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' akk · · ak0 � �� The determinant polynomial of P, written as dp (P) , is defined by dp (P) = |C (P) | Definition 7 (Multiplicity Discriminant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F = �n i=0 aixi where an ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , γs) ∈ M (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The the γ-discriminant of F, written as D (γ) , is defined by D (γ) = 1 an dp � �������������������� F (0)xγ0−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)x0 F (1)xγ1−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (1)x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)xγs−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)x0 � �������������������� where γ0 is the smallest so that the above matrix is square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' It is straightforward to show that γ0 = γ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 3 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let n = 5 and F = �n i=0 aixi and an ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then D (5) = dp � ������������� F (0)x3 F (0)x2 F (0)x1 F (0)x0 F (1)x4 F (1)x3 F (1)x2 F (1)x1 F (1)x0 � ������������� = 1 a5 ������������������ a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 ������������������ D (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = dp � ����������� F (0)x2 F (0)x1 F (0)x0 F (1)x3 F (1)x2 F (1)x1 F (1)x0 F (2)x0 � ����������� = 1 a5 ���������������� a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 ���������������� D (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 2) = dp � ��������� F (0)x1 F (0)x0 F (1)x2 F (1)x1 F (1)x0 F (2)x1 F (2)x0 � ��������� = 1 a5 �������������� a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 �������������� D (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = dp � ��������� F (0)x1 F (0)x0 F (1)x2 F (1)x1 F (1)x0 F (2)x0 F (3)x0 � ��������� = 1 a5 �������������� a5 a4 a3 a2 a1 a0 a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4 · 3a5 4 · 3 · 2a4 3 · 2 · 1a3 �������������� D (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = dp � ������� F (0)x0 F (1)x1 F (1)x0 F (2)x1 F (2)x0 F (3)x0 � ������� = 1 a5 ������������ a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4 · 3a5 4 · 3 · 2a4 3 · 2 · 1a3 ������������ D (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = dp � ������� F (0)x0 F (1)x1 F (1)x0 F (2)x0 F (3)x0 F (4)x0 � ������� = 1 a5 ������������ a5 a4 a3 a2 a1 a0 5a5 4a4 3a3 2a2 1a1 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4 · 3a5 4 · 3 · 2a4 3 · 2 · 1a3 5 · 4 · 3 · 2a5 4 · 3 · 2 · 1a4 ������������ D (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = dp � ����� F (1)x0 F (2)x0 F (3)x0 F (4)x0 F (5)x0 � ����� = 1 a5 ���������� 5a5 4a4 3a3 2a2 1a1 5 · 4a5 4 · 3a4 3 · 2a3 2 · 1a2 5 · 4 · 3a5 4 · 3 · 2a4 3 · 2 · 1a3 5 · 4 · 3 · 2a5 4 · 3 · 2 · 1a4 5 · 4 · 3 · 2 · 1a5 ���������� 4 Note that the last one D (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1) = 5544332211a4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Since a5 ̸= 0, we see that D (1, 1, 1, 1, 1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Theorem 9 (Main Result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F = �n i=0 aixi where an ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let M(n) = � µ0, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µp � where the entries are ordered in the lexicographically increasing order, that is, µ0 ≺lex µ1 ≺lex · · · ≺lex µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then we have the following conditions for the multiplicity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' mult(F) = � � � � � � � � � µ0 if D � µp � ̸= 0 µ1 else if D � µp−1 � ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' ̸= 0 µp else if D (µ0) ̸= 0 Equivalently, mult(F) = µi ⇐⇒ D � µp � = · · · = D � µp−i−1 � = 0 ∧ D � µp−i � ̸= 0 Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' We have the following condition for each multiplicity vector for degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' mult(F) = � � � � � � � � � � � � � � � � � � � (1, 1, 1, 1, 1) if D (5) ̸= 0 (2, 1, 1, 1) else if D (4, 1) ̸= 0 (2, 2, 1) else if D (3, 2) ̸= 0 (3, 1, 1) else if D (3, 1, 1) ̸= 0 (3, 2) else if D (2, 2, 1) ̸= 0 (4, 1) else if D (2, 1, 1, 1) ̸= 0 (5) else if D (1, 1, 1, 1, 1) ̸= 0 Equivalently, for instance, mult(F) = (2, 2, 1) ⇐⇒ D (5) = D (4, 1) = 0 ∧ D (3, 2) ̸= 0 Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Note that µ0 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , 1) and D (µ0) = 1 an ��������� nan · · 1a1 n (n − 1) an · · 2 · 1a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' n (n − 1) · · · 1an ��������� = n � i=1 ii · an−1 n ̸= 0 Hence the last condition is always satisfied and there is no need to check the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Note that µi and µp−i are conjugates of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 4 Proof of the Main Theorem Here is a high level view of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' We start with converting D (µ) into the equivalent symmetric polynomials in generic roots (though displayed as a ratio of two determinants) which is easier to embed the multiplicity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then by making use of the connection between divided difference with multiple nodes and the derivatives of higher orders at the nodes, we convert the expression in generic roots to that in distinct roots with multiplicity information integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' The theorem will be proved by eliminating the entries in the determinantal expression obtained from the second stage which may vanish under the given multiplicity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='1 Multiplicity discriminant in terms of roots We first understand what the multiplicity discriminants look like in terms of roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Notation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' V (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn) := ������� αn−1 1 · · αn−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' α0 1 · · α0 n ������� Lemma 13 (Multiplicity discriminant in generic roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F = an(x−α1) · · · (x−αn) and γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , γs) ∈ M(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then D(γ) = aγ1−2 n ������������������ F (1)(α1)αγ1−1 1 · · F (1)(αn)αγ1−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (1)(α1)α0 1 · · F (1)(αn)α0 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)(α1)αγs−1 1 · · F (s)(αn)αγs−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)(α1)α0 1 · · F (s)(αn)α0 n ������������������ V (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn) (1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Since γ1 ≥ · · · ≥ γs and γ0 = γ1 − 1, we have deg(F (0)xn−2) > · · · > deg(F (0)xγ1−1) > max(F (0)xγ0−1, F (1)xγ1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , F (s)xγs−1) Thus D(γ) = 1 an dp � �������������������� F (0)xγ1−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)x0 F (1)xγ1−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (1)x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)xγs−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)x0 � �������������������� = 1 an aγ1−n n dp � ��������������������������� F (0)xn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)xγ1−1 F (0)xγ1−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)x0 F (1)xγ1−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (1)x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)xγs−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)x0 � ��������������������������� = aγ1−n−1 n dp � �������������������� F (0)xn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)x0 F (1)xγ1−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (1)x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)xγs−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (s)x0 � �������������������� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Now we recall the following result from [6] which is the key for proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , Gn ∈ C [x]2n−2 where C [x]2n−2 consists of all the polynomials in x with degree no greater than 2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then dp � ��������� F (0)xn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' F (0)x0 G1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Gn � ��������� = an−1 n ������� G1(α1) · · G1(αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Gn(α1) · · Gn (αn) ������� V (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn) (2) 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' After specializing G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , Gn in (2) with F (1)xγ1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , F (1)x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , F (s)xγs−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , F (s)x0, respectively, we have D(γ) = aγ1−n−1 n an−1 n ����������������� � F (1)xγ1−1� (α1) · · � F (1)xγ1−1� (αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' � F (1)x0� (α1) · · � F (1)x0� (αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' � F (s)xγs−1� (α1) · · � F (s)xγs−1� (αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' � F (s)x0� (α1) · · � F (s)x0� (αn) ����������������� V (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn) which can be easily simplified into (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Remark 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' It is very important to note that the right hand side is a polynomial function in α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn, even though written as a rational function, since the numerator is exactly divisible by the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Hence the above definition should be read as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Treating α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn as distinct indeterminates, carry out the exact division obtaining a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Treating α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , αn as numbers, evaluate the resulting polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Lemma 15 (Multiplicity discriminant in multiple roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let F be of degree n with m distinct roots r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , rm, of multiplicities µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , µm, that is µ1 + · · · + µm = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Let γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' , γs) ∈ Γ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' Then we have D(γ) = c · ����������������� (F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' (F (1)x0)(0)(r1) · · (F (1)x0)(µ1−1)(r1) · · · · · (F (1)x0)(0)(rm) · · (F (1)x0)(µm−1)(rm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' (F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfd_jz/content/2301.00315v1.pdf'} +page_content=' (F (s)x0)(0)(r1) · · (F (s)x0)(µ1−1)(r1) · · · · · (F (s)x0)(0)(rm) · · (F (s)x0)(µm−1)(rm) ����������������� � 1≤i +0.5 +0.0 +M +K +q vector [A-1]Fe12 +Wave length 入 [A] +inf +11.6 +6.8 +7.0 +14.1 +inf +3.5 +40 +Energy +[8l] +35 +Fe magmom +3.0 + Spin spiral energy [meV] +Local norm magnetic moment [ +I magmom +30 +2.5 +25 +2.0 +20 +1.5 +15 +1.0 +10 +0.5 +5 +0.0 +0 +M +K +q vector [A-1]CoCI2 +Wave length 入 [A] +inf +10.6 +5.3 +6.1 +12.3 +inf +2.5 +50 +Energy +Local norm magnetic moment [lμbl +Co magmom +Spin spiral energy [meV] +Cl magmom +2.0 +40 +1.5 +30 +1.0 +20 +0.5 +10 +0.0 +K +M +q vector [A-1] \ No newline at end of file diff --git a/EtE4T4oBgHgl3EQffQ30/content/tmp_files/load_file.txt b/EtE4T4oBgHgl3EQffQ30/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aafbd345a11dcd62f4b539aabaecfc53d7845f8c --- /dev/null +++ b/EtE4T4oBgHgl3EQffQ30/content/tmp_files/load_file.txt @@ -0,0 +1,1651 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf,len=1650 +page_content='Type II multiferroic order in two-dimensional transition metal halides from first principles spin-spiral calculations Joachim Sødequist and Thomas Olsen∗ Computational Atomic-Scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, 2800 Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Lyngby, Denmark (Dated: January 13, 2023) We present a computational search for spin spiral ground states in two-dimensional transition metal halides that are experimentally known as van der Waals bonded bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Such spin spirals break the rotational symmetry of the lattice and lead to polar ground states where the axis of polarization is strongly coupled to the magnetic order (type II multiferroics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We apply the generalized Bloch theorem in conjunction with non- collinear density functional theory calculations to find the spiralling vector that minimizes the energy and then include spin-orbit coupling to calculate the preferred orientation of the spin plane with respect to the spiral vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find a wide variety of magnetic orders ranging from ferromagnetic, stripy anti-ferromagnetic, 120◦ non-collinear structures and incommensurate spin spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The latter two introduce polar axes and are found in the majority of materials considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spontaneous polarization is calculated for the incommensurate spin spirals by performing full supercell relaxation including spinorbit coupling and the induced polarization is shown to be strongly dependent on the orientation of the spiral planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also test the effect of Hubbard corrections on the results and find that for most materials LDA+U results agree qualitatively with LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' An exception is the Mn halides, which are found to exhibit incommensurate spin spiral ground states if Hubbard corrections are included whereas bare LDA yields a 120◦ non-collinear ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' INTRODUCTION The recent discovery of ferromagnetic order in two- dimensional (2D) CrI3 [1] has initiated a vast interest in 2D magnetism [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Several other materials have subsequently been demonstrated to preserve magnetic order in the mono- layer limit when exfoliated from magnetic van der Waals bonded compounds and the family of 2D magnets is steadily growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' A crucial requirement for magnetic order to persist in the 2D limit is the presence of magnetic anisotropy that breaks the spin rotational symmetry that would otherwise ren- der magnetic order at finite temperatures impossible by the Mermin-Wagner theorem [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is exemplified by the cases of 2D CrBr3 [6, 7] and CrCl3 [8, 9], which are isostructural to CrI3 and while the former remains ferromagnetic in the atomic limit due to easy-axis anisotropy (like CrI3) the lat- ter has a weak easy plane that forbids proper long range or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Other materials with persisting ferromagnetic order in the 2D limit include the metallic compounds Fe3/4/5GeTe2 [10–12] and the anisotropic insulator CrSBr [13], which has an easy-axis aligned with the atomic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Finally, FePS3 [14] and MnPS3 [15] constitute examples of in-plane anti- ferromagnets that preserve magnetic order in the monolayer limit due to easy-axis anisotropy, whereas the magnetic order is lost in monolayers of the isostructural easy-plane compound NiPS3 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The 2D materials mentioned above all consti- tute examples of rather simple collinear magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, the ground state of three-dimensional magnetic materials of- ten exhibit complicated non-collinear order that gives rise to a range of interesting properties [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Such materials, are so far largely lacking from the field of 2D magnetism and the discov- ery of new non-collinear 2D magnets would greatly enhance ∗ tolsen@fysik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='dk the possibilities of constructing versatile magnetic materials using 2D magnets as building blocks [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The ground state of the classical isotropic Heisenberg model can be shown to be a planar spin spiral characterised by a propagation vector Q [19] and such spin configurations thus comprise a broad class of states that generalise the concept of ferromagnetism and anti-ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In fact, spin spiral order is rather common in layered van der Waals bonded ma- terials [20] and it is thus natural to investigate the ground state order of the corresponding monolayers for spin spiral order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Moreover, for non-bipartite magnetic lattices the concept of anti-ferromagnetism is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is exemplified by the abundant example of the triangular lattice where one may con- sider the cases of anti-aligned ferromagnetic stripes or 120◦ non-collinear order, which can be represented as spin spirals of Q = (1/2,0) and Q = (1/3,1/3) respectively [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The concept of spin spirals thus constitute a general framework for specifying the magnetic order, which may or may not be com- mensurate with the crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Finite spin spiral vectors typically break symmetries inher- ent to the crystal lattice and may thus induce physical prop- erties that are predicted to be absent if one only considers the crystal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In particular, the spin spiral may yield a polar axis that lead to ferroelectric order [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Such materials are referred to as type II multiferroics and examples include MnWO4 [24], CoCr2O4 [25], LiCu2O2 [26] and LiCuVO4 [27] as well as the triangular magnets CuFeO2 [28], CuCrO2 [28], AgCrO2 [29] and MnI2 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition to these ma- terials, 2D NiI2 has recently been shown to host a spin spiral ground state that induces a spontaneous polarization [31] and 2D NiI2 thus comprises the first example of a 2D type II mul- tiferroic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The prediction of new materials with certain desired prop- erties can be vastly accelerated by first principles simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In general, the search for materials with spin spiral ground states is complicated by the fact that the magnetic order re- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='05107v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 2 quires large super cells in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, if one neglects spinorbit coupling, spin spirals of arbitrary wavevec- tors can be represented in the chemical unit cell by utilising the generalized Bloch theorem that encodes the spiral in the boundary conditions [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This method has been applied in conjunction with density functional theory (DFT) to a wide range of materials and typically produces results that are in good agreement with experiments [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the present work we use DFT simulations in the frame- work of the generalized Bloch theorem to investigate the mag- netic ground state of monolayers derived from layered van der Waals magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We then calculate the preferred orientation of the spiral plane by adding a single component of the spinorbit coupling in the normal direction of various trial spiral planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This yields a complete classification of the magnetic ground state for these materials under the assumption that higher or- der spin interactions can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' On the other hand, the effect of higher order spin interactions can be quantified by deviations between spin spiral energies in the primitive unit cell and a minimal super cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The results for all compounds are discussed and compared with existing knowledge from ex- periments on the parent bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Finally, we analyse the spontaneous polarization in all cases where an incommensu- rate ordering vector is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' II we summarise the theory used to obtain spin spiral ground states based on the generalized Bloch theorem and briefly outline the implemen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' III we present the results and summarise the magnetic ground states of all the investigated materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' IV provides a conclusion and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Generalized Bloch’s Theorem The Heisenberg model plays a prominent role in the the- ory of magnetism and typically gives an accurate account of the fundamental magnetic excitations as well as the thermo- dynamic properties of a given material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the isotropic case it can be written as H = −1 2 ∑ i j Ji jSi ·Sj, (1) where Si is the spin operator for site i and Ji j is the exchange coupling between sites i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In a classical treatment, the spin operators are replaced by vectors of fixed magnitude and it can be shown that the classical energy is minimised by a planar spin spiral [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Such a spin configuration is charac- terised by a wave vector Q, which is determined by the set of exchange parameters Ji j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spin at site i is rotated by an an- gle Q · Ri with respect to the origin and the wave vector may or may not be commensurate with the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In a first principles framework it is thus natural to search for planar spin spiral ground states that give rise to periodically modulated magnetisation densities satisfying mq(r+Ri) = Uq,Rimq(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (2) Here Ri is a lattice vector (of the chemical unit cell) and Uq,Ri is a rotation matrix that rotates the magnetisation by an an- gle q · Ri around the normal of the spiral plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the ab- sence of spinorbit coupling we are free to perform a global rotation of the magnetisation density and we will fix the spi- ral plane to the xy-plane from hereon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the framework of DFT, the magnetisation density (2) gives rise to an exchange- correlation magnetic field satisfying the same symmetry un- der translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' If spinorbit coupling is neglected the Kohn- Sham Hamiltonian thus commutes with the combined action of translation (by a lattice vector) and a rotation of spinors by the angle q·Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This implies that the Kohn-Sham eigenstates can be written as ψq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k(r) = eik·rU† q(r) � u↑ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k(r) u↓ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k(r) � (3) where u↑ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k(r) and u↓ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k(r) are periodic in the chemical unit cell and the spin rotation matrix is given by Uq(r) = � eiq·r/2 0 0 e−iq·r/2 � (4) This is known as the generalized Bloch Theorem (GBT) and the Kohn-Sham equations can then be written as HKS q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='kuq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k = εq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='kuq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k (5) where the generalized Bloch Hamiltonian: HKS q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='k = e−ik·rUq(r)HKSU† q(r)eik·r (6) is periodic in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Here k is the crystal momentum, q is the spiral wave vector and HKS is the Kohn-Sham Hamil- tonian, which couples to the spin degrees of freedom through the exchange-correlation magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the present work, we will not consider constraints be- sides the boundary conditions defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For a given q we can thus obtain a unique total energy Eq and the mag- netic ordering vector is determined as the point where Eq has a minimum (denoted by Q) when evaluated over the entire Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, if the chemical unit cell contains more than one magnetic atom there may be different local ex- trema corresponding to different intracell alignments of mag- netic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In order ensure that the correct ground state is obtained it is thus pertinent to perform a comparison be- tween calculations that are initialised with different relative magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' As a simple example of this, one may consider a honeycomb lattice of magnetic atoms where the ferromagnetic and anti-ferromagnetic configurations both cor- respond to q = 0, but are distinguished by different intracell orderings of the local magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We will discuss this in the context of CrI3 in section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also note that the true magnetic ground state is not nec- essarily representable by the ansatz (2) and one is therefore not guaranteed to find the ground state by searching for spin spirals based on the minimal unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In figure 1 we show four examples of possible magnetic ground states of the tri- angular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Three of these correspond to spin spirals of 3 Q = (1/3, 1/3) Q = (1/2, 0) Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14) Q = (0, 1/2) (a) Γ M/S K X Y (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (a) Examples of magnetic structures in the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The Q = (1/3,1/3) (corresponding to the high symmetry point K) is the classical ground state in the isotropic Heisenberg model with nearest neighbour antiferromagnetic exchange and is degenerate with Q = (−1/3,−1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The stripy antiferromagnetic Q = (1/2,0) (corresponding to the high symmetry point M) is only found for CoI2 in the present study and is degenerate with Q = (0,1/2) and Q = (1/2,1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The incommensurate spiral with Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14) corresponds to the prediction of NiI2 in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The rectangular cell with Q = (0,1/2) is a bicollinear antiferromagnet that corresponds to superpositions of (0, ±1/4) states in the primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (b) Brillouin zone of the hexagonal (blue) and rectangular (orange) unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The high symmetry band paths used to sample the spiral ordering vectors are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' the minimal unit cell while the fourth - a bicollinear antifer- romagnet - requires a larger unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The bicollinear state may arise as a consequence of higher order exchange interac- tions, which tend to stabilize linear combinations of degener- ate single-q states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spinorbit coupling In the presence of spinorbit coupling, the spin spiral plane will have a preferred orientation and the magnetic ground state is thus characterised by a normal vector ˆn0 of the spiral plane as well as the spiral vector Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spinorbit coupling is, however, incompatible with application of the GBT and has to be ap- proximated in a post processing step when working with the spin spiral representation in the chemical unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It can be shown that first order perturbation theory only involves contri- butions from the spinorbit components orthogonal to the plane [39] ⟨ψq,ˆn|L·S|ψq,ˆn⟩ = ⟨ψq,ˆn|(L· ˆn)(S· ˆn)|ψq,ˆn⟩, (7) and this term is thus expected to yield the most important con- tribution to the spinorbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Since (L · ˆn)(S · ˆn) com- mutes with a spin rotation around the axis ˆn, the spin spi- ral wavefunctions remain eigenstates when such a term is in- cluded in HKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This approach was proposed by Sandratskii [40] and we will refer to it as the projected spinorbit coupling (PSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For the spin spiral calculations in the present work we include spinorbit coupling non-selfconsistently by performing a full diagonalization of the HKS q,k including the PSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The mag- netic ground state is then found by evaluating the total energy at all normal vectors ˆn, which will yield ˆn0 as the normal vec- tor that minimizes the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Computational Details The GBT has been implemented in the electronic structure software package GPAW [41], which is based on the projector augmented wave method (PAW) and plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The im- plementation uses a fully non-collinear treatment within the local spin density approximation where both the interstitial and atom-centered PAW regions are handled non-collinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spinorbit coupling is included non-selfconsistently [42] as de- scribed in Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The implementation is described in de- tail in Appendix V A and benchmarked for fcc Fe in Appendix V B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find good agreement with previous results from the literature and we also assert that results from spin spiral cal- culations within the GBT agree exactly with supercell calcu- lations without spinorbit in the case of bilayer CoPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Finally, we compare the results of the PSO approximations with full inclusion of spinorbit coupling for both supercells and GBT spin spirals of the CoPt bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find exact agreement be- tween the PSO in the supercell and GBT spin spiral and the approximation only deviates slightly compared to full spinor- bit coupling for the supercell calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' All calculations have been carried out with a plane wave cutoff of 800 eV, a k-point density of 14 ˚A and a Fermi smear- ing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The structures and initial magnetic moments are taken from the Computational Materials Database (C2DB) [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='In order to find the value of Q, which describes the ground state magnetic order, we calculate Eq along a represen- tative path connecting high symmetry points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' While the true value of Q could be situated away from such high symmetry lines we deem this approach sufficient for the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' RESULTS A comprehensive review on the magnetic properties of lay- ered transition metal halides was provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Here we present spin spiral calculations and extract the magnetic properties of the corresponding monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition to the magnetic moments, the properties are mainly characterised by a spiral ordering vector Q and the normal vector to the spin spiral plane ˆn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The materials either have AB2 or AB3 stoi- chiometries and we will discuss these cases separately below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 4 Q Emin [meV] (θ,ϕ) Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' IP order BW [meV] PSO BW [meV] mΓ [µB] ∆εQ [eV] TiBr2 (1/3, 1/3) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='12 (90,90) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 TiI2 (1/3, 1/3) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='33 (90,90) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 NiCl2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='81 (90,31) FM ∥ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='81 NiBr2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='11) -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='62 (44,0) FM ∥, HM 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='62 NiI2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14) -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='48 (64,0) HM 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='28 VCl2 (1/3, 1/3) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='07 (90,0) 120◦ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='96 VBr2 (1/3, 1/3) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='21 (90,18) 120◦ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 VI2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14) -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='43 (6,0) stripe 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='96 MnCl2 (1/3, 1/3) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='48 (90,15) stripe or HM 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='92 MnBr2 (1/3, 1/3) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='13 (90,15) stripe ∥ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='76 MnI2 (1/3, 1/3) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='32 (0,0) HM 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='41 FeCl2 (0, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 (0, 0)∗ FM ⊥ 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 FeBr2 (0, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 (0, 0)∗ FM ⊥ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 FeI2 (0, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 (0, 0)∗ stripe ⊥ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 CoCl2 (0, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 (90,90)∗ FM ∥ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 CoBr2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='03) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='04 (0,0) FM ∥ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 CoI2 (1/2, 0) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='95 (90,90) HM 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Summary of magnetic properties of the AB2 compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The ground state ordering vector is denoted by Q and Emin is the ground state energy relative to the ferromagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The normal vector of the spiral plane is defined by the angles θ and ϕ (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also display the experimental in-plane order of the parent layered compound (Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' IP order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition we state the spin spiral band width BW, the magnetic moment per unit cell in the ferromagnetic state mΓ and the band gap at the ordering vector ∆εQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For the case of NiI2, mΓ deviates from an integer value because the ferromagnetic state is metallic in LDA (whereas the spin spiral ground state has a gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The cases of FeX2, CoCl2 and CoBr2 are half metals, which enforces integer magnetic moment despite the metallic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The asterisks indicate ferromagnets where full spinorbit coupling was included and the angles then refer to the direction of the spins rather that the spiral plane normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have performed LDA and LDA+U calculations for all materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In most cases, the Hubbard corrections does not make any qualitative difference although the spiral ordering vector does change slightly and we will not discuss these cal- culations further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The Mn halides comprise an exception to this where LDA+U calculations differ significantly from those of bare LDA and the LDA+U calculations will be dis- cussed separately for these materials below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For the AB2 materials, we find 12 that exhibit a spiral or- der that breaks the crystal symmetry and yields a ferroelec- tric ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For six of these compounds we have calcu- lated the spontaneous polarization by performing full relax- ation (including self-consistent spinorbit coupling) in super- cells hosting the spiral order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Magnetic ground state of AB2 materials The AB2 materials all have space group P¯3m1 correspond- ing to monolayers of the CdI2 (or CdCl2) prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The mag- netic lattice is triangular and a few representative possibilities for the magnetic order is illustrated in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnetic properties of all the considered compounds are summarized in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition to the ordering vector Q we provide the angles θ and φ, which are the polar and azimuthal an- gles of ˆn0 with respect to the out-of-plane direction and the ordering vector respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It will be convenient to consider three limiting cases of the orientation of spin spiral planes: the proper screw (θ = 90,ϕ = 0), the out-of-plane cycloid (θ = 90,ϕ = 90) and the in-plane cycloid (θ = 0,ϕ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also provide the ground state energy relative to the fer- romagnetic configuration (Q = (0,0)), the band gap, the spin spiral band width, which reflects the strength of the magnetic interactions and the PSO band width, which is the energy dif- ference between the easy and hard orientations of the spiral plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnetic moments are calculated as the total mo- ment in the unit cell using the ferromagnetic configurations without spinorbit interaction and thus yields an integer num- ber of Bohr magnetons for insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnitude of the local magnetic moments (obtained by integrating the magne- tization density over the PAW spheres) in the ground state are generally found to be very close to the moments in the ferro- magnetic configuration, unless explicitly mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spin spiral energy dispersions are provided for all AB2 materials in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The different classes of materials are described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' NiX2 The nickel halides all have ground states with incommen- surate spiral vectors between Γ and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Experimentally, both NiI2 and NiBr2 in bulk form have been determined to have in- commensurate spiral vectors [45–47] in qualitative agreement with the LDA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The case of NiCl2, however, have been found to have ferromagnetic intra-layer order whereas we find a rather small spiral vector of Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In bulk NiI2 the experimental ordering vector Qexp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1384,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='457) has an in-plane component in the ΓM- direction with a magnitude of roughly 1/7 of a recipro- cal lattice vector, while for the monolayer we find Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14,0), which is in the ΓK-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Evaluating the spin spiral energy in the entire Brillouin zone, however, re- veals a nearly degenerate ring encircling the Γ-point with a 5 Γ M K Γ −20 0 20 40 E(q) [meV] (a) K G M 28 18 8 0 10 20 30 40 Energy [meV] (b) IP Screw OoP IP −44 −42 −40 Esoc(θ, ϕ) [meV] θ ϕ θ (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spin spiral energy of NiI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Left: the spin spiral energy as a function of q without spinorbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Center: Spin spiral energy in evaluated in entire Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Right: spiral energy as a function of spiral plane orientation evaluated at the minimum Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spiral plane orientation is parameterized in terms of the polar angle θ and azimuthal angle ϕ (measured from Q) of the spiral plane normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' radius of roughly 1/5 of a reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The point qM = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='21,0) thus comprises a very shallow saddle point with an energy that exceeds the minimum by merely 2 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is illustrated in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also show a scan of the spin spiral energy (within the PSO approximation) as a func- tion of orientation of the spin spiral plane on a path that con- nects the limiting cases of in-plane cycloid, out-of-plane cy- cloid and proper screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' An unconstrained spin spiral calcu- lation using the rectangular unit cell of figure 1 does not re- veal any new minima in the energy, which implies that the ground state is well represented by a single-q spiral and that higher order exchange interactions are neglectable in NiI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The normal vector of the spiral makes an angle of 64◦ with the out-of-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This orientation is in good agree- ment with the experimental assignment of a proper screw (along Qexp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1384,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='457)), which corresponds to a tilt of 55◦±10◦ with respect to the c-axis [47], but disagrees with the model proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [31] where the spiral was found to be a proper screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' At low temperatures NiBr2 has been reported to exhibit Qexp = (x,x,3/2) where x changes continuously from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='027 at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 K to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='09 at 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 K and then undergoes first order transi- tion at 24 K to intra-layer ferromagnetic order [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The struc- ture predicted here is close to the one observed in bulk at 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The discrepancy could be due to the magnetoelastic defor- mation [49] that has been associated with the modulation of the spiral vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This effect could in principle be captured by relaxing the structure in supercell calculations, but the small wavelength spirals require prohibitively large supercells and are not easily captured by first principles methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is also highly likely that LDA is simply not accurate enough to de- scribe the intricate exchange interactions that define the true ground state in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Bulk NiCl2 is known to be an inter-layer antiferromag- net with ferromagnetically ordered layers [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find the ground state to be a long wavelength incommensurate spin spiral with Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='06), which is in rather close prox- imity to ferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The ground state energy is less than 1 meV lower than the ferromagnetic state, but we cannot say at present whether this is due to inaccuracies of LDA or if the true ground state indeed exhibits spiral magnetic order in the monolayer limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' VX2 The three vanadium halides are insulators and whereas VCl2 and VBr2 are found to form Q = (1/3,1/3) spiral struc- tures, VI2 has an incommensurate ground state with Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnetic ground state of VCl2 and VBr2 is in good agreement with experiments on bulk materials where both have been found to exhibit out-of-plane 120◦ order [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This structure is expected to arise from strong nearest neigh- bour anti-ferromagnetic interactions between the V atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The case of VI2 has a significantly smaller spiral band width, signalling weaker exchange interactions compared to VCl2 and VBr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' A collinear energy mapping based on the Perdew- Burke-Ernzerhof (PBE) exchange-correlation functional [44] yields a weakly ferromagnetic nearest neighbour interaction for VI2 and strong anti-ferromagnetic interactions for VCl2 and VBr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is in agreement with the present result, which indicate that the magnetic order of VI2 is not dominated by nearest neighbour interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Experimentally [52], the bulk VI2 magnetic order has been found to undergo a phase transition at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 K from a 120◦ state to a bicollinear state with Q = (1/2,0), where the spins are perpendicular to Q and tilted by 29◦ from the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Such a bicollinear state implies that the true ground state is a double- q state stabilized by higher order spin interactions and cannot be represented as a spin spiral in the primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' To check whether LDA predicts the experimental ground state we have therefore performed spiral calculations in the rectangular cell shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The result is shown in figure 3 along with the spiral calculation in the primitive cell and we do not find any new minima in the super cell calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have initalized angles in the super cell caluculation such that they corresponds to bicollinear order and the angles are observed to relax to the single-q spin spiral of the primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is likely that LDA is insufficient to capture the subtle higher order exchange interactions in this material, but it is possible that the monolayer simply has a magnetic order that differs 6 Γ K M Γ X S Y Γ S −4 −2 0 2 4 E(q) [meV/uc] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spin spiral energies of VI2 obtained from the primitive cell (black) and the rectangular super cell (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The dashed lines repeat the primitive cell results on the corresponding super cell path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' from the individual layers in the bulk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the PSO approximation we find that VCl2 and VBr2 pre- fer out-of-plane spiral planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The energy is rather insensitive to ϕ forming a nearly degenerate subspace of ground states with a slight preference of the proper screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The ground state of VI2 is found to be close to the in-plane cycloid with a nor- mal vector to the spiral plane forming a 6◦ angle with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spinorbit corrections in VI2 are also found to be the smallest compared to other iodine based transition metal halides stud- ied here and the ground state energy only deviates by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 meV per unit cell from the out-of-plane cycloid, which constitutes the orientation of the spin plane with highest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' MnX2 The manganese halides are all found to form 120◦ ground states, which is in agreement with previous theoretical studies [53] using PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In contrast to the other insulators studied in the present work, however, we find that the results are qualita- tively sensitive to the inclusion of Hubbard corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This was also found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [54], where the sign of the nearest neighbour exchange coupling was shown to change sign when a Hubbard U parameter was included in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' With U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 eV we find that all three compounds has spiral ground states with incommensurate spiral vector Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='11,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='11,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Moreover, spin spiral band width in the LDA+U calculations decrease by more than an order of magnitude compared to the bare LDA calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The experimental magnetic structure of the manganese halides are rather complicated, exhibiting several magnetic phase transitions in a range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 K below the initial order- ing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In particular MnI2 (MnBr2) has been found to have three (two) complex non-collinear phases [55], and MnCl2 has two complex phases that are possibly collinear [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The experimental ground state of bulk MnCl2 not unam- biguously known, but under the assumption of collinearity a possible ground state contains 15 Mn atoms in an extended stripy pattern [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Due to the weak and subtle nature of mag- netic interactions in the manganese compounds, however, it is not unlikely that the ground state in the monolayers can dif- fer from that of bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is corroborated by an experimental study of MnCl2 intercalated by graphite where a helimagnetic ground state with Qexp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='153,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='153) was found [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is rather close to our predicted ordering vector obtained from LDA+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Experimentally, bulk MnBr2 is found to exhibit a stripy bicollinear uudd order at low temperatures [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The order cannot be represented by a spiral in the minimal cell, but re- quires calculations in rectangular unit cells with spiral order Q = (0,1/2) similar to VI2 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have calcu- lated the high symmetry band path required to show this order and do not find any new minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is likely that the situation resembles MnCl2 where a single-q spiral has been observed for decoupled monolayers in agreement with our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' FeX2 We find all the iron halides to have ferromagnetic ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For FeCl2 and FeBr2 this is in agreement with the experimentally determined magnetic order for the bulk com- pounds [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In contrast, FeI2 has been reported to exhibit a bicollinear antiferromagnetic ground state [60] similar to the case of MnBr2 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is again possible that the ground state of the monolayer (calculated here) could differ from the magnetic ground state of the bulk compound as has been found for MnCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' LDA predict the three compounds to be half metals, mean- ing that the majority spin bands are fully occupied and only the minority bands have states at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This en- forces an integer number of Bohr magnetons (four) per unit cell at any q-vector in the spin spiral calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Thus longi- tudinal fluctuations are expected to be strongly suppressed in iron halides and it is likely that these materials can be accu- rately modelled by Heisenberg Hamiltonians despite the itin- erant nature of the electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The projected spin orbit coupling is not applicable to collinear structures and we therefore include full spin orbit coupling, which is compatible with the Q = (0,0) ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find that all the iron compounds have an out-of- plane easy axis, which is in agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The bandwidth provided in table I then simply corresponds to the magnetic anisotropy energy which is smallest for FeCl2 and increases for the heavier Br and I compounds as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' CoX2 We predict CoCl2 to have an in-plane ferromagnetic ground state in agreement with the experimentally determined mag- netic order of the bulk compound [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' CoBr2 is found to have a long wavelength spin spiral with Q = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spi- ral energy in the vicinity of the Γ-point is, however, extremely flat with almost vanishing curvature and the ground state en- ergy is merely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='04 meV lower than the ferromagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We regard this as being in agreement with the experimental re- port of intra-layer ferromagnetic order in the bulk compound [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 7 The case of CoI2 deviates substantially from the other two halides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' CoCl2 and CoBr2 are half-metals with m = 3 µB per unit cell, whereas CoI2 is an ordinary metal with m ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 µB per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find the magnetic ground state of CoI2 to be stripy anti-ferromagnetic with Q = (1/2,0), whereas ex- periments on the bulk compound have reported helimagnetic in-plane order with Qexp = (1/6,1/8,1/2) in the rectangular cell [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note, however, that the calculated local mag- netic moments vary strongly with q (up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 µB) in the spin spiral calculations, which signals strong longitudinal fluctu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This could imply that the material comprises a rather challenging case for DFT and LDA may be insufficient to treat this material properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spontaneous polarization of AB2 materials The materials in table I that exhibit spin spiral ground states are expected to introduce a polar axis due to spinorbit cou- pling and thus allow for spontaneous electric polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The stripy antiferromagnet with Q = (1/2,0) preserves a site- centered inversion center and remains non-polar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition, the case of Q = (1/3,1/3) with in-plane orientation of the spi- ral plane breaks inversion symmetry, but retains the three-fold rotational symmetry (up to translation of a lattice vector) and therefore cannot acquire components of in-plane polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' To investigate the effect of symmetry breaking we have constructed 7×1 supercells of VI2 and the Ni halides and per- formed a full relaxation of the q = (1/7,0) spin spiral com- mensurate with the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is not exactly the spin spi- rals found as the ground state from LDA, but we will use these to get a rough estimate of the spontaneous polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that this is very close to the in-plane component of Qexp for bulk NiI2, which is found to be nearly degenerate with the predicted ground state (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The other materi- als exhibit similar near-degeneracies, but the calculated polar- ization could be sensitive to which spiral ordering vector is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have chosen to focus on the incommensurate spi- rals, but note that all the Q = (1/3,1/3) materials of table II are expected to introduce a spontaneous polarization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Besides the incommensurate spirals we thus only include the cases of MnBr2 and MnI2 where the Q = (1/3,1/3) spirals may be represented in √ 3 × √ 3 supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The former case represents an example of a proper screw while the latter is an in-plane cycloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The experimental order in the Mn halides materials is complicated, and our LDA+U calculations yield an ordering vector that differs from that of LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, here we mostly consider these examples for comparison and to check the symmetry constraints on the polarization in the Q = (1/3,1/3) spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In order to calculate the spontaneous polarization we relax the atomic positions in the super cells both with and without spinorbit coupling (included self-consistently) and calculate the 2D polarization from the Berry phase formula [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We can separate the effect of relaxation from the pure electronic contribution by calcu- lating the polarization (including spin-orbit) of the structures that were relaxed without spinorbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' These numbers are stated in brackets in table II as well as the total polarization (including relaxation) and the angles that define the orienta- tion of the spiral plane with respect to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The self-consistent calculations yield the optimal orientations of the spiral planes without the PSO approximations and it is reassuring that the orientation roughly coincides with the results of the GBT and the PSO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnitude of polarization largely scales with the atomic number of ligands (as expected from the strength of spinorbit coupling) and the iodide compounds thus produce the largest polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The in-plane cycloid in MnI2 only give rise to out-of-plane polarization as expected from sym- metry and the Q = (1/3,1/3) proper screw in MnBr2 has po- larization that is strictly aligned with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The latter results is expected for any proper screw in the ΓK-direction because Q then coincides with a two-fold rotational axis and the ground state remains invariant under the combined action of this ro- tation and time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Since the polarization is not affected by time-reversal it must be aligned with the two- fold axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The polarization vectors of the remaining materials (except for NiCl2) are roughly aligned with the intersection between the spiral plane and the atomic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is interesting to note that the calculated magnitudes of to- tal polarization are 5-10 times larger than the prediction from the pure electronic contribution where the atoms were not re- laxed with spinorbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also tried to calculate the polarization by using the Born effective charge tensors (with- out spin-orbit) and the atomic deviations from the centrosym- metric positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, this approximation severely un- derestimates the polarization and even produces the wrong sign of the polarization in the case of NiBr2 and NiI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' To obtain reliable values for the polarization it is thus crucial to include the relaxation effects and take the electronic contribu- tion properly into account (going beyond the Born effective charge approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [31] a value of 141 fC/m was predicted in 2D NiI2 from the gKNB model [62] and this is comparable to the values found in table II without relaxation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' When relaxation is included we find a magnitude of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 pC/m for NiI2, which is an order of magnitude larger com- pared to the previous prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The results are, however, not directly comparable since Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [31] considered a spiral along the ΓK direction whereas the present result is for a spiral along ΓM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [31] finds the polarization to be aligned with Q in agreement with the symmetry considerations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Finally, the values for the spontaneous polarization in table II may be compared with those of ordinary 2D ferroelectrics, which are typically on the order of a few hundred pC/m for in-plane ferroelectrics and a few pC/m for out-of-plane ferro- electrics [63] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In all of these type II multiferroics, the orientation of the induced polarization depends on the direction of the ordering vector, which may thus be switched by application of an ex- ternal electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have checked explicitly that the sign of polarization is changed if we relax a right-handed instead of a left-handed spiral (corresponding to a reversed ordering vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The small values of spontaneous polarization in these materials implies that rather modest electric fields are required for switching the ordering vector and thus comprise an in- 8 (θ,ϕ) P∥ P⊥ Pz VI2 (11, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 (-31) 290 (96) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='11) NiCl2 (90, -30) -37 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4) 76 (15) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5(-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1) NiBr2 (69, -10) 12 (-6) 340 (32) 26 (37) NiI2 (70, 0) 8 (-48) 1890 (400) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='18 (12) MnBr2 (90, 0) 430 (38) 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='02) 0 (0) MnI2 (0, 0) 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 (-7) 260 (-105) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Orientation of spin planes, and 2D polarization (in fC/m) of selected transition metal halides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' P∥ denotes the polarization along Q, while P⊥ denotes the polarization in the atomic plane orthogonal to Q and Pz is the polarization orthogonal to the atomic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The numbers in brackets are the polarization values obtained prior to re- laxation of atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We have used 7×1 supercells for the V and Ni halides and √ 3× √ 3 supercells for the Mn halides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' All calcu- lations are set up with left-handed spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The numbers in brackets state the spontaneous polarization without relaxation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' teresting alternative to standard multiferroics such as BiFeO3 and YMnO3, where the coercive electric fields are orders of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Magnetic ground state of AB3 materials The AB3 materials all have space group P¯3m1 correspond- ing to monolayers of the BI3 (or AlCl3) prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The mag- netic lattice is the honeycomb motif, thus hosting two mag- netic ions in the primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Several materials of this pro- totype have been characterized experimentally, but here we only present results for the Cr compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is due to the fact that experimental data of in-plane order is missing for all but CrX3, FeCl3 and RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Moreover, all magnetic com- pounds were found to have a simple ferromagnetic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' RuCl3 is a well known insulator with stripy antiferro- magnetic in-plane order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' However, bare LDA finds a metallic state and both Hubbard corrections and self-consistent spinor- bit coupling are required to obtain the correct insulating state [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The latter is incompatible with the GBT approach and we have not pursued this further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Bulk FeCl3 is known to be an insulating helimagnet with Q = ( 4 15, 1 15, 3 2) [65], while we find the monolayer to be a metallic ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For CrI3 we compare the spin spiral dispersion to the spiral energy determined by a third nearest neighbour energy map- ping procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The prototype thus serves as a testing ground for applying unconstrained GBT to materials with multiple magnetic atoms in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We analyse the intracell an- gle between the Cr atoms of CrI3 and provide an expression for generating good initial magnetic moments for GBT calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We finally discuss the observed deviations from the classical Heisenberg model and to what extend the flat spiral spectrum can be used to obtain the magnon excitation spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' CrX3 The chromium trihalides are of considerable interest due to the versatile properties that arise across the three different halides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Monolayer CrI3 was the first 2D monolayer that were demonstrated to host ferromagnetic order below 45 K [1] and has spurred intensive scrutiny in the physics of 2D magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The magnetic order is governed by strong magnetic easy-axis anisotropy, which is accurately reproduced by first principles simulations [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In contrast, monolayers of CrCl3 exhibit ferromagnetic interactions as well, but no proper long range order due easy-plane anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Instead, these monolayers exhibit Kosterlitz-Thouless physics, which give rise to quasi long range order below 13 K [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The GBT is not really necessary to find the ground state of the monolayer chromium halides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' They are all ferromagnetic and insulating and only involve short range exchange interac- tions that are readily obtained from collinear energy mapping methods [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Nevertheless, the gap between the acoustic and optical magnons in bulk CrI3 has been proposed to arise from either (second neighbor) Dzyalosinskii-Moriya interac- tions [69] or Kitaev interactions [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The former could in principle be extracted directly from planar spin-spiral calcu- lations [40], while the latter requires conical spin spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The origin of this gap is, however, still subject to debate [72] and here we will mainly focus on the magnetic interactions that do not rely on spinorbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the following we will focus on CrI3 as a representative member of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The honeycomb lattice contains two magnetic atoms per unit cell and the magnetic moments at the two sites will in general differ by an angle ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Since we do not impose any con- straints except for the boundary conditions specified by q, the angle will be relaxed to its optimal value when the Kohn-Sham equations are solved self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The convergence of ξ, may be a tedious process since the total energy has a rather weak dependence on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For a given q the classical energy of the model (1) is minimized by the angle ξ 0 given by tanξ 0 = −ImJ12(q) ReJ12(q), (8) where J12(q) = ∑ i J12 0i e−iq·Ri (9) is the Fourier transform of the inter-sublattice exchange cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' If one assumes nearest neighbour interactions only, ξ 0 becomes independent of exchange parameters and the result- ing expression thus comprises a suitable initial guess for the inter-sublattice angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that the classical spiral en- ergy is independent of ξ (in the absence of spinorbit coupling) when J12(q) = 0 and the angle may be discontinuous at such q-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This occurs for example in the magnetic honeycomb lattice at the K-point (q = (1/3,1/3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In general, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (8) has two solutions that differ by π and only one of these minimzes the energy while the other maximizes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The maximum en- ergy constitutes an ”optical” spin spiral branch, which is if interest if one wishes to extract the exchange coupling con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spiral energies of CrI3 (with optimized intracell angles) are shown in figure 4, where we show both the ferromagnetic (ξ = 0) and the antiferromagnetic (ξ = π) results on the ΓK 9 Γ M K Γ 0 10 20 30 E(q) [meV] Mapping FM AFM Γ M K Γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='94 m [µB] Γ M K Γ 0 90 180 ξ [o] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (Left: spin spiral energies of CrI3 compared to third nearest neighbour energy mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Right: angles beteen the two magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spin spirals are initialised with angles determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (8) which are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The moments are collinear on the ΓK path and so the AFM solution is also quasi-stable in DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Center: the magnitude of local magnetic moments along the spiral path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also show the spiral energy obtained from the model (1) with exchange parameters calculated from a collinear en- ergy mapping using four differnet spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We get J1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='47 meV, J2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='682 meV and J3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='247 meV for the first, second and nearest neighbour interactions respectively, which is in good agreement with previous LDA calculations [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The model spiral energy is seen to agree very well with that obtained from the GBT, which largely validates such a three-parameter model (when spinorbit is neglected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We do, however, find a small deviation in the regions between high- symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This is likely due to higher order exchange interaction, which will deviate in the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For example, a biquadratic exchange term [38], will cancel out in any collinear mapping, but will influence the energies ob- tained from the GBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Biquadratic exchange parameters could thus be extracted from the deviation between the two calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In figure 4 we also show the calculated values of ξ and the magnitude of the local magnetic moment at the Cr sites along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The self-consistent intracell angles are found to match very well with the initial guess, except for a slight deviation on the Brillouin zone boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This corroborates the fact that exchange couplings beyond second neighbours are insignificant (the second nearest neighbor coupling is an intra-sublattice interaction and does not influence the angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It is also rather instructive to analyze the variation in the magnitude of local magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In general, the map- ping of electronic structure problems to Heisenberg types of models like (1) rests on an adiabatic assumption where it is assumed that the magnitude of the moments are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' How- ever, the present variation in the magnitude of moments does not imply a breakdown of the adiabatic assumption, but re- flects that DFT should be mapped to a quantum mechanical Heisenberg model rather than a classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In particu- lar, the ratio of spin expectation values between the ferromag- netic ground state and the (anti-ferromagnetic) state of highest energy is approximately ⟨Si⟩AFM/⟨Si⟩FM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='83 in the quan- tized model [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' While this ratio is somewhat smaller than the difference between ferromagnetic and anti-ferromagnetic moments found here, the result does imply that the magni- tude of moments should depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' And the fact that the q = 0 anti-ferromagnetic moments are smaller than the ferro- magnetic ones in a self-consistent treatments reflects that DFT captures part of the quantum fluctuations inherent to the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that the spin spiral energy Eq calculated from the isotropic Heisenberg model using the optimal angle given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (8) is related to the dynamical excitations (magnon ener- gies) by ω± q = E± q /S and the spiral energies thus comprise a simple method to get the magnetic excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' How- ever, even if a model like (1) fully describes a magnetic mate- rial (no anisotropy or higher order terms) there will be a sys- tematic error in the extracted exchange parameters (and re- sulting magnon spectrum) if the parameters are extracted by mapping to the classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The reason is, that the clas- sical energies correspond to expectation values of spin con- figurations with fixed magnitude of the spin, which is not ac- commodated in a self-consistent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This error is di- rectly reflected by the variation of the magnitude of moments in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The true exchange parameters can only be ob- tained either by mapping to eigenstates of the model [74] or by considering infinitesimal rotations of the spin, which may be handled non-selfconsistently using the magnetic force the- orem [37, 75–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Nevertheless, the deviations between ex- change parameters obtained from classical and quantum me- chanical energy mapping typically deviates by less than 5 % [74] and for insulators it is a good approximation to extract the magnon energies from planar spiral calculations although the mapping is only strictly valid in the limit of small q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK In conclusion, we have demonstrated the abundance of spi- ral magnetic order in 2D transition metal dichalcogenides from first principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The calculations imply that type II multiferroic order is rather common in these materials and we have calculated the spontaneous polarization in a se- lected subset of these using fully relaxed structures in super cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' While the super cell calculations does not correspond to the exact spirals found from the GBT, the calculations show that relaxation effects plays a crucial role for the induced po- larization and should be taken into account in any quantitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The spontaneous polarization in type II multifer- roics is in general rather small compared to what is found in ordinary 2D ferroelectrics and could imply that the chirality 10 of spirals are switchable by small electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' It would be highly interesting to calculate the coercive field for switch- ing in these materials, but due to the importance of relaxation effects and spin-orbit coupling this is a non-trivial computa- tion that cannot simply be obtained from the Born effective charges and force constant matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The GBT comprises a powerful framework for extract- ing the magnetic properties of materials from first princi- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition to the single-q states considered here, one may use super cells to extract the importance of higher order exchange interactions and unravel the possibility of having multi-q ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In addition, for non-centrosymmetric materials, the PSO approach may be readily applied to ob- tain the Dzyaloshinskii-Moriya interactions, which may lead to Skyrmion lattice ground states or stabilize other multi-q states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Implementation In the PAW formalism we expand the spiral spinors using the standard PAW transformation [79] ψq,k(r) = ˆ T ˜ψq,k(r) = ˜ψq,k(r)+∑ a ∑ i (φ a i (r)− ˜φ a i (r)) � dr[ ˜pa i (r)]∗ ˜ψq,k(r), (10) where ˜ψq,k(r) is a smooth (spinor) pseudo-wavefunction that coincides with ψq,k(r) outside the augmentation spheres and devi- ates from ψq,k(r) by the second term inside the augmentation spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The all-electron wavefunction ψq,k(r) is thus expanded in terms of (spinor) atomic orbitals φ a i inside the PAW spheres and the expansion coefficients are given by the overlap between the pseudowavefunction and atom-centered spinor projector functions ˜pa i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (3) we may write this as ψq,k(r) = eik·rU† q(r) ˜uq,k(r)+∑ a ∑ i (φ a i (r)− ˜φ a i (r)) � dr[ ˜pa i (r)]∗eik·rU† q(r) ˜uq,k(r) = eik·rU† q(r) ˜uq,k(r)+∑ a ∑ i (φ a i (r)− ˜φ a i (r)) � dr[e−ik·rUq(r) ˜pa i (r)]∗ ˜uq,k(r) = eik·rU† q(r) ˜uq,k(r)+∑ a ∑ i (φ a i (r)− ˜φ a i (r)) � dr[ ˜pa i,q,k(r)]∗ ˜uq,k(r) ≡ Tq,k ˜uq,k(r), (11) where Uq(r) was given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (4) and we defined ˜pa i,q,k(r) = e−ik·rUq(r) ˜pa i (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (12) The PAW transformed Kohn-Sham equations then read ˜Hq,k ˜uq,k(r) = εq,kSq,k ˜uq,k(r), (13) with ˜Hq,k = T † q,kHTq,k, Sq,k = T † q,kTq,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (14) Calculations in the framework of the GBT thus requires two modifications compared to the approach for solving the ordinary Kohn-Sham equations in the PAW formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 1) The k-dependence of the standard Bloch Hamiltonian is replaced by k → k ∓ q/2 for spin-up and spin down components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 2) Different spin dependent projector functions has to be applied when calculating the projector overlaps with the spin-up and spin-down components of the psudowavefunctions (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Benchmark The LDA implementation of the GBT have been tested by checking that our results agree with similar calculations from the literature and by verifying internal consistency by compar- ing with super cell calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The case of fcc Fe has been found to have a spin spiral ground state [81] and the calcu- lation of the ordering vector Q has been become a standard benchmark for spin spiral implementations [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In previous simulations the ordering vector was found to be rather sensi- tive to the lattice constant and in figure 6 we show the spin spi- ral energies along the ΓXW path using the experimental lattice constant as well as the lattice constant which has been found to reproduce the experimental ordering vector [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The cal- culated value of Q is in good agreement with previous reports in both cases [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also confirm a similar low energy bar- 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 qx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 E − E0 [eV] Spiral Supercell FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Comparison between GBT spin spiral calculations and su- percell calculations without spinorbit coupling in monolayer CoPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 100 120 140 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='58 ˚A (exp) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='50 ˚A Γ X W −20 −10 0 E(q) [meV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spin spiral energies of fcc Fe for the experimental lattice constant (red) and a strained latice constant, which is known to re- produce the experimental spin spiral order in (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The dashed vertical lines indicate the minima found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' rier between the two local minima, as is expected from LDA [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In order to check internal consistency we have investigated the case of monolayer CoPt [40] where we compare spin spi- ral energies calculated using the GBT with energies calculated from super cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We thus construct a 16x1 super cell of the CoPt monolayer and consider spirals with qc = ( n 16) in units of reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' This allows us to extract 16 different spiral energies in the supercell using standard non-collinear DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In order to compare the two methods we have used a k-point grid of 16×16×1 for the GBT and 1×16×1 for the supercell and a plane wave cutoff of 700 eV for both calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 5 we compare the results without spinorbit coupling and find excellent agreement between supercell and GBT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that when spinorbit coupling is neglected one has Eq = E−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Since spinorbit coupling is incompatible with the GBT one has to resort to approximate schemes to include it in the cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In the present work we have used the PSO method proposed by Sandratskii [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 7 we compare spin spi- ral calculations with supercell calculations where the spinorbit coupling has been included either fully or by the PSO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The PSO method is fully compatible with the GBT and we −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 qx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='20 E − E0 [eV] Spiral Proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' SOC Spiral Full SOC SC Full SOC SC Projected SOC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Comparison between GBT spin spiral calculations and super- cell calculations with projected and full spinorbit coupling in mono- layer CoPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' find excellent agreement between the spin spiral energies cal- culated with the GBT and with supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The PSO approach is, however an approximation and the correct result can only be obtained from the supercell using the full spinorbit cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We see that the PSO calculations are in good agreement with those obtained from full spinorbit coupling but overesti- mates the energies at the Brillouin zone boundary by a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' In contrast, if one tries to include the full spinor- bit operator in the GBT calculations (by diagonalizing HKS including spinorbit coupling on a basis of GBT eigenstates without spinorbit coupling) the energies are severely under- estimated with respect to the exact result (from the supercell calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that the spiral energies including spinor- bit coupling shows a slight asymmetry between points at q and −q, which can be related to the Dzyaloshinskii-Moriya inter- actions in the system [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Seyler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Zhong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Schmidgall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' McGuire, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' H.' metadata={'source': 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Newman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Lambert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Pruneda, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Ferrer, First principles simulations of the magnetic and struc- tural properties of iron, The European Physical Journal B- Condensed Matter and Complex Systems 40, 371 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' [84] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Kn¨opfle, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Sandratskii, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' K¨ubler, Spin spiral ground state of γ-iron, Physical Review B 62, 5564 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 15 SUPPLEMENTARY INFORMATION - TYPE II MULTIFERROIC ORDER IN TWO-DIMENSIONAL TRANSITION METAL HALIDES FROM FIRST PRINCIPLES SPIN-SPIRAL CALCULATIONS Joachim Sødequist1 and Thomas Olsen1,∗ 1CAMD, Department of Physics, Technical University of Denmark, 2820 Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Lyngby Denmark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' SPIN SPIRAL DISPERSIONS The entire spin spiral dispersion carry more information than just the energy minima was reported we reported in the main text, these are shown here for in figures 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' One can find not only the stability with respect to the ferromagnetic configuration, but also compare to any other configuration in the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Additionally, we can observe whether the remain magnetic moments are unchanged during self-consistent field cycle, and we find this is generally true except for CoI2 and perhaps the titanium compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We note that the local magnetic moments found here are the integrated inside the PAW spheres of the respective atoms, thus these local moments does not integrate to the total moments reported in the main text since interstitial magnetization density is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We also provide the projected spin orbit energies in figure 3 for the lowest energy state, naturally the shape will depend very much on the specific spiral, in some cases such as the q = K we find quite similar energies in the out-of-plane orientations, whereas incommensurate spirals tend to have more well defined minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The in-plane orientations reported here are related by a 90◦ phase shift, but the dashed line highlight that they are indeed degenerate as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' SPIN SPIRAL CONVERGENCE An example of convergence of a intracell angle in a rectangular supercell of the hexagonal VI2 system is represented in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We find that for all calculations which reach the convergence criteria on particularly the density, all converge the angle within some narrow region of the true angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' We observe that the number of iterations required increase dramatically, when the initial guess is further away from the true angle, hence highlighting the importance of choosing initial conditions according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' (8) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Convergence of the intracell angle ξ in spin spiral ground state calculation of VI2 at the spiral vector q = (1/4,0,0) at varying different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The calculations shown in red, did not reach the convergence criteria on the density within the time-wall of the calculation, while the blue were considered converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' The black horizontal line is the expected angle for a smooth spin spiral as it if it was an equivalent spin spiral in the primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Vl2convergence,g=(1/4,0,0),rect 175 150 125 5-angle 100 75 50 25 0 0 200 400 600 800 1000 SCF-count16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spin spiral energies for AB2 magnets and the local magnetic moments of the magnetic atoms and ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' For ferromagnetic refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' TiBr2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 inf 0 [] Energy 10 Ti magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 [meV] I norm magnetic moment 20 Br magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 Spin spiral energy [ 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 60 S 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 ocal 80 Lo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]Til2 Wave length ^ [A] inf 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 inf 0 n magnetic moment [lμsl] Energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 Ti magmom M 10 [me] I magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 Spin spiral energy 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 30 ocal norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 Lo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]VCI2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 inf 0 一 门 Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 10 V magmom [meV] moment Cl magmom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 20 Spin spiral energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 5 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]VBr2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 inf 0 门 : moment [lμl] Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 V magmom [meV] Br magmom 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 Spin spiral energy 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]V12 Wave length 入 [A] inf 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 inf ６ Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 V magmom [meV] 4 I magmom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 I energy 2 Local norm magnetic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 Spin spiral 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]MnCI2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 inf 0 Energy Mn magmom Local norm magnetic moment [ Spin spiral energy [meV] Cl magmom 10 2 15 20 K M q vector [A-1]MnBr2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 inf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 [|μB|] Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 Mn magmom [meV] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 Br magmom 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 I energy Spin spiral 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0 K M q vector [A-1]Mn12 Wave length ^ [A] inf 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 inf 0 Energy Mn magmom Spin spiral energy [meV] I magmom 10 15 20 K M q vector [A-1]CoBr2 Wave length 入 [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 inf Energy Co magmom [meV] 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 Br magmom Spin spiral energy 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 Local norm magnetic i 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0 K M q vector [A-1]Col2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 inf Energy 20 Co magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 Spin spiral energy [meV] I magmom 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]NiCI2 Wave length 入 [A] inf 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 inf 50 Energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 Ni magmom Spin spiral energy [meV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 40 Cl magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 Local norm magnetic 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]NiBr2 Wave length 入 [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 inf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 Energy n magnetic moment [lμbl 40 Ni magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 [meV] Br magmom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 30 Spin spiral energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 Local norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 K M q vector [A-1]Nil2 Wave length ^ [A] inf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 inf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 40 Energy magnetic moment [lμ] Ni magmom [meV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 30 I magmom 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 T energy 10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 Spin spiral 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 norm 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 20 ocal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 30 - K M q vector [A-1]17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Projected spin orbit energies of the ground state found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 2 TiBr2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 - [meV] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 E 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 - IP Screw OoP IP theta, phiTil2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 [meV] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 IP Screw OoP IP theta, phiVCI2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='70 [meV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='74 IP Screw OoP IP theta, phiVBr2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='31 [meV] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='32 Jos: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='33 E 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='36 IP Screw OoP IP theta, phiVI2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 E 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='9 IP Screw OoP IP theta, phiMnCI2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='780 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='785 [meV] 790 Jos: E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='795 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='800 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='805 IP Screw OoP IP theta, phiMnBr2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='22 IP Screw OoP IP theta, phiMn12 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='8 [meV] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='4 IP Screw OoP IP theta, phiCoBr2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='40 [meV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='42 soc E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='44 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='46 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='48 IP Screw OoP IP theta, phiCol2 40 41 [meV] 42 soc E 43 44 - 45 - IP Screw OoP IP theta, phiNiCI2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='050 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='055 [meV] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='060 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='065 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='070 IP Screw OoP IP theta, phiNiBr2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='65 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='70 [meV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='75 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='80 E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='85 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='90 IP Screw OoP IP theta, phiNil2 40 41 - [meV] 42 43 44 : IP Screw OoP IP theta, phi18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content=' Spin spiral energies for ferromagnetic AB2 magnets and the local magnetic moments on the atoms FeCI2 Wave length 入 [A] inf 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 40 Energy [8l] 35 Fe magmom 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 Spin spiral energy [meV] Local norm magnetic moment [ I magmom 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='5 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQffQ30/content/2301.05107v1.pdf'} +page_content='0 10 0.' metadata={'source': 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2023) +The construction of Hilbert spaces that are characterized by local constraints as the low-energy +sectors of microscopic models is an important step towards the realization of a wide range of quantum +phases with long-range entanglement and emergent gauge fields. Here we show that planar structures +of trapped atoms in the Rydberg blockade regime are functionally complete: Their ground state +manifold can realize any Hilbert space that can be characterized by local constraints in the product +basis. We introduce a versatile framework, together with a set of provably minimal logic primitives +as building blocks, to implement these constraints. As examples, we present lattice realizations of +the string-net Hilbert spaces that underlie the surface code and the Fibonacci anyon model. We +discuss possible optimizations of planar Rydberg structures to increase their geometrical robustness. +I. +INTRODUCTION +Recent advances in the control of single atoms and their +coherent manipulation [1–5] are the technological founda- +tion for applications such as quantum simulation [6–9], +high-precision metrology [10, 11] and, hopefully, future +quantum computers [12–15]. For any of these applica- +tions, suitable platforms must offer a fine-grained control +over of their degrees of freedom, dynamically tunable +interactions, and the possibility to decouple the environ- +ment. Promising in this regard are arrays of individually +trapped, neutral atoms that can be manipulated by opti- +cal tweezers [1, 3] and excited into Rydberg states [16, 17]. +These exhibit strong interactions which lead to the Ry- +dberg blockade mechanism where excited atoms prevent +their neighbors within a tunable radius from being excited +[18–22]. In this paper, we study on very general grounds +the theoretical capabilities of the Rydberg platform in +the blockade regime and demonstrate its versatility by +constructing the gauge-invariant Hilbert spaces of two +models with abelian and non-abelian topological order. +Encouraged by the fast development of the Rydberg +platform, there has been increased interest in identifying +promising near-term applications for the NISQ era [23]. +Among the many applications of two-dimensional arrays +of Rydberg atoms, the field of geometric programming and +the design of synthetic quantum matter have been identi- +fied as promising candidates to leverage the capabilities +of available and upcoming NISQ platforms. +The rationale of geometric programming is the solu- +tion of algorithmic problems by encoding them into the +geometry of the atomic array. This direction of research +is founded on the insight that due to the Rydberg block- +ade, the ground states of these systems naturally map to +maximum independent sets (MIS) on so called unit disk +graphs [24]; finding MIS is a long-known optimization +problem in graph theory that has been shown to be NP- +hard [25]. This makes the computation of ground state +∗ nicolai.lang@itp3.uni-stuttgart.de +energies of Rydberg arrangements NP-hard as well [26], +but also opens the possibility to tackle a variety of other +hard optimization problems [27, 28] by polynomial-time +reductions to the MIS problem [29]. First solutions of MIS +instances on various graphs in two and three dimensions +have been demonstrated in experiments recently [30–32], +and a quantitative comparison of experimental solutions +with classical algorithms suggest a superlinear quantum +speedup for some classes of graphs [32]. +A very different application of the Rydberg blockade +mechanism is the engineering of synthetic quantum matter +on the single-atom level. The potential of this approach +has been demonstrated recently by Verresen et.al. [33] +(related results were reported by Samajdar et.al. [34]), +who proposed the realization of topological spin liquids on +delicately designed lattice structures of atoms. In this sce- +nario, the Rydberg blockade enforces a dimer constraint +(the local gauge constraint of an odd Z2 lattice gauge +theory [35]) which, in combination with quantum fluctua- +tions, can give rise to long-range entangled many-body +states with abelian topological order. First experimental +results were reported shortly after [36], accompanied by a +theoretical study of the used quasiadiabatic preparation +scheme [37]. +This paper is written from and motivated by the syn- +thetic quantum matter perspective, but its results apply +to geometric programming as well. Our starting point is +the question whether other local constraints (besides the +dimer constraint) can be realized on the Rydberg plat- +form. To find an answer, we first formalize the problem +and then use this formulation to derive our main result, +namely that every local constraint that can be encoded +by a Boolean function can be implemented in the ground +state manifold of a planar arrangement of atoms in the +blockade regime. Crucial for this result is the existence of +a structure that implements the truth table of a NOR-gate +(“Not OR”) in its ground state manifold. While our proof +is constructive, it does typically not yield optimal (= +small) solutions. We therefore expand on our main result +and compile a comprehensive list of provably minimal +structures that realize all important primitives of Boolean +logic. Together with a structure that facilitates the cross- +arXiv:2301.01508v1 [quant-ph] 4 Jan 2023 + +2 +ing of two “wires” within the plane, these primitives +provide a toolbox to build structures that satisfy more +complicated constraints. As an example, we construct a +system with a ground state manifold that is locally iso- +morphic to the gauge-invariant Hilbert space of an even +Z2 lattice gauge theory, i.e., the charge-free sector of the +toric code [38]. With a similar construction, we tailor a +pattern of atoms with a ground state manifold isomorphic +to the string-net Hilbert space of the “golden string-net +model [39]”; a system that, with added quantum fluc- +tuations, could support non-abelian Fibonacci anyons. +Having constructed all these structures, we briefly discuss +possibilities to numerically optimize their geometries to +make them more robust against geometric imperfections +and the effects of long-range van der Waals interactions. +Note added. When finalizing this manuscript we be- +came aware of related results reported by Nguyen et. al +in Ref. [40]. The authors focus on optimization problems +on non-planar graphs and find the same structures for +some of the primitives discussed in this paper (especially +the implementation of the ring-shaped NOR-gate and the +crossing). The authors follow the rationale of geomet- +ric programming, so that their motivation, approach and +framework differ from ours. +II. +RATIONALE AND OUTLINE +Here we illustrate the rationale of the paper and pro- +vide a brief outline of its main results without technical +overhead. Readers interested in the details can skip for- +ward to Section III. Readers only interested in specific +applications can read this section first and then skip to +Section VII or Section IX. +In this paper, we consider two-dimensional arrange- +ments of trapped atoms that can either be in their elec- +tronic ground state or excited into a Rydberg state (Ryd- +berg structures). We focus on systems without quantum +fluctuations, where the ground states are determined by +local detunings and Rydberg blockade interactions (Sec- +tion III). The detunings lower the energy for atoms in +the Rydberg state by an atom-specific amount, and the +Rydberg blockade interaction forbids atoms closer than a +specific distance to be excited simultaneously. The inter- +play of these two contributions singles out ground states +that are characterized by excitations patterns where no +additional atom can be excited without violating the Ryd- +berg blockade, and where the sum of the detunings of the +excited atoms is maximal (so called maximum-weight inde- +pendent sets). There can be different configurations that +minimize the energy, hence the ground state manifold is +typically degenerate. In this paper, we ask which ground +state manifolds such structures can realize and, conversely, +how to tailor structures that realize a prescribed ground +state manifold (Section IV). +A simple example is given in Fig. 1a where the po- +sition of the atoms is shown in (i); the two atoms are +constrained by the Rydberg blockade (gray circles) and +Figure 1. Rationale. (a) Structure of two atoms (i) with local +detunings ∆ (blue vertices) that are in Rydberg blockade (gray +circles); the blockade is indicated by a black edge connecting +the atoms. The ground state manifold (ii) is given by patterns +of excited atoms (orange) that minimize the energy; here it +is two-fold degenerate. The two ground state configurations +realize the truth table (iii) of a NOT-gate Q = A. (b) Structure +of five atoms (i) with local detunings ∆ (blue) and 2∆ (green) +in a ring-like Rydberg blockade. The ground state manifold (ii) +is four-fold degenerate. If one selects the three labeled atoms +and identifies them with the columns of the table in (iii), the +four ground state configurations realize the truth table of a +NOR-gate Q = A ↓ B = A ∨ B. (c) Joining the output atom of +the NOR-gate with the input atom of the NOT-gate (and adding +their detunings) yields a new structure that realizes the truth +table of an OR-gate: Q = A ↓ B = A ∨ B. This construction is +called amalgamation. +cannot be excited simultaneously (indicated by the black +edge connecting them). The color of the atoms encodes +their detuning; here both atoms lower the energy of the +system by ∆ when excited into the Rydberg state (blue +nodes). In (ii) we show the two excitation patterns that +minimize the energy (orange nodes denote excited atoms). +Note that the atoms cannot be excited simultaneously +due to the Rydberg blockade. If one lists the ground state +configurations in a table, where each column corresponds +to an atom and each row to a ground state configuration, +we find the “truth table” of a Boolean NOT-gate Q = A. +Here we interpret one of the atoms as “input” (A) and +the other as “output” (Q). +This concept generalizes to more complicated Boolean +gates (Fig. 1b): Consider the five atoms in a ring-like +blockade (i). Three of the atoms (blue) lower the energy +by ∆, two (green) by 2∆ when excited. By inspection +one finds the four degenerate ground state configurations +in (ii). This is promising as truth tables of Boolean gates +that operate on two bits have four rows. However, they +only have three columns (two for the inputs of the gate +and one for its output). We therefore select three of the +five atoms by assigning labels to them: A and B play the + +3 +role of the inputs and Q is the output. We call atomic +structures with designated input/output atoms Rydberg +complexes (Section V A). If we list the four ground state +configurations of these three atoms, we find the truth +table of a NOR-gate Q = A ↓ B = A ∨ B in (iii). Note +that the remaining two atoms (we call them ancillas)— +while not contributing independent degrees of freedom— +are still necessary to realize this specific ground state +manifold. At this point things get interesting because it +is a well-known fact of Boolean algebra that the NOR-gate +is functionally complete (just like the NAND-gate): Every +Boolean function can be decomposed into a circuit build +from NOR-gates only. +To leverage this decomposition, we need a method +to combine “gate complexes” to form larger “circuit +complexes”; we call this procedure amalgamation (Sec- +tion V B). A simple example is shown in Fig. 1c where we +attach the NOT-gate from Fig. 1a to the output of the NOR- +gate in Fig. 1b (note that the detunings of the atoms that +are joined add up). Using the detunings and blockades in +(i) yields the four degenerate ground state configurations +in (ii). When we label the inputs of the NOR-gate again by +A and B, and now focus on the output Q of the attached +NOT-gate, we find indeed the truth table of an OR-gate +Q = A ↓ B = A ∨ B in (iii). Thus we can parallel the log- +ical composition of gates by a geometrical combination of +atomic structures such that the relation between ground +state configurations and truth tables remains intact. In +combination with the insight that every Boolean circuit +can be drawn in the plane without crossing lines (after +suitable augmentations), this allows us to show that the +truth table of any Boolean function can be realized as +the ground state manifold of a suitably designed atomic +structure. This functional completeness is our first main +result and motivates the title of the paper (Section VI). +For instance, the existence of a structure that realizes +the truth table of an OR-gate is a corollary of functional +completeness. However, the specific construction as the +combination of a NOR-gate and a NOT-gate in Fig. 1c raises +the questions whether this particular realization with +six atoms is unique and whether it is minimal (in the +sense that the same truth table could not be realized +with fewer atoms). The answer to the first question is +negative: There are geometrically different structures +that realize the same truth table in their ground state +manifold. The answer to the second question is positive, +though: We show that it is impossible to implement this +truth table with less than six atoms. Note that the func- +tional completeness implies the existences of structures +for all common gates of Boolean logic (such as AND, XOR, +etc.). We take this as motivation to construct provably +minimal structures for all these primitives (Sections VII +and VIII). Together with the procedure of amalgama- +tion, these equip our versatile toolbox to engineer more +complicated structures. +Our second important contribution is an application of +the functional completeness as a tool to engineer synthetic +quantum matter (Section IX). Many interesting quantum +phases in two dimension are characterized by hidden pat- +terns of long-range entanglement, known as topological +order. These patterns can give rise to anyonic excitations +which make such systems potential substrates for quantum +memories and even quantum computers. A large class of +entanglement patterns can be understood as condensates +of extended objects (like strings). A crucial first step for +the realization of these phases is therefore the prepara- +tion of Hilbert spaces spanned by states of such extended +objects. However, in experiments, we typically start from +Hilbert spaces with a local tensor product structure (for +example, an array of two-level atoms). Our only hope is +to make the extended objects emerge due to interactions +in the low-energy sector of a suitably designed physical +system. This often boils down to enforce local gauge sym- +metries which single out states that can be interpreted in +terms of extended objects. Such local constraints can be +reformulated as Boolean functions that must be satisfied +by the states of the local degrees of freedom of the underly- +ing system. For any constraint of this form, our functional +completeness result ensures the existence of a structure of +atoms, interacting via the Rydberg blockade mechanism, +that realizes this constraint in its ground state space. It is +then just a matter of copying and joining these structures +in a translational invariant way to tessellate the plane. +The ground state manifolds of such tessellations can there- +fore implement a large class of non-trivial Hilbert spaces +on which condensation (driven by quantum fluctuations) +might lead to topologically ordered many-body quantum +phases. Using our toolbox developed in the first part of +the paper, we demonstrate this construction explicitly +for the abelian toric code phase (Section IX A) and the +non-abelian, computationally universal Fibonacci anyon +model (Section IX B). +The truth tables realized by the ground states of all +atomic structures presented in this paper depend on the +positions of the atoms. (Because these positions define +which pairs are in blockade and which atoms can be ex- +cited simultaneously.) However, the exact placement is +often ambiguous. For example, consider the structure in +Fig. 1a (i) which realizes the NOT-gate. It is clear that the +blockade constraint (black edge) does not change if the +atoms are slightly shifted, as long as the blockade radii +(gray circles) encompass both atoms. We refer to the set +of atom positions as the geometry of a structure and argue +that “robust” geometries should avoid distances between +atoms that are close to the critical blockade distance. For +the complexes in Fig. 1, this translates into the geomet- +ric objective to maximize the distances between nodes +and gray circles. We formalize this notion by assigning a +number to geometries that quantifies their “robustness” +(Section X A) and numerically construct optimized geome- +tries that maximize this number (Section X B). +We conclude the paper with an outline of open ques- +tions, directions for further research (Section XI), and a +brief summary (Section XII). + +4 +III. +PHYSICAL SETTING +We consider planar arrangements of trapped atoms +with repulsive van der Waals interactions when excited +into the Rydberg state [2, 41]. Every atom is assigned +an index i ∈ V = {1 . . . N}, placed at position ri ∈ R2, +and described by a two-level system |n⟩i where n = 0 +corresponds to the electronic ground state and n = 1 the +excited Rydberg state. +The quantum dynamics of such systems is achieved +by coupling the electronic ground state to the Rydberg +state by external laser fields with Rabi frequency Ωi and +detuning ∆i for each atom [42–44]. Here we are mainly +interested in the regime Ωi → 0 where the Hamiltonian +reduces to +H[C] = − +� +i +∆ini + +� +i 0 the coupling strength of the van der Waals +interaction; we refer to H[C] with U = UvdW as the van +der Waals (vdW) model. However, in many situations +a simplified model U = U∞ with U∞(r ≥ rB) = 0 and +U∞(r < rB) = ∞ with blockade radius rB is a reasonable +approximation for the low-energy physics of Eq. (1); we +refer to H[C] with U = U∞ as the PXP model [33, 47]. In +this paper, we use the PXP model unless stated otherwise. +We discuss valid choices for the blockade radius rB in +Section X A where we optimize the geometry of structures +to limit the effects of residual van der Waals interactions. +In the PXP model, the effect of the van der Waals +interactions is approximated by a kinematic constraint +that is completely encoded by a blockade graph B = +(V, E), where an edge e = (i, j) ∈ E between atoms i, j ∈ +V indicates that they are in blockade, i.e., their distance +is smaller than the blockade radius rB. An abstract graph +that can be realized in this way is called a unit disk +graph (not every graph has this property); conversely, +a geometry GC that realizes a prescribed graph as its +blockade graph is a unit disk embedding of this graph (the +“unit” here is the blockade radius rB). Throughout the +paper, the blockade graph of a structure will be drawn +by black edges connecting atoms that are in blockade. +IV. +DEFINITION OF THE PROBLEM +The primary goal of this paper is to find structures C +such that there is a well-separated low-energy eigenspace +H0[C] of H[C] where H0[C] satisfies certain prescribed +Figure 2. Setting & Objective. A two-dimensional structure +C = (ri, ∆i)i∈V of atoms i ∈ V with position ri and detuning +∆i is governed by the Hamiltonian H[C] that describes the +Rydberg blockade interaction with blockade radius rB. The +Hamiltonian gives rise to a low-energy eigenspace H0[C] < H +of width δE, separated from the excited states by a gap ∆E. +The objective of this paper is the construction of a structure C +from a given target Hilbert space HT such that H0[C] ≃ HT. +properties that we describe in detail below. We quantify +the separation of H0[C] by its spectral width δE and +its gap ∆E to the rest of the spectrum (Fig. 2). Note +that the experimental prerequisites for the construction of +arbitrary structures C are already in place [4, 45, 46, 48]. +If one would switch on a weak drive δE < Ωi ≪ ∆E, this +would induce quantum fluctuations between the states of +the Hilbert space H0[C], potentially giving rise to many- +body states with interesting properties. In this paper, +we do not study such quantum effects but focus on the +implementation of the subspace H0[C]. We specify the +eigenspace to construct in terms of a target Hilbert space +HT: +H0[C] +!≃ HT . +(2) +Informally speaking, our goals is to “solve” this equation +for structures C for given HT. To make this possible, the +target Hilbert space HT must be specifiable in a form +that we define in the remainder of this section. +Formal languages. +Throughout the paper we make +use of the notion of (formal) languages [49] on the bi- +nary alphabet F2 = {0, 1}. A word x ≡ (x1x2 . . . xn) ≡ +x1x2 . . . xn ∈ F∗ +2 is a finite string of letters xi ∈ F2 (the +set of all such finite strings is denoted F∗ +2). +A (for- +mal) language L is then simply a collection of words: +L ⊆ F∗ +2. +Here we only consider uniform languages +with words that have all the same length. For exam- +ple, LCPY := {000, 111} ⊂ F∗ +2 is a uniform language of +words with length n = 3, x = (111) is a word in LCPY and +x1 = 1 is the first letter of x. The words y = (011) ∈ F∗ +2 +and z = (0000) ∈ F∗ +2 are not in this language: y, z /∈ LCPY. +The subscript “CPY” stands for “copy” and hints at the +role this language will play later. +Other examples are the class of languages generated by +the truth tables of Boolean functions. Let w : Fn−1 +2 +→ F2 +be an arbitrary Boolean function of n − 1 variables; then +L[w] := {x1 . . . xn−1y | y = w(x1, . . . , xn−1)} ⊂ F∗ +2 +(3) + +5 +Figure 3. +Tessellated language & target Hilbert space. +A +tessellated target Hilbert space HT is a subspace of the full +Hilbert space of K qubits placed on each edge of a square +lattice L; it is spanned by product states |x⟩ of bit patterns +x ∈ LL[fT]. The tessellated language LL[fT] comprises all bit +patterns x ∈ F∗ +2 that locally satisfy the Boolean check function +fT : Fg +2 → F2. The g = 4K arguments of the check function +on each site s are singled out by the bit-projector us. +is the language generated from the rows of the truth table +of w, where the first n−1 letters of each word correspond +to the input x and the last letter encodes the output +w(x). A language of this class always has 2n−1 words of +uniform length n. Note that the “copy” language LCPY is +not of the form Eq. (3). +Another special class is given by tessellated languages +on lattices. In the following, we introduce the concept ex- +emplarily for a finite square lattice L with periodic bound- +aries; the generalization to other lattices and boundary +conditions is straightforward. Start by associating K clas- +sical bits to every edge e ∈ E(L) of the lattice (Fig. 3). A +bit configuration of the system x ∈ XL = FK|E(L)| +2 +⊂ F∗ +2 +assigns every bit a Boolean value xi +e (i = 1 . . . K). We +focus on the family of uniform languages L ⊆ XL that +can be characterized by a Boolean function that is local +in the following sense: For a site s ∈ V (L) of the square +lattice, let the bit-projector us(x) = (x1 +e1, . . . , xK +e4) single +out the (ordered) set of g = 4K bits on the four edges +ei emanating from s. Let f : Fg +2 → F2 be an arbitrary +Boolean function of g arguments, henceforth referred to +as check function. The tessellated language of bit patterns +on L generated by f is then defined as +LL[f] := {x ∈ XL | ∀s ∈ V (L) : f(us(x)) = 1} . +(4) +In words: LL[f] is the set of bit patterns on the lattice L +that locally satisfy the constraints imposed by f. +Target Hilbert spaces. +To any uniform language +L ⊆ Fn +2 we can naturally associate the linear subspace of +states on n qubits (or spin-1/2) +H(L) := span { |x⟩ | x ∈ L } ⊆ (C2)⊗n . +(5) +For example, H(LCPY) = span { |000⟩ , |111⟩ } is the two- +dimensional subspace on three qubits spanned by product +states with configurations in LCPY = {000, 111}. By con- +trast, the Hilbert space H′ = span +� +(|000⟩ + |111⟩)/ +√ +2 +� +is not of the form (5). +We require the target Hilbert space HT, that we aim +to realize as ground state manifold H0[C], to be specified +by a language LT according to Eq. (5): +HT = H(LT) . +(6) +We are particularly interested in the special class of tes- +sellated target Hilbert spaces given in terms of tessellated +languages that are generated by a check function (Fig. 3): +HT = HL[fT] := H(LL[fT]) . +(7) +Recall that these languages come equipped with a spatial +structure (in the sense that the bits are located on the +edges of a lattice L). This spatial structure is inherited +by the Hilbert space HL[fT] viewed as state space of a +system where K qubits are placed on every edge of L. +For example, the Hilbert space HZ2 of the even Z2 +lattice gauge theory is a particular subspace of a Hilbert +space that describes a system of qubits on the edges of +a square lattice (i.e., K = 1 and g = 4). HZ2 is spanned +by the product states of patterns of qubits in the state +|1⟩ that form closed loops [50]. +HZ2 is an admissible +tessellated target Hilbert space because we can realize +HZ2 = HL[fZ2] with the check function +fZ2(x1, x2, x3, x4) = 1 ⊕ x1 ⊕ x2 ⊕ x3 ⊕ x4 +(8) +where ⊕ denotes modulo-2 addition (Exclusive-OR or +XOR); the bit-projector us(x) simply singles out the four +bits on edges emanating from site s: +us +� +� +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3, x1 +e4) . +(9) +Physically, Eq. (8) enforces Gauss’s law on a charge-free +background by forbidding strings of qubits in state |1⟩ +to end on a site. Further examples for tessellated target +Hilbert spaces are the more general “string-net” Hilbert +spaces that can describe a large variety of topological +orders and deconfined gauge theories [39]. +V. +RYDBERG COMPLEXES +Before we can tackle our main goal, namely the con- +struction of tessellated Rydberg structures C with H0[C] ≃ +HT = HL[fT] for a given check function fT, we first need +to specify the notion of a finite Rydberg complex as a pre- +liminary step. Specific examples for Rydberg complexes +can be found throughout the remainder of the paper. + +6 +A. +From structures to complexes +Consider the language LCPY = {000, 111} and let +HCPY = H(LCPY) = span { |000⟩ , |111⟩ } be our target +Hilbert space. Our goal is to realize HCPY as the ground +state manifold H0[CCPY] of a structure CCPY of n = 3 atoms. +This, however, is impossible: Since |111⟩ ∈ HCPY, none of +the three atoms can be in blockade with each other. Con- +sequently, H0[CCPY] cannot contain only the states |000⟩ +and |111⟩ (Appendix A 1). This problem is not specific +to the language LCPY but shared by many (though not all) +languages. The solution is to consider larger structures +of N ≥ n atoms and to identify the letters of words with +a subset of n distinguished atoms (we call them ports); +the remaining N − n atoms play then the role of ancillas. +A structure together with a distinguished set of ports will +be referred to as a (Rydberg) complex. +Let us formalize this notion. Consider a structure C +of N atoms and a language L ⊆ Fn +2 of words of uniform +length n ≤ N. Let L = {A, B, . . . } denote a set of n labels +where each label is associated with a fixed letter position +of words in L. (If one prints all words of L as rows of a +table, the labels correspond to the column headers.) Let +ℓ : L → V be an injective label function that assigns a +label to a subset of n atoms (the ports); the N − n atoms +without labels are the ancillas. We refer to the structure +C together with the labeling ℓ as a (Rydberg) L-complex +CL if the states that span H0[C] can be identified by the +configurations of the ports alone: +H0[CL] ≡ H0[C] = span { |x, a(x)⟩ ∈ H | x ∈ L } . +(10) +In |x, a(x)⟩, the state of ports is given by the first n bits x +(in some fixed order) and the state of ancillas by a N − n +bit-valued function a : L → FN−n +2 +. The ground state +space H0[CL] will be referred to as an L-manifold. An +important aspect of this definition is that the ancillas do +not introduce additional low-energy degrees of freedom; +they are only needed to unleash the full potential of the +blockade interactions. In this sense, we say that a complex +CT ≡ CLT realizes a target Hilbert space HT = H(LT) and +write +(C2)⊗N ⊇ H0[CT] ≃ HT = H(LT) ⊆ (C2)⊗n +(11) +with the isomorphism ≃ given by |x, a(x)⟩ ↔ |x⟩. If we +say that a complex realizes a language L, we mean that +it realizes the target Hilbert space HT = H(L) defined by +this language. +As an example, consider the again the “copy” language +LCPY = {000, 111} with n = 3; the ground state manifold +of a LCPY-complex CCPY ≡ CLCPY must be two-dimensional +(since |LCPY| = 2) and characterized by the property that +three distinguished atoms (the ones assigned labels by ℓ) +are always forced to be in the same state: +H0[CCPY] = span { |000, a(000)⟩ , |111, a(111)⟩ } . +(12) +Such a complex will be one of our primitives to implement +check functions for tessellated target Hilbert spaces. We +will discuss a specific realization CCPY that requires a single +ancilla in Section VI; that is, with N = n = 3 atoms the +target Hilbert space HCPY cannot be realized, whereas +with N = 4 it can. +As another example, consider the logical XOR-gate +wXOR(x1, x2) = x1 ⊕ x2 which may be needed as a prim- +itive for a check function like Eq. (8). We can ask for +a complex CXOR that realizes the target Hilbert space +HXOR = H(LXOR) given by the language LXOR ≡ L[wXOR] = +{000, 011, 101, 110} that is generated by this Boolean gate. +The ground state manifold of such a complex must be +spanned by four states, +H0[CXOR] = span +� +|000, a(000)⟩ , |011, a(011)⟩ , +|101, a(101)⟩ , |110, a(110)⟩ +� +(13) +where the configurations of potential ancillas are deter- +mined by the configurations of the three ports. We will +introduce a specific realization CXOR in Section VII; it re- +quires N = 7 atoms of which four are ancillas, and we +show that this is indeed the smallest complex that can +realize the language of a XOR-gate. +Since LXOR = L[wXOR] is generated from a Boolean gate, +we refer to complexes that realize a language of this form +as gates, too. Furthermore, we denote the atoms that map +to the input bits of the gate as input ports, and the atom +that corresponds to the output bit as the output port. We +also extend this nomenclature to Boolean functions w on +more than two inputs. Let us stress that these terms are +only inspired from the usual role played by such functions +as parts of Boolean circuits. In the present context, there +is no time evolution or dynamics involved (there is no +information “flowing into” the input ports, although it +might be sometimes helpful to use this picture). +The construction of an L-complex for a given language +L with word length n can be split into two steps: First, +one has to find a structure C on at least n atoms with an +|L|-fold degenerate ground state manifold. Then, one has +to identify a labeling ℓ of n atoms such that their states +in the ground state manifold map one-to-one to words +in L. The structure C together with the labeling then +yields an L-complex. Note that the same structure can be +interpreted as different complexes for different languages +by choosing different label functions. Furthermore, not +every structure with |L|-fold degenerate ground state +manifold allows for a valid labeling that realizes L. Hence +the construction is a quite non-trivial task in general. +This makes a reductionist approach seem most promising, +where one starts with a finite set of small “primitive” +complexes and constructs larger complexes by “gluing” +them together. +B. +Amalgamation +The process of combining two complexes by joining +(some of) their ports is referred to amalgamation. To +define the process formally, we first need a new concept +to combine two languages. + +7 +Consider two uniform languages L1 and L2 of words +of length n1 and n2, respectively. Let γ ⊆ {(p1, p2) | pi ∈ +{1, . . . , ni}} be a set of disjoint [51] pairs of letter positions +and set γi := {pi | p ∈ γ}. For a word x ∈ Li, let xγi +denote the word with all letters at positions in γi deleted. +Then, the γ-intersection of L1 and L2 is defined as +L1 +γ +∩L2 := +� +x yγ2 | x ∈ L1, y ∈ L2, ∀(a,b)∈γ xa = yb +� +which is a language of words of length n1 + n2 − |γ|. +L1 +γ +∩L2 is the set of concatenations of words from L1 and +L2 where the letters at the positions indicated by pairs +in γ coincide, and where the second copy of these letters +has been deleted. Analogously, we define the reduced +γ-intersection as +L1 +γ +∩ L2 := +� +xγ1 yγ2 | x ∈ L1, y ∈ L2, ∀(a,b)∈γ xa = yb +� +, +only that now both copies of identified letters are deleted; +hence this is a language of words with length n1+n2−2|γ|. +As an example, consider again the XOR-language +LXOR = {000, 011, 101, 110} and the CPY-language LCPY = +{000, 111}. We would like to copy the output of the XOR- +gate. To do this, we intersect the output bit (letter 3) of +the XOR-language with one of the bits (say letter 1) of the +CPY-language: γ = {(3, 1)}. The γ-intersection is the new +language +LXOR +γ +∩LCPY = {00000, 01111, 10111, 11000} +(14) +with words of length 3 + 3 − 1 = 5. The underscores indi- +cate the letters that derive from words of both languages. +If one drops these letters as well (by using the reduced +γ-intersection), the language describes a XOR-gate with +fan-out of two: +LXOR +γ +∩ LCPY = {0000, 0111, 1011, 1100} . +(15) +The above definitions on the level of languages are +useful because they are paralleled by a combination of +complexes called amalgamation: Consider two complexes +CL1 and CL2 that realize the languages L1 and L2 with +N1 and N2 atoms, respectively. Fix a set of pairs of ports +γ such that L′ = L1 +γ +∩L2 ̸= ∅, and then combine the two +complexes by identifying the atoms in γ: +CL′ = CL1 +γ +⊗ CL2 := += +. +(16) +The new complex CL′ has N1 + N2 − |γ| atoms. For this +construction, we assume that the ports that belong to +pairs in γ are located on the boundary of their complex +(we will show in Section VI why this is possible). The +Hamiltonian of the new complex is +H[CL′] = (H[CL1] + H[CL2] + δH) /γ +(17) +where the formal quotient •/γ indicates that pairs of +atoms in γ are identified; δH denotes additional interac- +tions between the two subcomplexes CLi that vanish in +the PXP model (in the vdW model they are finite but +strongly suppressed due to the quick decay of UvdW). +In a nutshell: H[CL′] is the sum of the Hamiltonians +of the original two complexes were the detunings of the +ports that are identified by γ add up. For example, let +n(1) and n(2) describe ports of CL1 and CL2, respectively, +and let γ identify these two ports. Then H[CL1] contains +a term −∆(1)n(1) and H[CL2] contains a term −∆(2)n(2). +The Hamiltonian (17) of the amalgamation contains the +term (−∆(1)n(1) − ∆(2)n(2))/γ = −(∆(1) + ∆(2))n′ where +n′ = n(1)/γ = n(2)/γ describes the atom that corresponds +to the identification of the two ports. +With δH = 0, it is straightforward to verify that the +amalgamation CL′ realizes the language L′ = L1 +γ +∩L2. +This is so because the ground state energy of H[CL′] is +lower-bounded by the sum of the ground state energies +of the summands H[CLi]; but this lower bound is realized +by configurations in L′ ̸= ∅. +The ports identified by +γ can be interpreted as ancillas of the new complex if +|L1 +γ +∩L2| = |L1 +γ +∩ L2|, i.e., if the states of these atoms +provide redundant information about the ground state +manifold; in this case, one would define L′ = L1 +γ +∩ L2 +instead. +An important special case of the above construction +is the amalgamation of gates where the input ports of +one gate are identified with the output ports of others. +For example, let w(x1, x2) and w′(x′ +1, x′ +2) be two Boolean +gates that are concatenated into the circuit on three inputs +˜w(x′ +1, x1, x2) := w′(x′ +1, w(x1, x2)). It is easy to see that +L[ ˜w] = L[w] +γ +∩ L[w′] with γ = {(3, 2)} where 3 labels the +third letter of words in L[w], which encodes the output +y = w(x1, x2), and 2 labels the second letter of words in +L[w′], which encodes the input x′ +2. Note that for Boolean +circuits without redundancies it is always |L[w] +γ +∩L[w′]| = +|L[w] +γ +∩ L[w′]| because all words are identified by the input +bits. This example demonstrates that the amalgamation +of gates is a crucial ingredient for the decomposition of +complex Boolean circuits into a small set of simple gates. +VI. +FUNCTIONAL COMPLETENESS +We have now all concepts and tools in place to formulate +the main result of this paper: +Theorem 1 (Functional completeness). For every tes- +sellated target Hilbert space HT = HL[fT] on some lattice +L that is generated by a check function fT, there exists a +structure CT in the PXP model such that +HT +loc +≃ H0[CT] , +(18) +with finite gap ∆E > 0 and perfect degeneracy δE = 0. + +8 +Figure 4. Decomposition of Boolean functions. (a) Any Boolean function fT can be represented by a graph GfT (a “Boolean +circuit”) with dedicated input vertices (blue squares), one output vertex (red square), and trivalent vertices (circles) of two +types (b): NOR-gates with two incoming and one outgoing edge (orange circles) and CPY-vertices with one incoming and two +outgoing edges (black circles); the edges themselves can be interpreted as trivial single-bit gates, here referred to as LNK-gates +(black edges). If the inputs (A,B) and outputs (Q,R) of all three primitives are assigned Boolean values that satisfy the truth +tables in (b), the value at the output vertex is y = fT(x1, . . . ) by construction. (c) The embedding Γ(GfT) (“drawing”) of the +abstract graph GfT in the plane R2 typically involves crossings (whenever GfT is non-planar); furthermore, input and output +vertices may lie in the interior of the graph. Since a crossing of wires can be implemented with the available vertices (d), the +graph can always be enhanced such that it becomes planar and input/output vertices lie on the perimeter of the embedding. (e) +Locally, the embedding Γ(GfT) decomposes into three primitives, namely the structures referred to as NOR, CPY, and LNK that +are functionally defined by the truth tables in (b) and geometrically by the sketches in (e). +In Eq. (18), +loc +≃ denotes an isomorphism of Hilbert +spaces like Eq. (11) that, in addition, preservers the lo- +cality structure: it maps local unitaries on HT to local +unitaries on H0[CT] and vice versa. +Here the locality +structure of H0[CT] is induced by the locality structure +of H which reflects the physical realization of the system. +The locality structure of HT = HL[fT] derives from the +lattice L and the bit-projector us that was used to define +the tessellated language LL[fT]; it is therefore part of +the defining properties of the Hilbert space HT. This +local isomorphism will be explicit for the examples in +Section IX. +Proof. The proof of Theorem 1 is constructive in principle +and best split into several steps: Steps 1 to 4 deal with the +construction of a Rydberg complex CfT=1 that implements +the constraint of the check function on a single site of +the lattice. In the final Step 5, the structure CT is then +constructed as the amalgamation of copies of CfT=1 on +the full lattice. +Step 1: Decomposition of fT. +The first goal is to con- +vert the check function fT : Fg +2 → F2 on g binary inputs +into a finite set of Boolean gates as “building blocks.” +There are many universal gate sets to choose from [52] +but the one that is most natural to the Rydberg platform +is the singleton {NOR} that contains only the NOR-gate [53] +A ↓ B := A ∨ B . +(19) +The idea behind this choice is simple: placing three atoms +A, C, B in a row such that the pairs (A, C) and (C, B) are +in blockade but the pair (A, B) is not naturally gives rise +to a constraint akin to C = A ↓ B (we discuss the details +below). The functional completeness of {NOR} allows us +to write +fT(x1, . . . , xg) = (. . . (xi ↓ xj) . . . (xk ↓ xl) . . . ) +(20) +where the expression on the right can be any (recursive) +combination of expressions built from the input variables +paired by NOR-gates. On an abstract level, this is a neat +result; however, in reality one has to be more careful +because variables can be used multiple times at different +locations in the NOR-expansion of fT. +To identify the true physical building blocks needed to +cast Eq. (20) into a structure of atoms, it is advisable +to translate the NOR-expansion into a graph GfT that +represents the underlying Boolean circuit and uses the +inputs xi only once at dedicated “input vertices” and +outputs the result fT(x1, . . . , xg) at a dedicated “output +vertex” (Fig. 4a). Otherwise, GfT is a trivalent graph with +two types of vertices, corresponding to CPY-operations +that copy a bit and NOR-gates that combine two bits +according to Eq. (19). If we assign arrows to the edges +to highlight the information flow, the two vertices are +distinguished by the number of in- and outgoing edges +(CPY: 1 in and 2 out, NOR: 2 in and 1 out). Furthermore, +we can interpret the edges themselves as trivial single-bit +gates (“LNK-gates”). If we assign Boolean values to the +inputs and outputs of these three primitives according to +the truth tables in Fig. 4b, the value of the output vertex +is given by y = fT(x1, . . . , xg). Without loss of generality, +we consider only circuits without redundancy, i.e., for +a given input {x1, . . . , xg} the state of the inputs and + +9 +outputs of all its primitives is uniquely determined. This +implies that there are exactly 2g such assignments that +are parametrized by the g inputs {x1, . . . , xg} (this can +be seen as a boundary condition; in a dynamical circuit, +one would call it an initial condition). +Step 2: Embedding of GfT. +The graph GfT represents +the Boolean circuit of fT on an abstract level (only the +connectivity of GfT is relevant). Our final goal is to trans- +late this graph into a functionally equivalent structure of +atoms in the plane. Thus we have to find an embedding +Γ(GfT) of GfT in R2; this embedding should be planar, +i.e., without crossing edges to avoid unwanted interac- +tions. Here we skip a formal definition of Γ(GfT) and +appeal to the intuition of the reader: Γ(GfT) describes a +drawing of GfT in the plane without crossing edges and +with well-separated vertices (Fig. 4c). Of course not every +graph GfT is planar, i.e., can be drawn without crossing +edges in the plane. However, it has been shown long +ago that every Boolean circuit can be made planar by +augmenting it with “crossover sub-circuits” whenever two +lines cross [54]. This crossover can be constructed with +various gate sets, including the NOR-singleton (Fig. 4d). +The embedding of the crossover then uses only the three +available primitives in Fig. 4b so that we can, without +loss of generality, assume Γ(GfT) to be planar. +Note +that the existence of a crossover also implies that we can +assume the input and output vertices to be located on +the perimeter of the embedding (as realized in Fig. 4c). +Translated into complexes, this will prove our claim in +Section V that we can assume the ports to sit on the +perimeter of a complex. +While Γ(GfT) may look very convoluted on a larger +scale, locally it decomposes into the three simple primi- +tives depicted in Fig. 4e, namely CPY, NOR, and LNK. The +next step is then to implement these three primitives as +complexes both geometrically (i.e., following the geome- +try in Fig. 4e) and functionally (i.e., following the truth +tables in Fig. 4b). An fT-complex can then be obtained +by amalgamation of these primitives according to the +geometric blueprint provided by Γ(GfT). +Step 3a: Implementing the LNK-complex. +The LNK- +complex is the physical counterpart of the “wires” in the +drawing of the circuit Γ(GfT). Logically, it corresponds to +the trivial gate w(x) = x with language LLNK = {00, 11}. +On the level of pure Boolean logic, wires are not entities +of their own but on the physical level, sending a bit from +one location to another requires dedicated machinery. +Before we discuss its construction, it is useful to intro- +duce a more fundamental complex that can be used to +construct two of the three primitives: the NOT-gate with +defining language L¬ = {01, 10}; it realizes the single-bit +gate w(x) = x and formalizes the core concept of the +Rydberg blockade. In the PXP model, it can be realized +naturally without ancillas by the Hamiltonian +H¬ = −∆(nA + nQ) +(21) +with a complex C¬ where |rA − rQ| < rB. The subscripts +denote the labels of the ports assigned by ℓ (we reserve +A, B, . . . for input ports and Q, R, . . . for output ports). +The ground state manifold is H0[C¬] = span { |01⟩ , |10⟩ } +with degeneracy δE¬ = 0 and gap ∆E¬ = ∆ > 0. +The elementary LNK-complex that translates a bit in +space can then be constructed as the amalgamation of +two NOT-gates (Fig. 5a) with Hamiltonian +HLNK = −∆nA − 2∆˜n1 − ∆nQ , +(22) +where adjacent atoms are in blockade but next-nearest +neighbors are not. Above and in the following we label +ancillas with a tilde and assign them numerical indices. +As for the NOT-gate, it is δELNK = 0 and ∆ELNK = ∆ with +the LNK-manifold +H0[CLNK] = span { |0(1)0⟩ , |1(0)1⟩ } . +(23) +Here and in the following we mark the states of ancillas by +parentheses. Repeated amalgamation of elementary LNK- +complexes results in LNK-complexes of arbitrary length +(always composed of an odd number of atoms and with +halved detuning at the endpoints). The two states in +H0[CLNK] of such chains correspond to the two ground +states of an antiferromagnetic Ising chain. +Step 3b: Implementing the CPY-complex. +The purpose +of the CPY-complex is to copy classical bits; it is defined by +the “copy” language LCPY = {000, 111}. The CPY-complex +is necessary because expansions in universal gates can +reuse inputs multiple times. Furthermore, circuits can +be simplified dramatically if intermediate results can be +reused. In conventional drawings of Boolean circuits, the +possibility to copy bits is silently assumed whenever one +splits up wires. Again, in a physical implementation one +has to provide the means to do so. +The implementation of the CPY-complex is detailed in +Fig. 5b. +It is easy to see (Appendix A 1) that there +cannot be a CPY-complex without ancillas because the +configuration 111 excludes a Rydberg blockade between +any of the three ports (which would automatically render +them completely uncorrelated). Adding a single ancilla +does the trick because the amalgamation of three NOT- +complexes on a single atom yields the desired complex +by construction. The four atoms are described by the +Hamiltonian +HCPY = −∆(nA + nQ + nR) − 3∆ ˜n1 , +(24) +and the geometry of the complex CCPY is chosen so that +the ancilla is in blockade with the three ports, but these +are not within blockade of each other. In combination +with Eq. (24), this implements the CPY-manifold +H0[CCPY] = span { |000(1)⟩ , |111(0)⟩ } +(25) +with δECPY = 0 and ∆ECPY = ∆ > 0. +Step 3c: Implementing the NOR-complex. +The NOR- +complex is crucial as it realizes a functionally complete +two-bit gate; it is specified by the language LNOR = +{001, 010, 100, 110}. +In contrast to the LNK- and CPY- +complexes, the NOR-complex cannot be bootstrapped from +the NOT-complex but must be constructed from scratch. + +10 +Figure 5. Complete set of logic primitives. (a) The (elementary) LNK-complex CLNK can be realized by a chain of three atoms +where adjacent atoms are in blockade (black edges). The detuning of the ports ∆ (blue squares, labeled by ℓ) is half that of the +ancilla 2∆ (green circle) in the bulk. The width δE and gap ∆E are shown together with a schematic spectrum that highlights +the logical manifold H0[CLNK] and one of the states orthogonal to H0[CLNK] that define the gap. The state of ancillas is shown +in parentheses. (b) The CPY-complex CCPY can be realized with a central ancilla (red circle) that is in blockade with the three +surrounding atoms (blue squares). To make the two logical states degenerate, the ancilla has a detuning of 3∆ if the other atoms +are detuned by ∆. (c) The NOR-complex CNOR can be realized with two ancillas (blue and green circles) that form a ring-like +blockade with the three ports (blue and green squares). To make the four logical states unique and degenerate, the detunings +cannot be chosen uniformly but must break the reflection symmetry about the axis through the output port Q. +In Appendix A 2 we show that a NOR-complex cannot be +realized with less than two ancillas in the PXP model. One +implementation of a NOR-complex is detailed in Fig. 5c. +The five atoms are governed by the Hamiltonian +HNOR = −∆(nA + nQ + ˜n1) − 2∆(nB + ˜n2) +(26) +which gives rise to the NOR-manifold +H0[CNOR] = span +� |001(01)⟩ , |010(10)⟩ , +|100(01)⟩ , |110(00)⟩ +� +(27) +with δENOR = 0 and ∆ENOR = ∆; this requires that the +atoms are arranged in a ring-like blockade, as depicted in +Fig. 5c. Note that the two ancillas are only necessary to +enforce the degeneracy of the logical states 010 and 100 +with 110. All remaining constraints come for free with the +Rydberg blockade. As we will show in Section VII, the +NOR-complex in Fig. 5c is not unique. We will also see that +the only fundamental Boolean gate that can be realized +with as few as five atoms is the NOR-gate, confirming our +intuition in Step 1 that the NOR-gate is the most natural +on the Rydberg platform. +Step 4: Constructing the fT-complex. +To construct +a complex CfT that implements the check function fT +(more precisely: the language L[fT]), one combines the +three primitives above according to an embedding Γ(GfT). +Since all vertices are (at most) trivalent, it is easy to +check that an amalgamation in the PXP model is possible +without geometrical obstructions, and that this procedure +yields an fT-complex with δEfT = 0 and ∆EfT ≥ ∆ > 0. +At this point, we have a complex with g = 4K input +ports on its boundary that outputs y = fT(x1 +e1, . . . ) on a +dedicated output port (also on its boundary, but this is +not important in the following): +(28) +To enforce the constraint fT(x1 +e1, . . . ) +!= 1, we only have +to add a local detuning on the output port to lower the +energy of valid configurations and gap out invalid ones. +This boils down to a simple modification of the check +function complex, +→ +(29) +where the output port is detuned and downgraded to +an ancilla. The ground state manifold of the modified +complex CfT=1 consists of all input configurations for +which fT(x1 +e1, . . . ) = 1. +Step 5: Constructing CT. +The complex CfT=1 enforces +the local constraint of the check function on a single site +of the lattice on which the tessellated target Hilbert space +HT = HL[fT] is defined. To construct CT for the full +system, place a copy CfT=1 �→ Cs +fT=1 on every site s ∈ +V (L) of the lattice, and amalgamate adjacent complexes +at the corresponding ports (possibly using LNK-complexes +to avoid unwanted interactions): + +11 +CT := +(30) +By construction, the ground states of this complex are in +one-to-one correspondence with words x ∈ LL[fT] (using +the ports on the edges denoted by blue squares). Note +that here we show the construction for a square lattice L; +the generalization to other lattices is straightforward. +This concludes the construction of CT such that HT +loc +≃ +H0[CT] in the PXP approximation. Note that the ancillas +do not introduce additional degrees of freedom in this +subspace and local unitaries on HT map to local unitaries +on H0[CT] (the latter involve the ancillas of the CfT=1 +complexes and can therefore be very complicated—but +they remain local on H). +■ +We conclude this section with a few remarks. First, +while the proof above is constructive, one should not +expect the resulting structures to be useful in real-world +applications, except for simple special cases. In particular, +we established no claims about optimality (in any sense) +of the constructed fT-complexes; on this we focus in the +next Section VII. Second, the modification in Eq. (29) +to construct CfT=1 from CfT is often straightforward to +implement and can simplify the complex considerably: +When there are no blockades between the output port and +some of the input ports, one simply deletes the output +port along with all ancillas that are in blockade with +it. This removes all configurations of input ports from +the ground state manifold where the output was not +excited (see Appendix B). And finally, the removal of the +output port may not be necessary at all if the constraint +fT(x1 +e1, . . . ) +!= 1 can be rewritten as an equality of the +form +f1(x1 +e1, . . . , x1 +e2, . . . ) +!= f2(x1 +e3, . . . , x1 +e4, . . . ) , +(31) +with Boolean functions f1,2 that take only 2K inputs each. +Then CfT=1 = Cf1 +γ +⊗ Cf2 where the two complexes are +amalgamated at their output ports: += +(32) +An example for this construction can be found in Sec- +tion IX A. +VII. +LOGIC PRIMITIVES +A crucial step of the proof in the previous section is to +show that every Boolean function f can be realized by a +Rydberg complex Cf in the sense that the language L[f] +of its truth table can be realized as ground state manifold. +As mentioned above, the complexes that arise from the +decomposition of f into LNK-, CPY- and NOR-primitives +are typically large and convoluted. +For example, the +decomposition of a simple AND-gate (∧) into NOR-gates +reads +A ∧ B = (A ↓ A) ↓ (B ↓ B) , +(33) +which would require two CPY- and three NOR-complexes, +wired together by a bunch of LNK-complexes so that the +resulting complex requires more than 20 atoms. As this +is way too much overhead for a simple gate, the question +arises whether important primitives of Boolean logic can +be realized by complexes that are much smaller than the +ones described by the NOR-decomposition in Section VI. +The answer is positive: In the following, we discuss +provably minimal complexes for the most important gates +of Boolean logic, all of which improve significantly over +the na¨ıve NOR-decomposition. Besides the usual gates of +Boolean algebra, NOT (¬ or •), AND (∧), and OR (∨), we +search for minimal complexes that realize the following +common logic gates (given in disjunctive normal form): +NOR: +A ↓ B = A ∧ B +(34a) +NAND: +A ↑ B := A ∨ B +(34b) +XOR: +A ⊕ B := (A ∧ B) ∨ (A ∧ B) +(34c) +XNOR: +A ⊙ B := (A ∧ B) ∨ (A ∧ B) . +(34d) +Of these gates, only NOR and NAND are universal on their +own. The following identities show that some of these +gates are simply inverted versions of others (we will use +this below): +A ∧ B = A ↑ B +(35a) +A ∨ B = A ↓ B +(35b) +A ⊕ B = A ⊙ B . +(35c) +Of the gates {¬, ∨, ∧, ↑, ↓, ⊕, ⊙}, we already know min- +imal complexes for NOT (2 atoms) and NOR (5 atoms), +recall Section VI. + +12 +A +Q +R +A +Q +R +1 +A +Q +R +2 +CPY +A Q R +1 +2 +1 +1 +1 +0 +0 +0 +A +Q +R +CPY +A +Q +A +Q +1 +A +Q +2 +LNK +A Q +1 +2 +0 +0 +1 +1 +A +Q +LNK +A +Q +A +Q +1 +A +Q +2 +NOT +A Q +1 +2 +0 +1 +1 +0 +A +Q +NOT (¬) +B +Q +A +B +Q +A +1 +B +Q +A +2 +B +Q +A +3 +B +Q +A +4 +AND +A B Q +1 +2 +3 +4 +1 +1 +1 +0 +0 +0 +0 +1 +0 +1 +0 +0 +A +B +Q +AND (∧) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +OR +A B Q +1 +2 +3 +4 +0 +1 +1 +1 +0 +1 +1 +1 +1 +0 +0 +0 +A +B +Q +OR (∨) +Q +A +B +Q +A +B +1 +Q +A +B +2 +Q +A +B +3 +Q +A +B +4 +NOR +A B Q +1 +2 +3 +4 +0 +1 +0 +1 +0 +0 +1 +1 +0 +0 +0 +1 +A +B +Q +NOR (↓) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +XOR +A B Q +1 +2 +3 +4 +0 +1 +1 +1 +0 +1 +1 +1 +0 +0 +0 +0 +A +B +Q +XOR (⊕) +B +A +Q +B +A +Q +1 +B +A +Q +2 +B +A +Q +3 +B +A +Q +4 +NAND +A B Q +1 +2 +3 +4 +1 +1 +0 +0 +0 +1 +0 +1 +1 +1 +0 +1 +A +B +Q +NAND (↑) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +XNOR +A B Q +1 +2 +3 +4 +0 +1 +0 +1 +0 +0 +1 +1 +1 +0 +0 +1 +A +B +Q +XNOR (⊙) +Figure 6. Common logic primitives. Rydberg complexes for the most common primitives of Boolean circuits. All complexes +are provably minimal, see Appendix A. Note that minimal complexes are not necessarily unique; e.g. the shown NOR-gate is +an alternative to the one in Fig. 5c, both of which are minimal. For each complex we show (1) the geometry with blockade +radii (gray dashed circles), (2) the complete ground state manifold (orange: |1⟩i, black: |0⟩i), and (3) the truth table of the +ports (labeled atoms) in the ground state manifold. The rows of the truth tables correspond to the numbered ground state +configurations. Colors of ancillas and ports in the geometry encode the detuning (see key). Atoms in blockade are connected by +black solid lines. + +13 +Using Eq. (35b), we can immediately construct an +OR-complex with six atoms by amalgamation of a NOT- +complex to the output port of a NOR-complex (remember +Fig. 1). +However, it is unclear whether this complex +is minimal, i.e., cannot be realized with fewer atoms. +Therefore we systematically devised proofs that a given +truth table cannot be realized with a given number N of +atoms, starting at N = 3 for each gate, and increasing the +number incrementally until the proof fails, i.e., realizations +can no longer be excluded. These arguments are quite +technical and can be found in Appendix A. However, this +approach has two benefits: First, it provides rigorous +lower bounds on how many atoms are needed to realize +a given gate, and second, it often provides a blueprint +for the construction of a minimal complex that saturates +this bound by carefully observing why one cannot exclude +realizations with a given number of atoms. +To complement this rigorous approach, we conducted a +brute force search on a computer that exhaustively scans +for (small) complexes that realize a given truth table. In +accordance with our proofs, we found solutions with the +minimal atom number for a given truth table (in addition, +we also found non-minimal complexes). +Interestingly, +there were alternative minimal solutions that we missed +in our manual approach; so minimal complexes are not +necessarily unique. +A selection of provably minimal complexes for all im- +portant Boolean primitives is shown in Fig. 6 (for the +sake of completeness, we include the NOT-, LNK- and CPY- +complexes discussed in Section VI). There are a few com- +ments in order. First, an example of non-unique minimal +complexes is the depicted NOR-complex built from five +atoms arranged in a triangular structure (cf. the ring- +shaped structure in Fig. 5c). Second, the six-atom OR- +complex we proposed above indeed is minimal, though not +unique either. Third, the selection of minimal complexes +in Fig. 6 for {∨, ∧, ↑, ↓, ⊕, ⊙} all build around the triangle- +based core of the NOR-complex, once again emphasizing +its central role in the context of Rydberg complexes. Fi- +nally, it turns out that the relations (35) are all reflected +in the minimal complexes, e.g., the amalgamation of a +NOT-complex and a XNOR-complex yields a minimal XOR- +complex; similar constructions hold for NAND and AND as +well as NOR and OR. If we recall the relation between NOT +and the minimal LNK-complex, the general picture emerges +that inverting complexes are simpler (by one atom) than +non-inverting ones. This is understandable in so far as +inversion is the most basic operation the Rydberg block- +ade is capable of, thus leading to the simplest complexes. +This is in contrast to the notation for Boolean circuits +known from electrical engineering where inverting gates +are represented by more complicated symbols than their +non-inverting counterparts (Fig. 6). +VIII. +CROSSING +The crossing complex realizes the somewhat surprising +feature of intersecting information channels in a strictly +two-dimensional setup of strongly interacting information +carriers (recall Step 2 in Section VI). The possibility +to realize such a planar crossing in a circuit with the +three primitives LNK, CPY and NOR was crucial for the +proof of Theorem 1. Note that the existence of such a +complex followed immediately from the existence of the +three aforementioned complexes and the well-known fact +that Boolean circuits can be made planar [54]. However, +just as for the Boolean gates in Section VII, the NOR-based +implementation of the circuit crossing in Ref. [54] is of +low practical value as it requires seven NOR-gates (if we +implement NOT-gates directly, Fig. 4d); even a simpler +crossing based on only three minimal XNOR-gates requires +∼ 27 atoms, see Fig. 7a. Thus we are again tasked with +finding a minimal complex that realizes the same function. +By systematically excluding the existence of crossing +complexes for N = 4, . . . , 9 atoms, we finally find the +minimal complex CCRS depicted in Fig. 7b comprising 10 +atoms. The proof for its minimality is very technical and +more complicated than for the logic primitives because +geometric constraints must be taken into account for the +crossing [55]. The structure with two dangling ports (Q +and R) immediately suggests the inverted crossing CICRS +in Fig. 7c with eight atoms, i.e., a complex that allows +two signals to pass each other while inverting both at +the same time. The minimality of the inverted crossing +complex CICRS with eight atoms follows as a corollary +from the minimality of the non-inverted crossing CCRS +with 10 atoms as the latter can be obtained from the +former by amalgamation of two NOT-complexes (thereby +adding two atoms). In line with our comment at the +end of the previous Section VII, the inverted variant of +the crossing is smaller than its non-inverted counterpart. +We note that the inverted crossing CICRS has also been +described in Ref. [40] were it plays an important role +in mapping non-planar optimization problems to planar +Rydberg structures. +IX. +EXAMPLES: SPIN LIQUID PRIMITIVES +In this part, we focus on our motivation outlined in the +introduction, namely the implementation of tessellated +target Hilbert spaces of systems that are characterized by +local gauge constraints. We discuss two models exemplar- +ily: the surface code with abelian Z2 topological order +and the non-abelian Fibonacci model. For the surface +code, we will be able to utilize the Boolean primitives +discussed in Section VII; by contrast, for the Fibonacci +model such a reduction will not be useful. + +14 +(a) +A +B +R +Q +(b) +CRS +A +Q +R +B +B +A +Q +R +B +A +Q +R +1 +B +A +Q +R +2 +B +A +Q +R +3 +B +A +Q +R +4 +CRS +A B Q R +1 +2 +3 +4 +0 +0 +0 +0 +0 +1 +0 +1 +1 +0 +1 +0 +1 +1 +1 +1 +(c) +ICRS +A +Q +R +B +B +A +R +Q +B +A +R +Q +1 +B +A +R +Q +2 +B +A +R +Q +3 +B +A +R +Q +4 +ICRS +A B Q R +1 +2 +3 +4 +0 +0 +1 +1 +0 +1 +1 +0 +1 +0 +0 +1 +1 +1 +0 +0 +Figure 7. Crossing. (a) The crossing constructed from the Boolean circuit crossing based on XNOR-gates (see Ref. [54] and Fig. 6); +it is an amalgamation of LNK-, CPY-, and XNOR-complexes. The ground state manifold (not shown) is 4-fold degenerate and +ensures A = Q and B = R. The complex requires ∼ 27 atoms and is therefore of no practical relevance. (b) By contrast, the +minimal crossing CCRS requires only 10 atoms; it was constructed by systematically excluding functionally equivalent complexes +with fewer atoms. The shown data is explained in the caption of Fig. 6. (c) The minimal inverted crossing CICRS is smaller than +the non-inverted crossing and requires only eight atoms. To construct CCRS from CICRS, two NOT-complexes must be amalgamated +to adjacent ports. This is a recurring scheme due to the inverting nature of the Rydberg blockade. +A. +Surface code +The toric code [38] is the prime example for a spin +liquid in two dimensions with long-range entangled ground +states that do not break any symmetries but instead +feature topological order. The toric code is referred to +as surface code if realized on surfaces with boundaries +[56]; we will stick to this name in the following. The +surface code describes a gapped phase with Z2 topological +order that is described by the mechanism of string-net +condensation [39]. It allows for localized excitations that +are abelian anyons [57] which, in turn, leads to ground +state degeneracies on topologically non-trivial surfaces +(including flat surfaces with non-trivial boundaries). As +a consequence, surface codes are promising candidates +for quantum memories that encode logical qubits reliably +into delocalized degrees of freedom [58]. This makes the +implementation of systems with this kind of topological +order interesting both from an academic and an applied +perspective [36, 59, 60]. +Here we consider the surface code on a finite square +lattice with “rough” boundaries (like the gray background +lattice in Fig. 8d); “rough” boundaries are terminated by +dangling edges that attach to four-valent vertices. The +Hamiltonian +H = −JA +� +Sites s +As − JB +� +Faces p +Bp +(36) +operates on qubits that live on the edges e of the square +lattice. The operators +As = +� +e∈s +σz +e +and +Bp = +� +e∈p +σx +e +(37) +are referred to as star and plaquette operators, respec- +tively. +Here, e ∈ s denotes edges that emanate from +site s and e ∈ p denotes sites that bound face p; σα +e +are Pauli matrices for α = x, y, z acting on the qubit +on edge e. +Since [As, Bp] = 0, the Hamiltonian (36) +is frustration-free and its ground state |G⟩ is character- +ized by As |G⟩ = Bp |G⟩ = |G⟩ for all sites s and faces +p (assuming JA, JB > 0). Due to the uniform “rough” +boundaries there is no ground state degeneracy and |G⟩ +is unique. +The construction of |G⟩ is straightforward: To satisfy +the constraint As |G⟩ = |G⟩ on sites s, one can choose +the product state |0⟩ with σz +e |0⟩ = |0⟩ for all edges. This +state does not satisfy the constraint Bp |G⟩ = |G⟩ on faces, +though. To fix this, one defines the multiplicative group +B = ⟨{Bp | Faces p}⟩ generated by all plaquette operators +(note that B2 +p = 1), and constructs the superposition +|G⟩ ∝ +� +C∈B +C |0⟩ . +(38) +The state |G⟩ is invariant under any Bp by construc- +tion since B is left-invariant under any Bp by defini- +tion. Furthermore, since [As, Bp] = 0, the site-constraint +As |G⟩ = |G⟩ is still satisfied. Thus Eq. (38) describes, +up to normalization, the unique ground state of Eq. (36). +The states |C⟩ ≡ C |0⟩ have a peculiar structure: each +C can be described as a collection of closed loops on the +lattice where the σx +e of products of Bp operators act (loops +that terminate on dangling edges at the boundary are +considered closed); this loop structure is then imprinted +on |0⟩ so that |C⟩ is a product state with a loop pattern +C of flipped qubits |1⟩. The ground state Eq. (38) is +therefore given by the equal-weight superposition of all +closed loop configurations on the square lattice—which +makes it an example of a string-net condensate [39] with +a non-trivial pattern of long-range entanglement [61, 62]. + +15 +(a) +C +A +D +B +CSCU +(c) +C +A +D +B +1 +C +A +D +B +2 +C +A +D +B +3 +C +A +D +B +4 +C +A +D +B +5 +C +A +D +B +6 +C +A +D +B +7 +C +A +D +B +8 +(b) +A B C D +1 +2 +3 +4 +5 +6 +7 +8 +0 +1 +0 +1 +1 +0 +0 +1 +0 +1 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +(d) +CLoop +Figure 8. Surface code. (a) Unit cell/vertex complex CSCU for the surface code (Z2 topological order). The complex is the +amalgamation and deformation of two XNOR-complexes [see Fig. 6 and Eq. (43)] and implements the check function constraint +fLoop = 1 defined in Eq. (42). The deformations are necessary to prevent an unwanted blockade of ancillas in the amalgamation. +Black edges denote blockades between atoms, gray edges illustrate the underlying square lattice. (b,c) Truth table and ground +state manifold of the complex. The manifold contains all configurations with an even number of labeled atoms excited, thereby +realizing Gauss’s law on the site (colored edges). This provides the local isomorphism between HT = HLoop and H0[CLoop]. (d) +Periodic tessellation CLoop of the vertex complex CSCU. The copies overlap on the edges and are amalgamated at these ports +(which makes the detunings uniform in the bulk). +To prepare this state in a real system, one could try to +implement the Hamiltonian (36) and cool the system into +its ground state. This is a challenging task due to the +four-body interactions (37) which are notoriously hard +to realize. On the Rydberg platform, an alternative and +more promising approach goes as follows: In a first step, +one prepares only the subspace +HLoop := { |Ψ⟩ | ∀ Sites s : As |Ψ⟩ = |Ψ⟩ } += span { |C⟩ | C ∈ B } +(39) +as the low-energy manifold of a suitably designed struc- +ture of atoms. (HLoop is the Hilbert space of a Z2 lattice +gauge theory with charge-free background [50]. The local +constraint As |Ψ⟩ = |Ψ⟩ corresponds to the gauge sym- +metry of this theory and is known as Gauss’s law.) The +Bp-terms in Eq. (36) induce quantum fluctuations on +this subspace which give rise to the string-net condensed +ground state in Eq. (38). On the Rydberg platform, quan- +tum fluctuations can be induced perturbatively by ramping +up the Rabi frequency Ωi. Such fluctuations can give rise +to interesting quantum phases, as shown in Ref. [33] for +a different model. This motivates the construction of a +Rydberg complex CLoop with +H0[CLoop] +loc +≃ HT = HLoop = span { |C⟩ | C ∈ B } , +(40) +i.e., a Rydberg complex the degenerate ground states +of which can be locally mapped one-to-one to loop con- +figurations on the square lattice. +H0[CLoop] is then a +subspace with dimension dim H0[CLoop] ∼ 2M where M +denotes the number of unit cells of the square lattice. +Note that H0[CLoop] cannot be decomposed into factors +of local Hilbert spaces (like, e.g., the full Hilbert space +H = (C2)⊗2M can). +To this end, we assign bits x1 +e to the edges of the square +lattice L (K = 1). Our goal is to specify the tessellated +“loop language” LL[fLoop]—which contains all bit patterns +that trace out closed loop configurations on the lattice +(closed in the sense defined above)—in terms of a local +check function fLoop and a local bit-projector us on each +site s of the square lattice. +The bit-projector simply +selects the four bits on edges adjacent to s, +us +� +� +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3, x1 +e4) +(41) +and the check function reads +fLoop(x1, x2, x3, x4) = (x1 ⊙ x2) ⊙ (x3 ⊙ x4) +(42) +with the XNOR-gate ⊙ defined in Eq. (34d), that is, A⊙B = +1 iff A = B. It is easy to verify by inspection that fLoop = 1 +if and only if the number of active bits is even, thereby +enforcing Gauss’s law on every site of the lattice (because +loops cannot terminate there). + +16 +We could now construct a complex as discussed in +Section V, using the minimal XNOR-complex depicted in +Fig. 6. For this construction, we would amalgamate three +of these complexes according to Eq. (42) and detune the +final output to enforce fLoop = 1; this would require at +least 16 atoms per site. We can do much better, though, +by rewriting the constraint as an equality: +fLoop = 1 +⇔ +x1 ⊙ x2 = x3 ⊙ x4 . +(43) +Indeed, Eq. (43) evaluates to true iff x1 + x2 + x3 + x4 +is even. In general, an implementation of an equality +constraint f1 = f2 of two functions on separate inputs is +achieved by amalgamation of their complexes Cf1 and Cf2 +at their output ports, as noted at the end of Section VI. +Therefore, the vertex complex CSCU ≡ CfLoop=1 (“Surface +Code Unit cell”) that realizes the constraint Eq. (43) is +that of only two XNOR-gates amalgamated at their outputs +(Fig. 8a) which requires only 11 atoms. Surprisingly, it +turns out that this realization is also minimal, see Ap- +pendix C 1 for a proof. (Note that typically the construc- +tion of larger complexes from minimal primitives does not +yield minimal complexes.) The two XNOR-complexes that +make up the vertex complex are geometrically deformed +variants of the XNOR-complex shown in Fig. 6. This is +necessary to prevent unwanted blockades between ancillas +in the amalgamation. +In Fig. 8b we show the configurations of the four la- +beled ports (A, B, C, and D) of the complex in the 8-fold +degenerate ground state manifold. In Fig. 8c we illus- +trate the excitation patterns of these eight ground states +(atoms excited to the Rydberg state are colored orange). +Highlighting the edges of the square lattice whenever the +labeled ports associated to them are excited yields the +local mapping (40) to the loop structure of states in HLoop. +Note that the ancillas do not add additional degrees of +freedom in the ground state manifold. +For the tessellation (Fig. 8d) the vertex complex is +copied and shifted periodically along the basis vectors of +the square lattice. The labeled ports are then amalga- +mated to the corresponding ports of complexes on adjacent +sites. Quite remarkably, due to the amalgamation, the +detunings in the bulk become uniform, which makes this +tessellation interesting under the constraints of current +platforms [32, 36]. (Note that imposing periodic bound- +ary conditions on the lattice, i.e., going back to the toric +code, would render the detunings completely uniform.) +Finally, we briefly comment on the modifications of +the surface code patch in Fig. 8d that would be neces- +sary to use it as a quantum code. It is well-known [56] +that a surface code patch encodes a single logical qubit +if its four sides alternate in boundary types: top and +bottom remain “rough” but left and right are modified to +“smooth” boundaries by cutting of the dangling edges of +the square lattice. On these boundaries, the sites become +trivalent “T”-shaped with the same Gauss’s law (i.e., the +number of active edges must be even). On these sites, the +4-valent complex in Fig. 8a must be replaced by a trivalent +one. Conveniently enough, this is just the XOR-complex +in Fig. 6 as the truth table of XOR contains exactly the +four assignments of three Boolean variables such that +x1 + x2 + x3 is even. As a bonus, closing of the left and +right sides of the patch with XOR-complexes leads to com- +pletely uniform detunings along these boundaries. The +simplicity of the vertex complex on trivalent sites suggests +a definition of the surface code on the Honeycomb lattice +(which is perfectly possible [39]). However, because of the +two sites per unit cell, this does not reduce the number +of required atoms per unit cell to implement the check +function. Indeed, the realizations with minimal Rydberg +complexes on both lattices are essentially equivalent, as +can be seen in Fig. 8d by rotating the tessellation by 45°. +B. +Fibonacci model +The surface code only supports abelian anyons, which +are not sufficient for universal topological quantum com- +putation, where gates are implemented fault tolerantly by +braiding of localized excitations and measurements corre- +spond to their fusion [63–65]. The simplest anyon model +that supports universal computation by braiding is known +as Fibonacci model due to the role the Fibonacci numbers +play in the fusion rules [66–68]; it may be realized in some +fractional quantum Hall states [69, 70]. As quasiparticles, +the properties of Fibonacci anyons are a consequence of +and encoded in the entanglement pattern of the ground +state on which they live. The latter turns out to have +a representation as a string-net condensate with weights +and “string-net” patterns that differ from the surface code +[cf. Eq. (38)]. If we consider a Honeycomb lattice with +qubits on its edges, the fixed-point ground state of the +Fibonacci model has the form [39] +|G⟩ = +� +S +Φ(S) |S⟩ , +(44) +where the sum goes over all patterns (“string-nets”) S of +flipped qubits |1⟩ on the edges of the Honeycomb lattice +where no single string ends on a vertex. That is, in con- +trast to the loop patterns C of the surface code, vertices +with three fusing strings are allowed. The coefficients +Φ(S) of the superposition are non-trivial functions of the +pattern S, so that the condensate is no longer an equal- +weight superposition [39, 71, 72]. It is possible to write +down a solvable, local Hamiltonian like Eq. (36) with the +exact ground state (44) which is, however, so complicated +that it is essentially useless for implementations [39]. This +complication, together with the potential usefulness of +the model for quantum computation, motivates again the +construction of a Rydberg complex CFib that implements +the tessellated target Hilbert space +H0[CFib] +loc +≃ HT = span { |S⟩ | String-net S } +(45) +which has the dimension dim H0[CFib] ∼ (1 + ϕ2)M + +(1 + ϕ−2)M where M is the number of unit cells of the + +17 +(a) +B +A +E +D +C +CFMU +CfFib=1 +(c) +B +A +E +D +C +1 +B +A +E +D +C +2 +B +A +E +D +C +3 +B +A +E +D +C +4 +B +A +E +D +C +5 +B +A +E +D +C +6 +B +A +E +D +C +7 +B +A +E +D +C +8 +B +A +E +D +C +9 +B +A +E +D +C +10 +B +A +E +D +C +11 +B +A +E +D +C +12 +B +A +E +D +C +13 +(b) +A B C D E +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +1 +1 +1 +1 +1 +1 +1 +0 +1 +1 +1 +1 +1 +0 +1 +0 +1 +1 +1 +1 +0 +1 +0 +1 +1 +0 +1 +1 +0 +1 +1 +0 +1 +1 +1 +1 +0 +0 +1 +1 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +1 +1 +0 +0 +0 +1 +1 +1 +1 +0 +(d) +CFib +Figure 9. Fibonacci model. (a) Unit cell complex CFMU for the Fibonacci model that implements two copies of the single-site +check function constraint fFib = 1 defined in Eq. (47). The complex is the amalgamation of two equivalent 8-atom complexes +CfFib=1 on the two trivalent sites that make up the basis of the honeycomb unit cell. Black edges denote blockades between +atoms, gray edges illustrate the underlying Honeycomb lattice. (b,c) Truth table and ground state manifold of the unit cell +complex. The manifold contains all configurations with closed strings and, in addition, configurations with three strings fusing +on a site. This provides the local isomorphism between the string-net Hilbert space HT and H0[CFib]. (d) Periodic tessellation +CFib of the complex CFMU. The copies overlap on the edges and are amalgamated at the corresponding ports. +Honeycomb lattice and ϕ = (1+ +√ +5)/2 is the golden ratio +[73, 74]. As for the surface code, H0[CFib] is a Hilbert +space that cannot be decomposed into factors of local +Hilbert spaces. +Since the Honeycomb sites are trivalent, the bit- +projector takes now the form +us +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3) +(46) +and the check function that specifies the allowed string- +nets can be written in the compact form +fFib(x1, x2, x3) = (x1 ⊕ x2 ≡ x3) ∨ (x1 ∧ x2 ∧ x3) +(47) +where the first clause (x1 ⊕ x2 ≡ x3) realizes the loop +constraint (≡ denotes the logical equivalence which is +equivalent to the XNOR-gate as a connective) and the +second clause (x1 ∧ x2 ∧ x3) allows for the fusion of three +strings. Note that without the second clause we fall back +to the loop constraint of the surface code (now on the +honeycomb lattice). +Since there are five assignments with fFib = 1, this +check function cannot be realized by a single logic gate +(despite having three ports) but must be decomposed +into a circuit. Furthermore, since the amalgamation of +two logic gates always results in a complex with an even +number of ports, at least three gates would be necessary +to realize the Fibonacci constraint. This already leads +into the territory of ≳ 15 atoms which we deem too much +overhead for a single site. Therefore we follow the same +approach as for the logic primitives in Section VII: We +systematically exclude the existence of complexes CfFib=1 +for N = 3, 4, . . . , 7 atoms (Appendix C 2). The approach +fails for N = 8 and we find the minimal complex in Fig. 9a +(dashed box). The amalgamation of two of the complexes, +one mirrored horizontally, yields the complex CFMU (“Fi- +bonacci Model Unit cell”) for the two-site unit cell of +the Honeycomb lattice, which can then be tessellated as +shown in Fig. 9d. In contrast to the surface code, the +detunings are not uniform in this case. The full ground +state manifold of the unit cell is shown in Fig. 9b. The +colored edges in Fig. 9c for each ground state configura- +tion establish the local mapping in Eq. (45). Note how +all string-net configurations are allowed except for single +strings terminating at a site. +Finally, let us mention that the complex for the hexag- +onal unit cell with 15 atoms in Fig. 9a can be interpreted +as the complex on a tilted square lattice (by virtually +contracting the vertical edges of the honeycomb lattice). +This complex, however, is not minimal as we know of a 12 +atom complex that realizes the Fibonacci check function +constraint on 4-valent sites. + +18 +X. +GEOMETRIC OPTIMIZATION +So far we optimized complexes only in terms of their +size (number of atoms) for a given language. As a result, +we ended up with minimal complexes that are defined +by their blockade graph B, local detunings {∆i}, and an +assignment of ports ℓ, i.e., atoms that realize the desired +language in the ground state manifold. Remember that +in a blockade graph B = (V, E) an edge e = (i, j) ∈ E +between atoms i, j ∈ V indicates that they are in blockade, +i.e., cannot be excited simultaneously. An abstract graph +that can be realized in this way by placing atoms in the +plane which are in blockade if and only if their distance +is smaller than some blockade radius rB is called a unit +disk graph, and a geometry GC that realizes a prescribed +graph as its blockade graph is a unit disk embedding +of this graph. So far, the actual geometry GC of our +minimal complexes was only taken into account insofar +as a unit disk embedding of the required blockade graph +B must exist. (Note that there are graphs that cannot be +realized as blockade graphs of planar geometries, so that +this “geometric realizability” is a non-trivial condition; +deciding whether a given graph can be realized in this +way is unfortunately NP-hard [75].) +Whenever there exists a planar geometry GC += +(ri)i∈V ∈ R2N ≡ CN that realizes a prescribed block- +ade graph, there typically exist many such geometries: +In most cases, there is a bit of “wiggle room” around a +given geometry without changing the blockade graph. In +addition, there can be geometrically distinct realizations +of the same blockage graph that cannot be continuously +deformed into each other without violating the blockade +constraints. For example: +This can lead to disconnected regions in the configuration +space CN that realize a given blockade graph. +To optimize the geometry of a complex in CN, we have +to quantify what we mean by a “good” complex. To this +end, we define an objective function Γ : CN → R that +quantifies the quality of the complex and that we seek to +minimize. One example is +˜Γ(GC) = δE +∆E +(48) +where δE and ∆E are the width of the ground state +manifold and the gap (recall Fig. 2). The problem with +Eq. (48) is that its evaluation scales exponentially with the +number of atoms N because the computation of δE and +∆E in principle requires access to the complete spectrum +of Eq. (1) (which is in general an NP-hard problem [26]). +While this is feasible for small complexes, it becomes +quickly a bottleneck as ˜Γ must be evaluated repeatedly +when iteratively optimizing a geometry. Furthermore, in +the PXP approximation, interaction energies are either +infinite or zero so that ˜Γ vanishes whenever the blockade +constraints are satisfied. Thus we need a simpler, heuristic +quantity that can be directly computed from the geometry +of the complex. +A. +Geometric robustness +To motivate the quantity we propose as objective func- +tion below, we first have to review the role of the blockade +radius rB in the PXP model. In the limit of vanishing driv- +ing, the blockade radius rB is the distance from an atom +where the van der Waals interaction matches its detuning: +C6 rB +−6 +i +!= ∆i. As the detunings can vary from atom to +atom in a generic structure C, so does the blockade radius +rBi (this dependence is quite weak, though). However, +as outlined in Section III, we would like to work in the +approximate framework of the PXP model with a unique +blockade radius rB, because then the effects of interactions +between atoms simplify to kinematic constraints encoded +in a blockade graph. In the following, we interpret a given +blockade graph B as the encoding of the constraints we +would like to realize with a structure C of yet unknown +geometry GC. +We can now introduce two dimensionless quantities. +First, the robustness of a structure w.r.t. a given blockade +graph B = (V, E) is defined as +ξB(C) := +min +(i,j)/∈E d(ri, rj) − max +(i,j)∈E d(ri, rj) +min +(i,j)/∈E d(ri, rj) + max +(i,j)∈E d(ri, rj) , +(49) +where d(ri, rj) denotes the Euclidean distance. The ro- +bustness is a scale-invariant, finite number ξB(C) ∈ [−1, 1] +where ξB(C) > 0 indicates a valid unit disk embedding GC +that realizes the prescribed blockade graph B for block- +ade radii in some finite interval. Larger positive values of +ξB(C) indicate more robust embeddings with more “wiggle +room” around the positions without changing the block- +ade graph, or, equivalently, a wider range of blockade +radii that yield the same blockade graph. If ξB(C) < 0, +the unit disk graph induced by GC does not match the +prescribed blockade graph B. +Similarly, the spread of a structure C is defined as +s(C) := maxi rBi − mini rBi +maxi rBi + mini rBi += (maxi ∆i)1/6 − (mini ∆i)1/6 +(maxi ∆i)1/6 + (mini ∆i)1/6 . +(50) +The spread s ∈ [0, 1] quantifies the relative variations +in blockade radii of a structure (a system with uniform +detuning ∆i ≡ ∆ has vanishing spread). Just as Eq. (49) +does not depend on the length scale, Eq. (50) is inde- +pendent of the C6 coefficient, i.e., the strength of the +interaction. + +19 +(a) +ξ(Copt +NOR△) = 0.268 +ξ(CNOR△) = 0.088 +s(C(opt) +NOR△ ) = 0.058 +ξ(Copt +NOR◦) = 0.236 +ξ(CNOR◦) = 0.111 +s(C(opt) +NOR◦ ) = 0.058 +(b) +ξ(Copt +SCU ) = 0.133 +ξ(CSCU) = 0.117 +s(C(opt) +SCU +) = 0.058 +Figure 10. +Optimization (Examples). +(a) Comparison of +perturbed (black) and optimized (red) geometries for the two +minimal NOR-complexes. Maximum distance blockades are +highlighted yellow, minimum distances of unblocked atoms +are indicated by dashed blue edges. The optimal geometries +are highly symmetric and match the manually constructed +ones in Fig. 6 and Fig. 5c. The robustness for each complex is +printed below the geometries and the spread on the bottom +of each column (we omit blockade graph indices). Note that +ξ(Copt +NOR△) > ξ(Copt +NOR◦) which makes the triangular version NOR△ +potentially more robust than the ring-shaped NOR◦. For all +geometries the validity constraint s(C) < ξ(C) is satisfied. (b) +Comparison of the optimized geometry for the vertex complex +CSCU of the surface code (red) and the manually constructed +geometry (black) from Fig. 8a; the robustness increases by +∆ξ = 0.016. Due to unconstrained atoms, the optimization can +break the symmetry and produce slightly skewed geometries. +We can now take into account the variability of the +blockade radius without abandoning the PXP model as +follows. We call a structure C a valid implementation of +a blockade graph B if +s(C) < ξB(C) . +(51) +This condition ensures that the geometry GC can be scaled +such that all distances of atoms that should (not) be in +blockade according to B, are smaller (larger) than the +smallest (largest) blockade radius of the structure C. As +this condition is scale-invariant, we do not have to specify +rB in the following. Note that all structures presented in +this paper are valid in the sense of Eq. (51). +B. +Numerical optimization +These considerations suggest the robustness ξB as a +measure for the quality of geometries. We therefore set +Γ = −ξB to maximize this quantity by minimizing Γ. +The blockade graph B and the detunings {∆i} are fixed +and define the functional properties of the complex; in +particular, the spread s(C) is constant. Thus we optimize +for geometries that satisfy the validity constraint (51) +with a maximal margin between robustness and spread. +We call a complex C globally (locally) optimal if ξB(C) > +0 and its geometry is a global (local) minimum of Γ in +CN. To minimize Γ on the high-dimensional space CN, +we employ the SciPy implementation [76] of generalized +simulated annealing [77, 78] in combination with a local +optimization based on the Nelder-Mead algorithm [79, 80], +see Appendix D for details. Remember that the robustness +is a scale-invariant quantity, so that the scale of the +optimized geometry is arbitrary. For normalization, we +rescale the geometries by setting the blockade radius +rB := 1 +2 +� +max +(i,j)∈E d(ri, rj) + min +(i,j)/∈E d(ri, rj) +� +!= 1 . +(52) +First, we initialized the algorithm with the hand-crafted +geometries of all primitives in Sections VII and VIII and +the vertex complexes in Section IX to optimize their ro- +bustness (we believe the results to be globally optimal but +we did not prove this). With these initial configurations, +the optimizer already started with a valid unit disk embed- +ding of B (ξB > 0) and tried to maximize the robustness +further. The results were typically only slightly deformed +versions of the manually constructed complexes, confirm- +ing our intuition. Some of the primitives (in particular the +ring-shaped NOR-complex in Fig. 5c) were already optimal +due of their high symmetry. In Fig. 10a we demonstrate +this by comparing slightly perturbed geometries (black) +to the subsequently optimized versions (red) for both +minimal realizations of the NOR-complex. In particular, +we find +ξBNOR△(Copt +NOR△) = 0.268 > 0.236 = ξBNOR◦(Copt +NOR◦) +(53) +and conclude that the triangular version NOR△ (Fig. 6) +is potentially more robust than the ring-shaped NOR◦ +(Fig. 5c). For both, the validity constraint (51) is safely +satisfied (x ∈ {◦, △}): +s(C(opt) +NORx ) = 0.058 < ξBNORx(Copt +NORx) . +(54) +Since the robustness depends only on the maximum (min- +imum) distance of atoms that are (not) in blockade, there +can be atoms with positions that are unconstrained in +small regions of the plane. These positions can be chosen +by the optimization algorithm at will, leading to slightly +skewed geometries that break the natural symmetry of +the complex; an example is given by the optimized surface +code unit cell complex in Fig. 10b. This is an artifact of +our particular objective function that can be eliminated +by more sophisticated choices for Γ (e.g. motivated by +specific experimental requirements). All optimized com- +plexes are accessible online [81], normalized according to +Eq. (52). +In a second run, we went one step further and initial- +ized the optimization with geometries that violated the +prescribed blockade graphs (by placing the atoms ran- +domly). In this case, the algorithm started with ξB < 0 +and first had to identify valid unit disk embeddings by +stochastic jumps in the configuration space. These runs + +20 +typically rediscovered the geometries we already knew. +In some cases, alternative geometries were found (which +turned out to be local maxima of robustness, though). +We conclude that it is not only possible to optimize given +geometries but also to find them (if they exist), at least +for small complexes. +As a final remark, we stress that geometric optimization +is in general not reducible, i.e., optimizing the primitives +of a larger circuit does not necessarily optimize the whole +circuit as constraints between primitives are not taken +into account by this approach. This is particularly im- +portant for tessellated complexes of quantum phases like +the spin liquids in Section IX, where one should optimize +the complete tessellation to minimize unwanted residual +interactions that are not present in the optimization of a +single-site or unit cell complex. +XI. +OUTLOOK +We conclude with a few comments on open questions +and directions for future research. +Minimality. +To find and prove the minimality of +complexes we systematically excluded realizations with +fewer atoms. While this approach is more efficient than +a brute force search (by exploiting constraints from the +language, the detunings, and the planar geometry), it +is still far from trivial and cannot be easily automated. +It would be both interesting and useful to develop an +algorithm that, given a uniform language, constructs a +minimal graph with weighted nodes, and a labeled node +for each letter position of the language, such that each +maximum-weight independent set [82] is in one-to-one +correspondence with a word of the language. We are +neither aware of such an algorithm nor of statements on +the complexity to find minimal solutions. (Note that a +solution of this problem might not even be a unit disk +graph, i.e., realizable by the blockade graph of a planar +Rydberg complex.) +Optimization. +It is clear that our treatment of op- +timization in Section X only scratches the surface. First, +our choice of the objective function Γ is heuristic and +other functions may be more appropriate for specific ex- +perimental settings. This would change the “optimal” +geometries of complexes, of course. Second, there is a +plethora of alternative numerical algorithms available that +could be used to minimize the objective function more ef- +ficiently. In particular the existence of distinct geometries +that are separated by complexes that violate the block- +ade graph may require more sophisticated algorithms to +escape locally optimal configurations and find the global +optimum. The algorithms also should scale well with +the size of the complex because, as mentioned previously, +tessellations should be optimized as a whole to take into +account constraints between its primitives. +If we go one step further and ask for an algorithm that +constructs geometries from a given blockade graph, we +quickly enter complexity hell: Deciding whether a given +blockade graph can be realized as a unit disk graph is +known to be NP-hard [75]. Even if we are promised to +be given a unit disk graph as blockade graph, there is +no efficient algorithm that outputs the geometry of a +complex that realizes it. This is so because there are +unit disk graphs that require exponentially many bits to +specify the positions of the nodes [83]. To add insult to +injury, even finding certain approximations of unit disk +graph embeddings are known to be NP-hard [84]. None +of these statements prevent us from looking for heuristic +algorithms to solve these problems for specific cases, of +course (as we demonstrated in Section X). +Uniformity. +Most of the complexes discussed in this +paper make use of atom-specific detunings (e.g. Fig. 6 +and Fig. 9d). Only the surface code tessellation in Fig. 8d +is uniform in detunings, at least in the bulk. While it is +possible to realize atom-specific detunings [45, 46], single- +site addressability adds significant experimental overhead. +Thus it is reasonable to ask whether complexes with +non-uniform detunings can be replaced by (potentially +larger) complexes with uniform detunings (without adding +additional degrees of freedom). For instance, there is a +third minimal NOR-complex with uniform detuning ∆i ≡ +∆. However, in amalgamated circuits this uniformity is +often destroyed—on the contrary, it is the non-uniformity +of the XNOR-complex (Fig. 6) that made the bulk of the +surface code uniform (Fig. 8d). The quest for uniformity is +therefore best formulated on the level of complete circuits +or tessellations. +Beyond planarity. +We focused completely on planar +Rydberg complexes to comply with the restrictions of +current experimental platforms: For the addressability of +single atoms it is simply convenient to have a dimension +of unimpeded access. However, technologically, three- +dimensional structures of Rydberg atoms are possible and +have been experimentally demonstrated [5, 31]. Releasing +the planarity constraint drastically changes the rules for +the construction of Rydberg complexes. For instance, +ports that are located inside a 2D complex (and would +require expensive crossings to be routed to the perimeter) +can be directly accessed from the third dimension, possibly +simplifying certain functional primitives. Note, however, +that at least the logic primitives in Fig. 6 do not profit +from a third dimension. (This follows from the proofs in +Appendix A.) +Beyond the PXP approximation. +Our construc- +tion of Rydberg complexes was based on the assumption +that atoms within the blockade radius can never be simul- +taneously excited, while atoms separated by more than +the blockade radius do not interact at all; this “PXP +approximation” implements the dynamical effect of the + +21 +interactions as a kinematic constraint. In reality, how- +ever, the atoms interact via the van der Waals interaction +UvdW = C6 r−6 which contributes also beyond the block- +ade radius, can lift the degeneracy δE of the ground state +manifold, and reduce the gap ∆E that separates it from +excited states. +One therefore expects that complexes +with δE ≈ 0 in the vdW model are geometrically more +constrained than in the PXP model. This has an effect +on the geometrical optimization of complexes (see above) +and the appropriate choice of the objective function: To +take into account residual interactions properly, heuristic +functions like the robustness should be replaced by realis- +tic functions like Eq. (48), at least for small complexes +where they can be computed exactly. +We checked that the three primitives in Fig. 5 can +be realized with perfect degeneracy δE = 0 and gap +∆E > 0 in the vdW model by small adjustments of the +detunings to balance residual interactions. In principle, a +NOR-complex can even be realized with only three atoms, +arranged in a triangle with precisely defined shape. This +is possible, because the two ancillas in Fig. 5c were only +necessary to balance the energies of states with one and +two input ports active; in the vdW model, the same can +be achieved by exploiting the residual interaction between +the two input ports. Which version of the NOR-complex +is more useful for implementations is an open question. +Quantum phase diagrams. +In this paper, we only +studied the ground state manifold of the Hamiltonian +(1) without quantum fluctuations (Ωi = 0). As has been +demonstrated in Refs. [33, 34], the interplay of quantum +fluctuations (Ωi > 0) and the strong blockade interac- +tions can give rise to interesting many-body quantum +phases at zero temperature. Thus it seems natural to +explore the quantum phase diagrams of the proposed spin- +liquid tessellations in Section IX, for example numerically +using density matrix renormalization group (DMRG) tech- +niques. Analytically, one could derive the effective Hamil- +tonians on the constructed low-energy manifolds for finite +but small Rabi frequencies Ωi ≪ ∆E in perturbation +theory [85]. Note that in general one expects the relative +strengths of the effective terms to depend on the specific +complex used to implement the local constraints. This +raises the subsequent question whether these couplings +can be tuned by modifications of the used complexes. +Dynamical preparation. +In recent experiments +[36], dynamical preparation schemes have been used to +prepare long-range entangled many-body states out-of- +equilibrium [37]. The idea is to use “quasiadiabatic” pro- +tocols Ωi(t) and ∆i(t) where the detuning increases con- +tinuously to its target value while a finite Rabi frequency +ensures the coupling of different excitations patterns. This +allows for the preparation of non-trivial superpositions of +states in the low-energy subspace of the classical Hamilto- +nian (1). It would be interesting to explore the states of +the proposed tessellations that can be prepared by such +dynamical protocols numerically, and study the effects of +defects in the intended logic of the complexes due to local +excitations. Similar questions arise for the primitives in +Sections VII and VIII and circuits built from these by +amalgamation. +XII. +SUMMARY +In this paper, we developed a framework to design +planar structures of atoms which can be excited into Ryd- +berg states under the constraint of the Rydberg blockade +mechanism (“Rydberg complexes”). Our framework tar- +gets the preparation of degenerate ground state manifolds +that are characterized locally by arbitrary Boolean con- +straints. We proved that the truth table of an arbitrary +Boolean function can be realized as ground state manifold +by decomposing its circuit representation into three prim- +itives that leverage the Rydberg blockade. Motivated by +this existence claim, we then presented provably minimal +complexes that realize the most important primitives of +Boolean circuits, including a crossing complex that is +needed to embed non-planar circuits into the plane. As +an application of our framework, we constructed periodic +Rydberg complexes with degenerate ground state mani- +folds that map locally on the non-factorizable string-net +Hilbert spaces of the surface code (with abelian topolog- +ical order) and the Fibonacci model (with non-abelian +topological order). In combination with quantum fluc- +tuations, these structures may be the starting point to +prepare topologically ordered states in upcoming quantum +simulators. We concluded the paper with a discussion of +the geometric optimization of Rydberg complexes using +numerical algorithms to increase their robustness against +geometric imperfections and the effects of long-range van +der Waals interactions. +Our results highlight the versatility of planar struc- +tures of atoms that interact via the Rydberg blockade +mechanism. We provide a conceptual foundation for the +rationales of geometric programming, the encoding and +solution of problems by tailoring the geometry of atomic +systems, and synthetic quantum matter, the goal-driven +design of quantum materials on the atomic level. Due to +the noisiness of near-term experimental platforms, the lat- +ter seems particularly promising because quantum phases +come with an inherent robustness against a finite den- +sity of excitations. 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Gong, Efficiency of generalized sim- +ulated annealing, Physical Review E 62(3), 4473 (2000), +doi:10.1103/physreve.62.4473. + +25 +Appendix A: Minimality of logic primitives +Here we prove the claims in Section VI and Section VII about the minimality of the logic primitives. The proofs in +this section do not require geometric arguments (i.e. whether a given blockade graph is a unit disk graph or not). This +makes the claims independent of the embedding dimension; in particular, they remain valid for three-dimensional +complexes. +We start with a few general remarks. First, the languages we seek to implement as ground state manifolds (GSM) +are irreducible in the sense that they cannot be written as a product of two smaller languages. [The product of two +formal languages is simply the set of all words from the first concatenated with all words from the second.] This is +easy to check for all Boolean gates by inspecting their truth tables. The crucial point is that irreducible languages can +only be implemented by complexes with connected blockade graphs. +Second, because we are only interested in GSM of PXP models, all detunings can be assumed to be strictly positive, +∆i > 0. Indeed, atoms with negative detuning cannot be excited in the GSM so that they can be deleted from the +complex without changing the GSM (and without closing the gap). The argument against atoms with vanishing +detuning is more subtle. If such an atom is not excited in any of the GSM states, it can be deleted without changing +the GSM. If it is excited in some of the GSM states, there is always an otherwise identical state in the GSM where it +is not excited. Such an atom therefore must be a port because as an ancilla it would add internal degrees of freedom +that are not accessible via the ports (this follows from our definition of a complex). The language that corresponds to +a complex with a zero-detuning port therefore has the property that for every word with a “1” at the corresponding +position, there must be an otherwise identical word with a “0”. (This does not imply that the language is reducible; +for example, L = {111, 011, 000} has this property for the first letter but is irreducible.) While such languages do +exist, they cannot be truth tables of Boolean functions because such a port cannot be used as an input or an output +(assuming we forbid “dummy” inputs that have no effect on the output). All languages discussed and implemented in +this paper (also the ones for the vertex complexes of spin liquids) do not have this property, hence we can assume +non-vanishing detunings. +Because of the positivity of all detunings, ground states are always given by maximal independent sets (MIS*) +of the blockade graph. [A maximal independent set is a subset of vertices such that (1) no two vertices of the set +are connected by an edge of the graph and (2) no vertex can be added to the set without violating (1). Maximum +independent sets (MIS) are the largest maximal independent sets.] The inverse is not necessarily true: Depending on +the detunings, not every MIS* describes a ground state configuration (an example is the ring-like NOR-complex). +1. +CPY-complex +Lemma 1. A CPY-complex cannot be realized with less than 4 atoms (1 ancilla). +Proof. Assume there is a complex without ancillas described by +H = −∆1n1 − ∆2n2 − ∆3n3 =: En1n2n3 . +(A1) +Since (n1n2n3) = (111) must be a ground state of the complex, none of the pairs of the atoms can be in blockade so +that there is no kinematic constraint on the configurations (n1n2n3). To be a CPY-complex, it must be +−(∆1 + ∆2 + ∆3) = E111 +!= E000 = 0 +and +En1n2n3 > 0 +for all +(n1n2n3) ̸= (000), (111) . +(A2) +The finite-gap condition requires in particular ∆i > 0 for all i = 1, 2, 3 which leads to −(∆1 + ∆2 + ∆3) < 0 and +thereby contradicts the degeneracy condition. +Alternative argument: The copy language LCPY = {000, 111} is irreducible. Since (111) must be in the GSM, the +only admissible blockade graph is the trivial graph on three vertices without edges: B = (V = {1, 2, 3}, E = ∅). But a +disconnected blockade graph cannot implement an irreducible language. +■ +2. +NOR-complex +Lemma 2. A NOR-complex cannot be realized with less than 5 atoms (2 ancillas). +Proof. We show that a NOR-complex cannot be realized with one ancilla or less. First, assume there is no ancilla so +that the Hamiltonian is again +H = −∆1n1 − ∆2n2 − ∆3n3 =: En1n2n3 , +(A3) + +26 +now with potential kinematic constraints due to the Rydberg blockade. The conditions for a NOR-complex demand the +equality of the following energies: +E001 = −∆3 +(A4a) +E010 = −∆2 +(A4b) +E100 = −∆1 +(A4c) +E110 = −∆1 − ∆2 . +(A4d) +It follows immediately ∆1 = ∆2 = ∆3 and ∆1 = 0 so that all detunings must vanish. But then (n1n2n3) = (000) +is—independent of the configuration and its implied kinematic constraints—degenerate with the four states that belong +to the NOR-manifold (which it must not be). +Alternative argument: The NOR-language LNOR = {001, 010, 100, 110} is irreducible and forbids a blockade between +the two input ports [because of (110)]. The only consistent blockade graph B is therefore the line graph of three +vertices. But this graph has only two maximal independent sets, whereas we need at least four to realize LNOR. +So let us assume a system with one additional ancilla, +H = −∆1n1 − ∆2n2 − ∆3n3 − ∆4˜n4 , +(A5) +and an arbitrary geometry that may lead to kinematic constraints on the allowed configurations. Let now ε(n1n2n3) +denote the minimal energy of the system without the contribution from the ports under the “boundary condition” that +these are in the state (n1n2n3) and under the kinematic constraints imposed by the Rydberg blockade; furthermore, set +En1n2n3 := −∆1n1−∆2n2−∆3n3+ε(n1n2n3). In the current situation with only one ancilla, it is either ε(n1n2n3) = 0 +if the minimum is obtained by ˜n4 = 0, or ε(n1n2n3) = −∆4 if ˜n4 = 1 minimizes the energy (and this is consistent with +the configuration (n1n2n3)). With this notation, the conditions to be a NOR-complex take the following form. First, +the degeneracy of the NOR-manifold demands the equivalence of the following expressions: +E001 = −∆3 + ε(001) +(A6a) +E010 = −∆2 + ε(010) +(A6b) +E100 = −∆1 + ε(100) +(A6c) +E110 = −∆1 − ∆2 + ε(110) , +(A6d) +which immediately implies +∆1 = ε(110) − ε(010) +(A7a) +∆2 = ε(110) − ε(100) +(A7b) +∆3 = ε(110) + ε(001) − ε(100) − ε(010) . +(A7c) +Second, the gap condition requires (among other conditions) +ε(000) = E000 +!> E100 = −∆1 + ε(100) = ε(010) − ε(110) + ε(100) +(A8a) +⇔ +ε(000) + ε(110) > ε(010) + ε(100) +(A8b) +because a state with (n1n2n3) = (000) is not allowed in the NOR-manifold. Note that the only kinematic constraints +on the ancilla in Eq. (A8b) can come from the two input vertices since n3 = 0 for all four terms. We show now that +Eq. (A8b) cannot be satisfied with a single ancilla. +Consider first the case where ∆4 ≤ 0. Then the minimal energy under any condition (n1n2n3) is reached by switching +the ancilla off, ˜n4 = 0 (this is possible for all kinematic constraints), so that 0 + 0 > 0 + 0 leads to a contradiction. +Thus we have to assume ∆4 > 0 (this we could have anticipated from the arguments above). Now the energy can be +lowered by switching the ancilla on, but this might be forbidden by the kinematic constraints for certain boundary +conditions (n1n2n3). We consider three cases: +(i) No blockade between the two inputs and the ancilla. In this case, the ancilla will be switched on in all four terms +of Eq. (A8b) so that −∆4 − ∆4 > −∆4 − ∆4 violates the gap condition. +(ii) The ancilla is in blockade with one of the inputs. W.l.o.g. let n1 be in blockade with ˜n4. Then Eq. (A8b) reads +−∆4 + 0 > −∆4 + 0 which again violates the gap condition. +(iii) The ancilla is in blockade with both inputs. Now Eq. (A8b) reads −∆4 + 0 > 0 + 0 which is in contradiction with +the assumption ∆4 > 0. + +27 +n1 +˜n2 +n3 +˜n4 +n5 +(n1n3n5) = (111) +(n1n3n5) = (001) +(n1n3n5) = (100) +(n1n3n5) = (000) +Figure 11. The line graph is the only connected blockade graph on five vertices with (at least) four maximal independent sets +(orange vertices), (at least) one of which has (at least) three vertices. To realize the state (111), the vertices {1, 3, 5} must be +chosen as ports, with 3 as output; the four maximal independent sets then realize the truth table of AND (these four states +cannot be made degenerate while maintaining a gap, see text). +In conclusion, we showed that it is impossible to satisfy the gap condition with a single ancilla. +Alternative argument: Of the six connected graphs on four vertices, only the “tetrahedron graph” has four maximal +independent sets (the others have at most three), which is necessary to realize the four words in LNOR. But none of +these four maximal independent sets contain more than one vertex [which would be necessary for (110)]. +■ +3. +AND-complex, OR-complex and XNOR-complex +Lemma 3. AND-, OR- and XNOR-complexes cannot be realized with less then 6 atoms (3 ancillas). +Proof. All these complexes contain the state (111) such that no two ports can be in blockade with each other. This +implies that no realization of these gates is possible with four or less atoms as the only connected blockade graph +which fulfills this constraint is the star graph of the CPY-complex (which has only two MIS*). +The number of vertices is still small enough to systematically screen the 21 connected graphs on five vertices and +select the 11 relevant ones with at least four maximal independent sets. One can check that only the chain graph has +a MIS* with (at least) three vertices, which is needed to realize the port configuration (111) (Fig. 11). This MIS* +contains the vertices {1, 3, 5} of the chain, which we therefore must choose as ports: (n1n3n5) = (111). With these +ports, the set of four MIS* then realizes the language L = {111, 100, 001, 000} which we identify as the truth table of +the AND-gate if we choose the port on the central atom 3 as output. This proves that the OR- and XNOR-complex cannot +be realized with five atoms (even if another port is declared as output). +So far the arguments were purely kinematic insofar as only the blockade constraints and the knowledge that the +GSM is generate by maximal independent sets were used. To exclude the AND-gate, this is not enough, and we have to +use energetic arguments by studying possible choices for detunings. The degeneracy of the GSM requires the following +four expressions to be equal: +E111 = −∆1 − ∆3 − ∆5 +(A9a) +E100 = −∆1 − ∆4 +(A9b) +E001 = −∆2 − ∆5 +(A9c) +E000 = −∆2 − ∆4 , +(A9d) +which immediately implies ∆4 = ∆5 and therefore ∆3 = 0, which is not allowed (remember that vanishing detunings +are forbidden). This proves that also the AND-complex cannot be realized with five atoms. +■ +4. +NAND-complex and XOR-complex +Lemma 4. NAND- and XOR-complexes cannot be realized with less than 7 atoms (4 ancillas). +Proof. The truth tables of both NAND and XOR contain the states (110), (101) and (011) so that no two ports can be in +blockade with each other. This excludes a realization with less than four atoms (see Appendix A 3). If two ancillas are +available, we can switch one of the input ports on; this switches (at least) one ancilla off. The remaining two ports and +(at most) one ancilla then must realize the NOT-language L¬ = {01, 10}. This is impossible since the two ports cannot +be directly connected and the only blockade graph with a single ancilla realizes the LNK-language LLNK = {00, 11}. So +let us assume that the complexes can be realized with three ports and three ancillas. For the following arguments, +only the edges between ports and ancillas are of importance; potential blockades between ancillas can be ignored. We +consider three cases: +(i) There is at least one port that connects to all three ancillas. If this port is on, all ancillas are off, hence the two +remaining ports must be on as well; but then at least two of the three states (110), (101) and (011) cannot be +realized in the GSM. + +28 +∆1 +˜∆1 +∆2 +˜∆3 +∆3 +˜∆2 +(a) +(b) +(c) +Figure 12. The three bipartite graphs between three ports (red) and three ancillas (blue) where all ports have degree two. Note +that these do not represent complete blockade graphs as we omit blockades between ancillas. The detunings in (c) are used in +Appendix A 5. +(ii) There is at least one port that connects to a single ancilla. This edge can be interpreted as an amalgamated +NOT-complex. If we delete the port, subtract its detuning from the connected ancilla, and declare the latter as a +new port, the new complex of five atoms realizes the truth table of the original complex with one column inverted +(w.l.o.g. the first one). For both gates, this new manifold contains the states (010), (001) and (111) (plus another +one that depends on the gate). The only blockade graph on five vertices with at least four MIS*, one of which +contains at least three vertices [needed for (111)], has been identified in Appendix A 3 as the line graph. There it +has also been shown that there is no assignment of detunings that realizes a four-fold degenerate GSM. +(iii) All inputs are connected with exactly two ancillas. There are three possibilities to connect three ports with +two ancillas each (Fig. 12). By inspection one shows that in all three cases there is a pair of ports that, when +activated, forces all ancillas connected to the third port to be off; as this forces the third port to be on, at least +one of the states (110), (101) and (011) cannot be realized in the GSM. +This proves that the NAND- and XOR-complex cannot be realized with six atoms. +■ +Note: Removing a NOT-complex by deleting the port, subtracting its detuning from its ancilla, and declaring +the ancilla as new port, is the inverse of amalgamation; let us call it amputation. One has to make sure that the +subtraction of the detuning of the port ∆p from the detuning of its adjacent ancilla ∆a does not lead to negative (or +vanishing) detunings on the ancilla (= new port). Indeed, if ∆p > ∆a, the port would be always on in all ground state +configurations; this makes the port superfluous and the language of the GSM reducible. If ∆p = ∆a, the language of +the original complex would have the property that for every word with a “0” at the corresponding position, there is a +otherwise identical word with a “1”. This is the dual property of the one discussed at the beginning of Appendix A +and no language discussed in this paper has this property. +5. +Uniqueness of the blockade graph of the minimal XNOR-complex +In contrast to the minimal NOR-complexes (for which there are different blockade graph realizations), there is only +one realization of the minimal XNOR-complex. This will be useful in Appendix C 2 to prove the minimality of the vertex +complex of the Fibonacci model. +Lemma 5. The blockade graph of the minimal XNOR-complex with 6 atoms (Fig. 6) is unique. +Proof. We showed in Appendix A 3 that a XNOR-complex needs at least six atoms; so let us assume we have six atoms +at our disposal. We now try to contrive a complex that realizes the language L⊙ = {001, 010, 100, 111} systematically: +(i) Assume there exists such a complex with at least one port that connect to only one ancilla. If this port is +amputated, the remaining 5 atoms realize the XOR-language L⊙ = {101, 110, 000, 011}, which is impossible as +shown in Appendix A 4. +(ii) Assume at least one port connects to all three ancillas. If this port is switched on, all ancillas are switched off and +therefore the other two ports must be active. This is inconsistent with one of the states (001), (010) and (100). +(iii) Because of (i) and (ii), only the case where all ports connect to two ancillas remains. There are three classes of +blockade graphs that satisfy this, Fig. 12. The first two graphs in Fig. 12 can be immediately excluded as they +are inconsistent with the states (001), (010) and (100) (= only one port activated). Only the “hexagon graph” in + +29 +Fig. 12 remains as a possible blockade structure between ports and ancillas. Without additional blockades between +the ancillas, the maximal independent sets of this graph allow for the states {000, 001, 010, 100, 111} ⊃ L⊙. +Let ∆1,2,3 denote the detunings of the three ports and ˜∆1,2,3 the detunings of the three ancillas (where ˜∆i +describes the ancilla opposite of port i, Fig. 12). In the state (100), only the first port is excited. So the opposite +ancilla must be excited as well to block the two other ports (if this ancilla were off, one could lower the energy by +switching the other two ports on). To balance this state energetically with the state (111), the detuning of the +ancilla must equal the sum of the detunings of its two adjacent ports. Due to the permutation symmetry of L⊙ +and the rotation symmetry of the “hexagon graph”, this argument is valid for all three ancillas: +˜∆1 = ∆2 + ∆3 , +˜∆2 = ∆1 + ∆3 , +and +˜∆3 = ∆1 + ∆2 . +(A10) +Because all detunings must be positive, this implies for any pair of ancillas +− ˜∆i − ˜∆j < −∆1 − ∆2 − ∆3 = E111 . +(A11) +Since (111) must be in the GSM (i.e., E111 must be the lowest allowed energy), there must be an additional +blockade between all pairs of ancillas to prevent them from being excited simultaneously. This yields the blockade +graph of the XNOR-complex depicted in Fig. 6. It has only four maximal independent sets that realize the language +L⊙ = {001, 010, 100, 111}. The choices of the port detunings ∆i > 0 are arbitrary; the ancilla detunings are then +given by Eq. (A10). +We conclude that the blockade graph of the minimal realization of a XNOR-complex with six atoms is unique (there is +only freedom in choosing the detunings). In addition, we proved that no strict superset of L⊙ can be realized by a +complex with six atoms or less (this is used in Appendix C 2). +■ +Appendix B: Constructing subcomplexes +Here we discuss a method to construct subcomplexes by fixing a port in the active state and deleting its adjacent +ancillas in the blockade graph. This method is used in the proofs of Appendix C and the final remark of Section VI. +Consider a complex C that realizes a language L with ports that are not in blockade with each other. We can select one +of the ports p and define the sublanguage Lp ⊂ L of words x ∈ L with xp = 1. Our goal is to construct a L′ +p-complex +C′ +p where L′ +p is obtained from Lp by deleting the constant letter at position p that corresponds to the fixed port. The +simplest solution is to keep the geometry of the complex C and increase the detuning of the fixed port ∆p, thereby +creating a gap between states of the original GSM where the port is on and states where it is off; the port can then be +downgraded to an ancilla. In all states of the new GSM this ancilla is active, while its adjacent ancillas are inactive. +This suggests that one can delete these atoms to obtain a smaller complex C′ +p that realizes the same language L′ +p: +Lemma 6. Let the finite complex C realize the irreducible language L with ports that are not in blockade with each +other (with δE = 0 and ∆E > 0). Consider one of the ports p with detuning ∆p > 0 and let the languages Lp and L′ +p +be defined as above. Then the structure C′ +p obtained from C by deleting the port p and all its adjacent ancillas is a +L′ +p-complex if the ports of C′ +p are inherited from C in the natural way. +Proof. First, note that since L is irreducible, it is Lp ̸= ∅, i.e., there are configurations in the GSM of C where the port +p is active. We have to show two things: (a) the structure C′ +p together with the inherited ports is a complex (i.e., its +ground states can be labeled by the configurations of the ports), and (b) the language that describes this GSM is L′ +p. +Let the GSM of the new structure C′ +p be defined by δE = 0 (since the structure is finite, it is automatically ∆E > 0). +Every kinetically allowed (= admissible) configuration in this GSM can be extended to an admissible configuration +of C by setting the deleted ancillas to off and the port p to on. If E0(C′ +p) denotes the ground state energy of C′ +p +and E0(C) the same for C, this implies that E0(C) ≤ E0(C′ +p) − ∆p. Conversely, because Lp ̸= ∅, there are admissible +configurations in the GSM of C where the port p is on and, consequently, all adjacent ancillas are off. By truncating +the configurations of the adjacent ancillas and the port p, this yields a admissible configuration for C′ +p with energy +E0(C) + ∆p so that E0(C′ +p) ≤ E0(C) + ∆p. In combination, we have +E0(C′ +p) = E0(C) + ∆p +(B1) +for the ground state energy of the new structure C′ +p. Using this result and the mappings of extension and truncation, +we can draw two conclusions: +(1) Every configuration in the GSM of C′ +p can be extended to a configuration in the GSM of C which corresponds to a +word in Lp. We can immediately conclude two things: + +30 +(i) Since the extended configurations must be distinguishable by the ports of the complex C ignoring port p (this +port is always on for configurations in Lp), and because these ports are inherited by the structure C′ +p, we can +conclude that the configurations of the GSM of C′ +p can also be distinguished by these ports. This makes C′ +p a +complex that realizes some language L?. +(ii) Every word in L? is mapped by the extension to a word in Lp which implies L? ⊆ L′ +p. +(2) Conversely, every configuration in the GSM of C which corresponds to a word in Lp can be truncated to an +admissible configuration of C′ +p with energy E0(C) + ∆p = E0(C′ +p), which implies L′ +p ⊆ L?. +In conclusion, we showed that L? = L′ +p and therefore that C′ +p is indeed a L′ +p-complex. +■ +Appendix C: Minimality of spin liquid primitives +1. +Vertex/Unit cell complex for the surface code (CSCU) +Lemma 7. The vertex complex (unit cell complex) CSCU of the surface code on the square lattice cannot be realized +with less than 11 atoms. +Proof. Here we show that the vertex complex of the surface code on the square lattice requires at least 11 atoms; to +this end, we use and expand on the tricks introduced in Appendix A 4. First, note that the GSM is symmetric under +the permutation of ports (Fig. 8b) and includes the state (1111), i.e., no two ports can be in blockade with each other. +In addition, the GSM is symmetric under the simultaneous inversion of an even number of letters in all words (= +columns): +LSCU ≡ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1111 +1100 +0011 +1001 +0110 +0101 +1010 +0000 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 4. letter +−−−−−−−−→ LSCU ≡ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1110 +1101 +0010 +1000 +0111 +0100 +1011 +0001 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 3. letter +−−−−−−−−→ LSCU = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1100 +1111 +0000 +1010 +0101 +0110 +1001 +0011 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 2. letter +−−−−−−−−→ LSCU = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1000 +1011 +0100 +1110 +0001 +0010 +1101 +0111 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +· · · (C1) +Let us now systematically exclude the existence of surface code complexes with N ≤ 10 atoms: +• N < 8: If one fixes one port of a surface code complex as active, the remaining three ports realize a XNOR-complex +with at least two atoms less than the surface code complex (because the active port deactivates at least one +ancilla permanently). Since we proved in Appendix A 3 that XNOR-complexes require at least six atoms, this +implies immediately that the surface code complex cannot be realized with N < 8 atoms. +• N = 8: If there are at least two ports that are connected to only one ancilla each, we can consider these as +amalgamated NOT-complexes and amputate two of them (see the note in Appendix A 4), thereby creating a +complex with only six atoms that realizes the same GSM due to the inversion symmetry detailed in Eq. (C1). +Since this is not possible, there can be at most one port that connects to only one ancilla. Choose one of the +other ports that connect to at least two ancillas and again fix it in the active state (here we use the permutation +symmetry of LSCU). This produces a complex with at most five atoms (the fixed port plus at least two ancillas +are removed from the surface code complex) that realizes again the XNOR-manifold, which is impossible. Hence +the surface code complex cannot be realized with eight atoms. +• N = 9: To show that the complex cannot be realized with nine atoms we consider three cases: +(i) Assume there is at least one port connected to three or more ancillas. If this port is fixed as active, it blocks +at least three ancillas. The resulting XNOR-complex on the three remaining ports has at most five atoms, +which is impossible. +(ii) Assume at least one port connects to a single ancilla. Amputating this port yields a LSCU-complex with +eight atoms. If an arbitrary port of this complex is fixed as active, the resulting complex has at most six +atoms. Inspection of the language LSCU [Eq. (C1)] shows that this complex realizes the truth table of a +XOR-gate, which, however, requires at least seven atoms (as shown in Appendix A 4). + +31 +(a) +(b) +(c) +1 +2 +3 +4 +5 +6 +7 +8 +(d) +(e) +(f) +Figure 13. The remaining six classes of blockade graphs for the surface code vertex complex on nine atoms with ports of degree +2 and without disconnected ancillas. Ports (ancillas) are colored red (blue) and connections between ancillas are omitted. The +only class that cannot be excluded kinematically is the “cross” graph (c) with atom labels {1, . . . , 8} and ports {1, 2, 3, 4}, see +text. +(iii) Because of (i) and (ii), only the case that all ports connect to exactly two ancillas remains. There are six +non-isomorphic bipartite graphs that connect sets of four (ports) and five vertices (ancillas), where all ports +have degree 2, Fig. 13. We exclude graphs with disconnected ancillas because these are typically covered by +the analogous step for N = 8. (Above we omitted this step to simplify the prove, so in principle here one +has to check the graphs with disconnected ancillas too. The result is the same, though.) By inspection, one +shows that all these graphs (except for the “cross” in Fig. 13c) allow for a pair of ports that, when activated, +block all ancillas of a third port (which then must be switched on as well). This, however, is inconsistent +with the language LSCU which includes for all triples of ports states where two are on and one is off. +The “cross” graph in Fig. 13c cannot be excluded with this type of kinematic reasoning because the set +of maximal independent sets (with the convention of ports shown in Fig. 13c) induces a superset of LSCU. +Therefore we have to use energetic arguments instead. With the atom indices shown in Fig. 13c, the gap +condition requires +E1111 = −∆1 − ∆2 − ∆3 − ∆4 +!< −∆1 − ∆2 − ∆3 − ∆8 = E1110 +⇒ +∆8 < ∆4 , +(C2a) +E1100 = −∆1 − ∆2 − ∆7 − ∆8 +!< −∆1 − ∆2 − ∆7 − ∆4 = E1101 +⇒ +∆8 > ∆4 . +(C2b) +Hence this graph cannot realize the LSCU-manifold. +• N = 10: To show that the surface code complex cannot be realized with N = 10 atoms, one follows the same +procedure as detailed above for the case of N = 9 atoms (here we only briefly summarize the necessary steps): +First, one excludes the case with ports that connect to a single ancilla (where one has to use that a N = 9 +realization of a LSCU-complex can have only ports that connect to at least two ancillas). Then, one excludes the +existence of ports that connect to at least four ancillas by using that XNOR-complexes cannot be realized with +five atoms or less. Finally, one must exclude blockade graphs with ports of degree three or two by the same +procedure as in Step (iii) above. In this case, there are 20 graph classes to cover of which 15 can be kinematically +excluded and 5 can be energetically ruled out. These arguments show that the surface code vertex complex +cannot be realized with 10 atoms. +■ +2. +Vertex complex for the Fibonacci model (CfFib) +Lemma 8. The vertex complex CfFib of the Fibonacci model on the Honeycomb lattice cannot be realized with less +than 8 atoms. + +32 +(a) +(b) +(c) +Figure 14. The remaining three classes of blockade graphs for the Fibonacci vertex complex on seven atoms with ports of +degree 2. Ports (ancillas) are colored red (blue) and connections between ancillas are omitted. Only the graph in (b) must be +energetically excluded. +Proof. The Fibonacci language LFib = {000, 011, 110, 101, 111} contains the three states (011), (110) and (101) which +we used in Appendix A 4 to show (with purely kinematic arguments) that XOR- and NAND-complexes cannot be realized +with less than seven atoms. As the argument only relied on these three states (and the existence of at least one other +state), it extends to the Fibonacci complex, which therefore also requires at least seven atoms. Furthermore, the three +states forbid blockades between any two ports. So assume a realization with seven atoms exists. We distinguish three +cases: +(i) At least one port connects to a single ancilla. Amputation (see the note in Appendix A 4) of this port yields a +complex with six atoms that realizes the manifold LFib = {100, 111, 010, 001, 011} obtained from LFib by inverting +the first letter (note that LFib is symmetric under permutations of ports). This language contains all XNOR states: +L⊙ ⊂ LFib. In Appendix A 5 we showed that there is only one blockade graph on 6 vertices that can realize these +states; this graph has only four maximal independent sets and therefore cannot realize the additional state (011) +in LFib. +(ii) At least one port connects to at least three ancillas. First, a port that connects to all four ancillas is inconsistent +with two of the three states (011), (110) and (101). So assume there is a port that connects to three of the four +ancillas. If this port is activated, it deactivates three ancillas and the remaining two ports (together with one +ancilla) realize the irreducible language L = {01, 10, 11} (the blockade graph of these three atoms must therefore +be connected). But there is no graph on three vertices with (at least) three maximal independent sets of which +(at least) one has (at least) two vertices. +(iii) Because of (i) and (ii), only the case where all ports connect to two ancillas remains. The possible classes of +bipartite graphs are shown in Fig. 14. With a similar line of arguments as used for the surface code [Case (iii) for +N = 9 in Appendix C 1], one can exclude two of the three graphs (a and c) with kinematic arguments [using the +states (011), (110) and (101)]. The set of maximal independent sets for the graph in Fig. 14b includes LFib as a +subset and can again be excluded by energetic arguments. +■ +Appendix D: Numerical approach for geometric optimization +To minimize the objective function Γ on the high-dimensional configuration space CN, we used the SciPy method +scipy.optimize.dual annealing [76, 88, 89] that implements generalized simulated annealing [77, 78] in combination +with a local optimization based on the Nelder-Mead algorithm [79, 80]. The stochastic algorithm starts from an initial +geometry (which can be chosen randomly), followed by iterations of jumps in CN with probabilities that depend on +the distance of the jump and the variation of the objective function Γ; following each random jump, the Nelder-Mead +algorithm optimizes the new configuration locally. After ≲ 2000 iterations we stop the algorithm and compute the +robustness of the final geometry. More technical details and all obtained optimal complexes can be found in Ref. [55]; +the data of the optimized complexes can also be accessed online [81]. + diff --git a/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0e690e0d53203c6cd24676ce11c16d1d9ca8fd4 --- /dev/null +++ b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt @@ -0,0 +1,2612 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf,len=2611 +page_content='Functional completeness of planar Rydberg structures Simon Stastny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Hans Peter B¨uchler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' and Nicolai Lang∗ Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 70550 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Germany (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 2023) The construction of Hilbert spaces that are characterized by local constraints as the low-energy sectors of microscopic models is an important step towards the realization of a wide range of quantum phases with long-range entanglement and emergent gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Here we show that planar structures of trapped atoms in the Rydberg blockade regime are functionally complete: Their ground state manifold can realize any Hilbert space that can be characterized by local constraints in the product basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We introduce a versatile framework, together with a set of provably minimal logic primitives as building blocks, to implement these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' As examples, we present lattice realizations of the string-net Hilbert spaces that underlie the surface code and the Fibonacci anyon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We discuss possible optimizations of planar Rydberg structures to increase their geometrical robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' INTRODUCTION Recent advances in the control of single atoms and their coherent manipulation [1–5] are the technological founda- tion for applications such as quantum simulation [6–9], high-precision metrology [10, 11] and, hopefully, future quantum computers [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For any of these applica- tions, suitable platforms must offer a fine-grained control over of their degrees of freedom, dynamically tunable interactions, and the possibility to decouple the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Promising in this regard are arrays of individually trapped, neutral atoms that can be manipulated by opti- cal tweezers [1, 3] and excited into Rydberg states [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' These exhibit strong interactions which lead to the Ry- dberg blockade mechanism where excited atoms prevent their neighbors within a tunable radius from being excited [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In this paper, we study on very general grounds the theoretical capabilities of the Rydberg platform in the blockade regime and demonstrate its versatility by constructing the gauge-invariant Hilbert spaces of two models with abelian and non-abelian topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Encouraged by the fast development of the Rydberg platform, there has been increased interest in identifying promising near-term applications for the NISQ era [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Among the many applications of two-dimensional arrays of Rydberg atoms, the field of geometric programming and the design of synthetic quantum matter have been identi- fied as promising candidates to leverage the capabilities of available and upcoming NISQ platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The rationale of geometric programming is the solu- tion of algorithmic problems by encoding them into the geometry of the atomic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This direction of research is founded on the insight that due to the Rydberg block- ade, the ground states of these systems naturally map to maximum independent sets (MIS) on so called unit disk graphs [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' finding MIS is a long-known optimization problem in graph theory that has been shown to be NP- hard [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This makes the computation of ground state ∗ nicolai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='lang@itp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='de energies of Rydberg arrangements NP-hard as well [26], but also opens the possibility to tackle a variety of other hard optimization problems [27, 28] by polynomial-time reductions to the MIS problem [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' First solutions of MIS instances on various graphs in two and three dimensions have been demonstrated in experiments recently [30–32], and a quantitative comparison of experimental solutions with classical algorithms suggest a superlinear quantum speedup for some classes of graphs [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A very different application of the Rydberg blockade mechanism is the engineering of synthetic quantum matter on the single-atom level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The potential of this approach has been demonstrated recently by Verresen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [33] (related results were reported by Samajdar et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [34]), who proposed the realization of topological spin liquids on delicately designed lattice structures of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In this sce- nario, the Rydberg blockade enforces a dimer constraint (the local gauge constraint of an odd Z2 lattice gauge theory [35]) which, in combination with quantum fluctua- tions, can give rise to long-range entangled many-body states with abelian topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' First experimental results were reported shortly after [36], accompanied by a theoretical study of the used quasiadiabatic preparation scheme [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This paper is written from and motivated by the syn- thetic quantum matter perspective, but its results apply to geometric programming as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our starting point is the question whether other local constraints (besides the dimer constraint) can be realized on the Rydberg plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' To find an answer, we first formalize the problem and then use this formulation to derive our main result, namely that every local constraint that can be encoded by a Boolean function can be implemented in the ground state manifold of a planar arrangement of atoms in the blockade regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Crucial for this result is the existence of a structure that implements the truth table of a NOR-gate (“Not OR”) in its ground state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' While our proof is constructive, it does typically not yield optimal (= small) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We therefore expand on our main result and compile a comprehensive list of provably minimal structures that realize all important primitives of Boolean logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Together with a structure that facilitates the cross- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='01508v1 [quant-ph] 4 Jan 2023 2 ing of two “wires” within the plane, these primitives provide a toolbox to build structures that satisfy more complicated constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' As an example, we construct a system with a ground state manifold that is locally iso- morphic to the gauge-invariant Hilbert space of an even Z2 lattice gauge theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=', the charge-free sector of the toric code [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' With a similar construction, we tailor a pattern of atoms with a ground state manifold isomorphic to the string-net Hilbert space of the “golden string-net model [39]”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' a system that, with added quantum fluc- tuations, could support non-abelian Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Having constructed all these structures, we briefly discuss possibilities to numerically optimize their geometries to make them more robust against geometric imperfections and the effects of long-range van der Waals interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' When finalizing this manuscript we be- came aware of related results reported by Nguyen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' al in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The authors focus on optimization problems on non-planar graphs and find the same structures for some of the primitives discussed in this paper (especially the implementation of the ring-shaped NOR-gate and the crossing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The authors follow the rationale of geomet- ric programming, so that their motivation, approach and framework differ from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' RATIONALE AND OUTLINE Here we illustrate the rationale of the paper and pro- vide a brief outline of its main results without technical overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Readers interested in the details can skip for- ward to Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Readers only interested in specific applications can read this section first and then skip to Section VII or Section IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In this paper, we consider two-dimensional arrange- ments of trapped atoms that can either be in their elec- tronic ground state or excited into a Rydberg state (Ryd- berg structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We focus on systems without quantum fluctuations, where the ground states are determined by local detunings and Rydberg blockade interactions (Sec- tion III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The detunings lower the energy for atoms in the Rydberg state by an atom-specific amount, and the Rydberg blockade interaction forbids atoms closer than a specific distance to be excited simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The inter- play of these two contributions singles out ground states that are characterized by excitations patterns where no additional atom can be excited without violating the Ryd- berg blockade, and where the sum of the detunings of the excited atoms is maximal (so called maximum-weight inde- pendent sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' There can be different configurations that minimize the energy, hence the ground state manifold is typically degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In this paper, we ask which ground state manifolds such structures can realize and, conversely, how to tailor structures that realize a prescribed ground state manifold (Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A simple example is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a where the po- sition of the atoms is shown in (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' the two atoms are constrained by the Rydberg blockade (gray circles) and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (a) Structure of two atoms (i) with local detunings ∆ (blue vertices) that are in Rydberg blockade (gray circles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' the blockade is indicated by a black edge connecting the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The ground state manifold (ii) is given by patterns of excited atoms (orange) that minimize the energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' here it is two-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The two ground state configurations realize the truth table (iii) of a NOT-gate Q = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (b) Structure of five atoms (i) with local detunings ∆ (blue) and 2∆ (green) in a ring-like Rydberg blockade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The ground state manifold (ii) is four-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' If one selects the three labeled atoms and identifies them with the columns of the table in (iii), the four ground state configurations realize the truth table of a NOR-gate Q = A ↓ B = A ∨ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (c) Joining the output atom of the NOR-gate with the input atom of the NOT-gate (and adding their detunings) yields a new structure that realizes the truth table of an OR-gate: Q = A ↓ B = A ∨ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This construction is called amalgamation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' cannot be excited simultaneously (indicated by the black edge connecting them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The color of the atoms encodes their detuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' here both atoms lower the energy of the system by ∆ when excited into the Rydberg state (blue nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In (ii) we show the two excitation patterns that minimize the energy (orange nodes denote excited atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note that the atoms cannot be excited simultaneously due to the Rydberg blockade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' If one lists the ground state configurations in a table, where each column corresponds to an atom and each row to a ground state configuration, we find the “truth table” of a Boolean NOT-gate Q = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Here we interpret one of the atoms as “input” (A) and the other as “output” (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This concept generalizes to more complicated Boolean gates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1b): Consider the five atoms in a ring-like blockade (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Three of the atoms (blue) lower the energy by ∆, two (green) by 2∆ when excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' By inspection one finds the four degenerate ground state configurations in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This is promising as truth tables of Boolean gates that operate on two bits have four rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, they only have three columns (two for the inputs of the gate and one for its output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We therefore select three of the five atoms by assigning labels to them: A and B play the 3 role of the inputs and Q is the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We call atomic structures with designated input/output atoms Rydberg complexes (Section V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' If we list the four ground state configurations of these three atoms, we find the truth table of a NOR-gate Q = A ↓ B = A ∨ B in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note that the remaining two atoms (we call them ancillas)— while not contributing independent degrees of freedom— are still necessary to realize this specific ground state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' At this point things get interesting because it is a well-known fact of Boolean algebra that the NOR-gate is functionally complete (just like the NAND-gate): Every Boolean function can be decomposed into a circuit build from NOR-gates only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' To leverage this decomposition, we need a method to combine “gate complexes” to form larger “circuit complexes”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' we call this procedure amalgamation (Sec- tion V B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A simple example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1c where we attach the NOT-gate from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a to the output of the NOR- gate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1b (note that the detunings of the atoms that are joined add up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Using the detunings and blockades in (i) yields the four degenerate ground state configurations in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' When we label the inputs of the NOR-gate again by A and B, and now focus on the output Q of the attached NOT-gate, we find indeed the truth table of an OR-gate Q = A ↓ B = A ∨ B in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Thus we can parallel the log- ical composition of gates by a geometrical combination of atomic structures such that the relation between ground state configurations and truth tables remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In combination with the insight that every Boolean circuit can be drawn in the plane without crossing lines (after suitable augmentations), this allows us to show that the truth table of any Boolean function can be realized as the ground state manifold of a suitably designed atomic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This functional completeness is our first main result and motivates the title of the paper (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For instance, the existence of a structure that realizes the truth table of an OR-gate is a corollary of functional completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, the specific construction as the combination of a NOR-gate and a NOT-gate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1c raises the questions whether this particular realization with six atoms is unique and whether it is minimal (in the sense that the same truth table could not be realized with fewer atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The answer to the first question is negative: There are geometrically different structures that realize the same truth table in their ground state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The answer to the second question is positive, though: We show that it is impossible to implement this truth table with less than six atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note that the func- tional completeness implies the existences of structures for all common gates of Boolean logic (such as AND, XOR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We take this as motivation to construct provably minimal structures for all these primitives (Sections VII and VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Together with the procedure of amalgama- tion, these equip our versatile toolbox to engineer more complicated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our second important contribution is an application of the functional completeness as a tool to engineer synthetic quantum matter (Section IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Many interesting quantum phases in two dimension are characterized by hidden pat- terns of long-range entanglement, known as topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' These patterns can give rise to anyonic excitations which make such systems potential substrates for quantum memories and even quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A large class of entanglement patterns can be understood as condensates of extended objects (like strings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A crucial first step for the realization of these phases is therefore the prepara- tion of Hilbert spaces spanned by states of such extended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, in experiments, we typically start from Hilbert spaces with a local tensor product structure (for example, an array of two-level atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our only hope is to make the extended objects emerge due to interactions in the low-energy sector of a suitably designed physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This often boils down to enforce local gauge sym- metries which single out states that can be interpreted in terms of extended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Such local constraints can be reformulated as Boolean functions that must be satisfied by the states of the local degrees of freedom of the underly- ing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For any constraint of this form, our functional completeness result ensures the existence of a structure of atoms, interacting via the Rydberg blockade mechanism, that realizes this constraint in its ground state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' It is then just a matter of copying and joining these structures in a translational invariant way to tessellate the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The ground state manifolds of such tessellations can there- fore implement a large class of non-trivial Hilbert spaces on which condensation (driven by quantum fluctuations) might lead to topologically ordered many-body quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Using our toolbox developed in the first part of the paper, we demonstrate this construction explicitly for the abelian toric code phase (Section IX A) and the non-abelian, computationally universal Fibonacci anyon model (Section IX B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The truth tables realized by the ground states of all atomic structures presented in this paper depend on the positions of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (Because these positions define which pairs are in blockade and which atoms can be ex- cited simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=') However, the exact placement is often ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For example, consider the structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a (i) which realizes the NOT-gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' It is clear that the blockade constraint (black edge) does not change if the atoms are slightly shifted, as long as the blockade radii (gray circles) encompass both atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We refer to the set of atom positions as the geometry of a structure and argue that “robust” geometries should avoid distances between atoms that are close to the critical blockade distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For the complexes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1, this translates into the geomet- ric objective to maximize the distances between nodes and gray circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We formalize this notion by assigning a number to geometries that quantifies their “robustness” (Section X A) and numerically construct optimized geome- tries that maximize this number (Section X B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We conclude the paper with an outline of open ques- tions, directions for further research (Section XI), and a brief summary (Section XII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' PHYSICAL SETTING We consider planar arrangements of trapped atoms with repulsive van der Waals interactions when excited into the Rydberg state [2, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Every atom is assigned an index i ∈ V = {1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' N}, placed at position ri ∈ R2, and described by a two-level system |n⟩i where n = 0 corresponds to the electronic ground state and n = 1 the excited Rydberg state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The quantum dynamics of such systems is achieved by coupling the electronic ground state to the Rydberg state by external laser fields with Rabi frequency Ωi and detuning ∆i for each atom [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Here we are mainly interested in the regime Ωi → 0 where the Hamiltonian reduces to H[C] = − � i ∆ini + � i ρc there is no physical evo- +lution since H2 < 0. One then finds that the effect of holonomies leads to a non-singular +evolution where the classical Big Bang singularity is replaced by a non-singular quantum +bounce where ρ = ρc and H = 0. This bouncing point constitutes a transitioning point +between a contracting (H < 0) and an expanding phase (H > 0). +3 +The threshold of primordial black hole formation in loop quantum +gravity +Having introduced before the fundamentals of LQG, we estimate in this section the PBH +formation threshold δc accounting for effects of loop quantum gravity at the level of the +background cosmic evolution. +To do so, we assume that the collapsing overdensity region is described by a homo- +geneous core (closed Universe) described by the following fiducial metric: +ds2 = −dt2 + a2(t) +� +dχ2 + sin2 χdΩ2� +, +(3.1) +– 5 – + +where dΩ2 is the line element of a unit two-sphere and a(t) is the scale factor of the +perturbed overdensity region. +For this type of close homogeneous and isotropic spacetime foliations one can show +that following the procedure as described in Sec. 2 the modified Friedmann equation in +k = 1 LQC accounting only for the holonomy corrections will read as [79, 80]: +H2 = +� ˙a +a +�2 += +1 +3M2 +Pl +(ρ − ρ∗) +� +1 − ρ − ρ∗ +ρc +� +, +(3.2) +where ρ is the energy density of the overdense region and ρ∗ = ρc +� +(1 + γ2)D2 + sin2 D +� +with D ≡ λ(2π2)1/3/v1/3, λ2 = 4 +√ +3 πγℓ2 +Pl [81], ℓPl being the Planck length and v = +2π2a3 being the physical volume of the unit sphere spatial manifold [81]. Since v increases +with time one can expand Eq. (3.2) in the limit u ≫ 1 [79]. At the end, keeping terms +up to O(1/v2/3) one can show that Eq. (3.2) takes the following form: +H2 = +� ˙a +a +�2 += +1 +3M2 +Pl +� +ρ +� +1 − ρ +ρc +� +− 3M2 +Pl +a2 +� +1 − 2 ρ +ρc +�� += F(ρ) +3M2 +Pl +− G(ρ) +a2 , +(3.3) +where F(ρ) = ρ(1 − ρ/ρc) and G(ρ) = 1 − 2ρ/ρc. In the limit where γ = 0, ρc → ∞ and +one recovers the standard GR k = 1 Friedmann equation H2 = +ρ +3M2 +Pl − 1 +a2 . +As regards now the background, the latter it will behave as the standard homoge- +neous and isotropic FLRW background whose fiducial metric reads as +ds2 = −dt2 + a2 +b(t) +� +dr2 + r2dΩ2� +(3.4) +and whose modified Friedmann equation within LQC will read as +H2 +b = +� ˙ab +ab +�2 += +ρb +3M2 +Pl +� +1 − ρ +ρc +� +. +(3.5) +In this setup, the collapsing overdense region corresponds to the region where 0 ≤ +χ ≤ χa and the areal radius at the edge of the ovedensity will read as +Ra = a sin χa. +(3.6) +At this point, we need to stress that the characteristic size of the overdensity is +initially super-horizon and will reenter the cosmological horizon when the areal radius +of the overdensity becomes equal to the cosmological horizon H−1, i.e. +1 +Hhc += ahc sin χa, +(3.7) +where the index “hc” denotes quantities at the horizon crossing time. Writing now the +energy density of the overdensity as ρ = ρb(1 + δ), where δ ≡ δρ +ρb , one can plug ρ into +Eq. (3.3) and working within the uniform Hubble gauge where H = Hb they can recast +Eq. (3.7) as +sin2 χa = +1 +Gδ +hc +�F δ +hc +Fhc +− 1 +� +, +(3.8) +– 6 – + +where F δ +hc = F [ρb,hc(1 + δ)] and Gδ +hc = G [ρb,hc(1 + δ)]. +Once then the overdensity region crosses the cosmological horizon will initially +follow the cosmic expansion and at some point it will detach from it starting to collapse +to form a black hole horizon. This basically happens at the time of maximum expansion +of the overdensity, when the Hubble parameter in Eq. (3.5) becomes zero, i.e. Hm = 0, +or equivalently when +am = 3M2 +PlGm +Fm +, +(3.9) +with the subscript “m” denoting quantities at the maximum expansion time. +Having derived above the horizon crossing time and the time at maximum expan- +sion we establish below a criterion for PBH formation by investigating the necessary +conditions for the triggering of the gravitational collapse process. Doing so, we confront +the gravitational force which pushes matter inwards and enhances in this way the black +hole gravitational collapse with the sound wave pressure force which pushes matter out- +wards, thus disfavoring the collapse of the overdensity. In particular, the criterion which +we adopt is the requirement that the time at which the pressure sound wave crosses +the radius of the overdensity region should be larger than the time at the maximum +expansion, which is actually the time of the onset of the gravitational collapse. Thus, +the sound pressure force will not have time to disperse the collapsing fluid matter to +the background medium and prevent in this way the collapsing process. Equivalently, +we require that the proper size of the overdensity χa is larger than the sound crossing +distance by the time of maximum expansion χs, i.e. +χa > χs. +(3.10) +To compute now the sound crossing distance by the time of maximum expansion we +assume matter in terms of a perfect fluid characterized by a constant equation-of-state +(EoS) parameter w, defined as the ratio between the pressure p and the energy density +ρ of the fluid, w ≡ p/ρ. Within this framework, the sound wave propagation equation +reads as +adχ +dt = √w , +(3.11) +where we used the fact that for a perfect fluid with a constant EoS parameter the square +sound wave c2 +s is equal to w, i.e. c2 +s = w. At the end, χs can be recast in following form: +χs = √w +� tm +tini +dt +a = +√w +3 +� ρini +ρb,m +dρb +(1 + w)ρb +�� ρb,m +ρb +� +2 +3(1+w) GmF(ρb) +Fm +− G(ρb) +, +(3.12) +where we have assumed that for a perfect fluid ρb ∝ a−3(1+w) and used Eq. (3.9) to +express am in terms of Fm and Gm. +At the end, using Eq. (3.8) the criterion for PBH formation reads as +1 +Gδ +hc +�F δ +hc +Fhc +− 1 +� +> sin2 χs. +(3.13) +– 7 – + +To determine therefore the PBH formation threshold, one can follow the following +procedure: From Eq. (3.8), one should firstly determine the the ratio ρhc/ρm for a given +value of χa and then solve numerically the inequality Eq. (3.13) in order to extract the +value of the critical energy density contrast δc required for the overdensity region to +collapse and form a PBH. +Practically, one should compute χs from Eq. (3.12) for a given value of ρm and +equate χs with χa. Then, solving Eq. (3.8) they will extract the ratio ρhc/ρm and then +plugging it into Eq. (3.13) they can extract numerically δc. At the end, given the fact +that ρb,hc < ρc, i.e. PBHs form after the quantum bounce, and that δ < 1 since we want +to be within the perturbative regime, one can show from Eq. (3.8) that +δc ≃ sin2 χs, +(3.14) +with χs given by Eq. (3.12). +At this point, we should stress that the above expression for the value of δc is a +lower bound estimate of its true value since it assumes the homogeneity of the collapsing +overdensity region which in general is not the case when one is met with strong pressure +gradients. Thus, it is strictly valid for regimes where w ≪ 1. However, PBH formation +was never studied before in a rigorous way within the context of LQG through numerical +simulations. To that end, Eq. (3.14) provides a reliable estimate for the value of δc. +4 +Results +Following the procedure described above, we calculate here the PBH formation threshold +δc within the framework of LQG and we compare it with its value in GR. In particular, +in the left panel of Fig. 1 we show the PBH formation threshold as a function of the +energy density at the time of maximum expansion ρm by fixing the EoS parameter +w = 1/3 since we study PBH formation during the RD era and the value of the Barbero- +Immirzi parameter γ = 0.2375 obtained from the computation of the entropy of black +holes [82]. Interestingly, we see a deviation from GR for high energies at the time of +maximum expansion which correspond to very small mass PBHs forming close to the +quantum bounce. This behavior is somehow expected since in this high energy regime, +one expects to see a quantifiable effect of the quantum nature of gravity. In particular, +we observe a drastic reduction of the value of δc in this region of high values of ρm up +to 50% compared the GR case. +This reduction of δc should be related with a smaller cosmological/sound horizon +in LQC compared to GR as it can be speculated from Eq. (3.14). To see this, let us find +the necessary conditions to get a cosmological horizon in LQC smaller than that in GR. +Doing so, one should require that +H2 +LQC > H2 +GR ⇔ +ρ +3M2 +Pl +� +ρ − ρ +ρc +� +− 1 +a2 +� +1 − 2 ρ +ρc +� +> +ρ +3M2 +Pl +− 1 +a2 ⇔ ρ < 6M2 +Pl +a2 . +(4.1) +For ρ = ρm and a = am as given by Eq. (3.9) one can verify that the inequality 4.1 is +identically satisfied. Thus, indeed the cosmological horizon in LQC is smaller than in +that GR leading to a reduction of δc compared to its GR value. +– 8 – + +10−13 +10−11 +10−9 +10−7 +10−5 +10−3 +10−1 +101 +ρm/M 4 +Pl +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +δc +w = 1/3 +LQG +GR +10−5 +10−3 +10−1 +101 +103 +105 +107 +MPBH in grams +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +δc +w = 1/3 +γ=10−1 +γ=1 +γ=101 +γ=102 +γ=103 +Figure 1: Left Panel:The PBH formation threshold in the uniform Hubble gauge in the +radiation-dominated era (w = 1/3) as a function of the energy density at the onset of the +PBH gravitational collapse in LQG (green curve) and in GR (Eq. (1.1)) (black dashed +curve). +Right Panel: The PBH formation threshold in the uniform Hubble gauge in +the radiation-dominated era (w = 1/3) as a function of the primordial black hole mass +MPBH in LQG for different values of the Barbero-Immirzi parameter γ. The vertical +black dashed line corresponds to MPBH = MPl. +Then, we consider the Barbero-Immirzi parameter γ as a free parameter of the +underlying quantum theory in the context of LQG. In particular, despite the fact that +the Bekenstein-Hawking entropy has been standardly used as a way to fix the value of γ, +the dependence of the entropy calculation on γ is controversial, and the value γ ≃ 0.2375, +calculated using thermodynamical arguments, is not broadly accepted [83–85]. In fact, +the choice to vary this parameter is motivated by the fact that γ is actually a coupling +constant with a topological term in the gravitational action, with no consequence at the +level of the classical equations of motion [86–91]. Thus, we vary the Barbero-Immirzi +parameter within the range of 0.1 < γ < 1000 accounting for observational constraints +for the duration of inflation after a quantum bounce [92, 93]. At the end, we plot in +the right panel of Fig. 1 the PBH formation threshold δc as a function of the PBH +mass for different values of the parameter γ within the observationally allowed range +γ ∈ [0.1, 1000]. We set the lower bound on the PBH mass equal to the Planck mass as +predicted within the quantum gravity approach [94] (See vertical black dashed line in +the right panel of Fig. 1). +In order to get the PBH mass, we account for the fact that the PBH mass is of +the order of the cosmological horizon mass at horizon crossing time. Solving at the end +numerically Eq. (3.8) we found that ρhc/ρm ∼ 10. This corresponds to +N = ln +� am +ahc +� += 1 +4 ln +�ρhc +ρm +� +∼ 0.6 e − folds +(4.2) +passing from horizon crossing time up to the onset of the gravitational collapse process at +– 9 – + +ρ = ρm, thus being in agreement with the results from PBH numerical simulations [95]. +As expected, when we increase the value of γ the overall mass range moves to higher +masses given the fact that higher values of γ are equivalent with lower values of ρc, thus +the quantum bounce happens at later times. Consequently, PBHs if formed will form at +later times, thus will acquire larger masses. +Interestingly, independently on the value of the Barbero-Immirzi variable δc is re- +duced on the low mass region, which for γ < 1000 corresponds to masses MPBH < 103g. +In particular, this reduction in δc in this very small PBH mass range will entail an +enhancement in their abundances with tremendous consequences on the associated to +them phenomenology. Indicatively, we mention here that these ultra-light PBHs can +trigger early PBH-matter dominated eras [20, 21, 96] before BBN and reheat the Uni- +verse through their evaporation [97] while at the same time they can account for the +Hubble tension through the injection to the primordial plasma of light dark radiation +degrees of freedom [98, 99] while at the same time they can produce naturally the baryon +assymetry through CP violating out-of-equilibrium decays of their Hawking evaporation +products [100, 101]. +Consequently, one can constrain the above mentioned observa- +tional/phenomenological signatures by studying PBH formation within the context of +LQG while vice-versa given the above mentioned phenomenology one can constrain the +Barbero-Immirzi parameter γ which is the fundamental parameter within LQG. In this +way, PBHs are promoted as a novel probe to constrain the potential quantum nature of +gravity. +5 +Conclusions +PBHs firstly introduced in ’70s are of great significance, since they can naturally account +for a part or all of the dark matter sector, while at the same time they might seed the +formation of large-scale structures through Poisson fluctuations. Moreover, they can also +offer the seeds for the progenitors of the black-hole merging events recently detected by +LIGO/VIRGO as well as for the supermassive black holes present in the galactic centers. +Their formation was mainly studied within the context of general relativity using both +analytic and numerical techniques. +In this work, we studied PBH formation within the context of LQG by investigating +the impact of the potential quantum character of spacetime on the critical PBH forma- +tion threshold δc, whose value can crucially affect the abundance of PBHs, a quantity +which is constrained by numerous observational probes. In particular, by comparing +the gravitational force with the sound wave pressure force during the process of the +gravitational collapse we obtained a reliable estimate on the value of δc. +Interestingly, we found that for low mass PBHs formed close to the quantum +bounce, the value of δc is drastically reduced up to 50% compared to the general rel- +ativistic regime with tremendous consequences for the observational/phenomenological +footprints of such small PBH masses. In this way, we quantified for the first time to the +best of our knowledge how quantum effects can influence PBH formation in the early +Universe within a quantum gravity framework. +– 10 – + +Finally, by treating the Barbero-Immirzi parameter γ as the free parameter of LQG +we varied its value by studying its effect on the value of the PBH formation threshold. +As expected, we found an overall shift of the PBH masses affected by the choice of γ. +Very interestingly, we showed as well that using the observational and phenomenological +signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, one +can constrain the quantum parameter γ. 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Lewicki, Doubly peaked induced stochastic gravitational +wave background: testing baryogenesis from primordial black holes, JHEP 07 (2022) 130, +[2205.06260]. +– 16 – + diff --git a/HNFJT4oBgHgl3EQfEyy7/content/tmp_files/load_file.txt b/HNFJT4oBgHgl3EQfEyy7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..948635f68541a450a9afcab01c93dcbbe47c8773 --- /dev/null +++ b/HNFJT4oBgHgl3EQfEyy7/content/tmp_files/load_file.txt @@ -0,0 +1,931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf,len=930 +page_content='Primordial black holes in loop quantum gravity: The effect on the threshold Theodoros Papanikolaoua aNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece E-mail: papaniko@noa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='gr Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Primordial black holes form in the early Universe and constitute one of the most viable candidates for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' The study of their formation process requires the determination of a critical energy density perturbation threshold δc, which in general depends on the underlying gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Up to now, the majority of analytic and numerical techniques calculate δc within the framework of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In this work, using simple physical arguments we estimate semi-analytically the PBH formation threshold within the framework of quantum gravity, working for concreteness within loop quantum gravity (LQG), which constitutes a non-perturbative and background- independent quantization of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, for low mass PBHs formed close to the quantum bounce, we find a reduction in the value of δc up to 50% compared to the general relativistic regime quantifying for the first time to the best of our knowledge how quantum effects can influence PBH formation within a quantum gravity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Finally, by varying the Barbero-Immirzi parameter γ of LQG we show its effect on the value of δc while using the observational/phenomenological signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, we propose the PBH portal as a novel probe to constrain the potential quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Keywords: primordial black holes, quantum gravity, loop quantum gravity arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='11439v1 [gr-qc] 26 Jan 2023 Contents 1 Introduction 1 2 The fundamentals of loop quantum gravity 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1 The classical dynamics 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2 The quantum dynamics 4 3 The threshold of primordial black hole formation in loop quantum gravity 5 4 Results 8 5 Conclusions 10 1 Introduction PBHs, firstly proposed in early ’70s [1–3], form in the early universe, typically during the Hot Big Bang (HBB) radiation-dominated (RD) era out of the gravitational collapse of enhanced cosmological perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' According to recent arguments, PBHs can naturally account for a part or the totality of dark matter [4, 5], seed the large-scale structures through Poisson fluctuations [6–9] as well as the primordial magnetic fields through the presence of disks around them [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the same time, they are associated with a plethora of gravitational-wave (GW) signals from black-hole merging events [12– 16] up to primordial scalar induced GWs [17–22] (for a recent review see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, through the aforementioned GW portal, PBHs can act as well as a novel probe shedding light on the underlying gravity theory [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Other hints in favor of PBHs can be found here [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In the standard PBH formation scenario, where PBHs form from the collapse of lo- cal overdensity regions, the PBH formation threshold δc depends in general on the shape of the energy density perturbation profile of the collapsing overdensity [27–30] as well as on the equation-of-state parameter at the time of PBH gravitational collapse [30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This critical threshold value is very important since it can affect significantly the abun- dance of PBHs, a quantity which is constrained by numerous observational probes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' From a historic perspective, after a first analytic calculation of δc by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Carr and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Hawking in 1975 [31, 35], δc was studied mostly through numerical hydrodynamic simulations by [36–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Within the last decade, there has been witnessed a remarkable progress regarding the determination δc both at the analytic as well as at the numerical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, at the analytic level, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='Harada, C-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Yoo & K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Kohri (HYK) in 2013 [32] refined the PBH formation threshold value obtained by Carr in 1975 by comparing the time at which the pressure sound wave crosses the overdensity collapsing – 1 – to a PBH with the onset time of the gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Their expression for δc in the uniform Hubble gauge reads as: δc = sin2 � π√w 1 + 3w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1) At this point, it is very important to stress that very recently there was exhibited a rekindled interest in the scientific community regarding the effect of non-linearities [41– 45] and non-Gaussianities [46–50] on the value of δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In addition, some first research works were also performed regarding the dependence of the PBH formation threshold on non sphericities [51, 52], on anisotropies [53], on the velocity dispersion of the collapsing matter [54] as well as within the context of modified theories of gravity [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In this work, we study semi-analytically the effect of the potential quantumness of spacetime on the determination of the PBH formation threshold by using simple physical arguments studying whether the PBH portal can act as a novel way to probe the quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' For concreteness, we work within the framework of loop quantum gravity (LQG) [56, 57] which constitutes a nonperturbative and background-independent quantization of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Very interestingly, LQG is able to solve the problem of past and future singularities [58] and provide the initial conditions for inflation, solving in this way naturally the flatness and the horizon cosmological problems [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' It can also account for the large scale structure formation [60] as well as for the currently observed cosmic acceleration [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' PBHs were studied firstly within the context of LQG in [64] where the PBH evolution was explored accounting for the effects of Hawking radiation and accretion in a LQG background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In the present work, we investigate the effect of LQG on the PBH formation process and in particular at the level of the determination of the PBH formation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' The paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 2 we revise the basics of loop quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 3 we determine semi-analytically the PBH formation threshold δc by comparing the gravity and the sound wave pressure forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Followingly, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 4 we present our results while Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 5 is devoted to conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 2 The fundamentals of loop quantum gravity Loop quantum gravity brings conceptually together the two fundamental pillars of mod- ern physics, namely General Relativity (GR) and Quantum Mechanics (QM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' It consti- tutes actually a non-perturbative and background-independent quantization of general relativity [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, it is based on a connection-dynamical formulation of GR defined on a spacetime manifold M = R × Σ, where Σ stands for the 3D spatial manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1 The classical dynamics Working within the Hamiltonian framework, the classical phase space consists of the Ashtekar-Barbero variables which are actually the two canonically conjugate variables – 2 – of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' These variables are the densitized triad Ea i and the Ashtekar connection Ai a defined as follows [65–68]: Ea i = |det(eb j)|−1ea i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1) Ai a = Γi a + γKi a, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2) where ea i is the triad field, Γi a is the spin connection, Ki a is the extrinsic curvature and γ is the so-called Barbero-Immirzi parameter which allows the quantisation procedure to be performed on a compact group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Such a setup is based on a 3+1 decomposition of the metric written in the following form: ds2 = N2dt2 − qab(dxa + Nadt)(dxb + Nbdt), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='3) where qab = ei aeib is the spatial metric, N is the lapse function and Ni is the shift vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This metric choice simplifies the quantisation process and is chosen for conve- nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' However, as said before, the LQG background equations will not depend on the choice of the spacetime metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This independence of the background on the choice of the spacetime foliation is associated to some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Firstly, one should require the diffeomorphism constraint which renders the theory independent of the choice of the spatial geometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' shift vector, and secondly the Hamiltonian constraint which ensures the theory to be invariant under the choice of temporal coordinates1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' lapse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' These two constraints conserve the general spacetime covariance of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thirdly, one imposes the Gaussian constraint which makes the theory invariant under any rotations of the triad fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the classical level, the two canonically conjugate variables Ea i and Ai a will be related with the following non-vanishing Poisson bracket: {Ai a(x), Ea i (y)} = 8πGγδb aδi jδ(3)(x − y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='4) while the dynamics of the theory will be governed by the following Hamiltonian acting on the canonical variables [69, 70]: H[N] = 1 8πG � Σ d3xN � F j ab − (1 + γ2)ϵjmnKm a Kn b � ϵjklEa kEb l √q , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='5) where F j ab is the curvature of the Ashtekar connection defined as F j ab ≡ ∂aAj b − ∂bAj a + ϵijkAj aAk b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Working now within the spatially flat Friedman-Lemaˆıtre-Robertson-Walker (FLRW) model, one introduces a fiducial cell V connected to a fiducial metric oqab and a fiducial orthonormal triad and co-triad (oea i ,o ωi a) such as oqab =o ωi a oωi b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, the reduced Ashtekar connection and densitized triad read as [67] Ai a = cV −1/3 0 oωi a, Eb i = pV −2/3 0 � det(oq) oeb i, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='6) 1Here, we conventionally denote as temporal coordinates the ones perpendicular to the 3D spatial slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This notation is convenient but does not preassume a preferred time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' As a consequence, general covariance is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' – 3 – where V0 is the fiducial volume as measured by the fiducial metric oqab and c, p are functions of the cosmic time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In order now, to identify an internal clock of our theory, we introduce a dynamical massless scalar field described by the Hamiltonian: Hφ = p2 φ 2|p|3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='7) At the end, the cosmological classical phase space is composed of two congugate pairs (c, p) and (φ, pφ) which obey the following Poisson brackets: {c, p} = 8πG 3 γ, {φ, pφ} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) with |p| = a2V 2/3 0 and c = γ ˙aV 1/3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Finally, using he Hamiltonian constraint one obtains the usual Friedmann equation within GR for a flat FRLW model, H2 = 8πG 3 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2 The quantum dynamics Working now at the quantum level, the classical phase space variables and the clas- sical Hamiltonian will be promoted to quantum operators while the Poisson brackets will be replaced by commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' However, within quantum field theory, the commutation algebra of quantum operators requires integration over the 3D space, thus assuming a well pre-defined background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Nevertheless, this setup cannot be applied within the framework of LQG since we want a background independent theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' For this reason, the quantisation process is performed at the level of two new canonical variables, namely the holonomy of the Ashtekar connection he(A) along a curve e ⊂ Σ and the flux of the densitized triad FS(E) along a 2-surface S defined as [67] he(A) ≡ Pexp �� e τiAi adxa � , FS(E) ≡ � S τiEi anad2y, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='10) where τ = −iσi/2 (σi are the Pauli matrices) with [τi, τj] = ϵijkτ k, na is the unit vector vertical to the surface S and P is a path-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' These functions constitute non-trivial SU(2) variables satisfying a unique holonomy-flux Poisson algebra [71–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Working within this representation one can then construct a kinematical Hilbert space for the gravity sector which is actually the space of the square integrable func- tions on the Bohr compactification of the real line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Hgrav kin ≡ L2(RBohr, dµBohr) [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Regarding the matter sector, the respective kinematical Hilbert space is defined like in the standard Shrondigner picture as Hmatter kin ≡ L2(R, dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, the whole kinematical Hilbert space of the theory is defined as Hkin ≡ Hgrav kin ⊗ Hmatter kin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Focusing now on the homogeneous and isotropic FLRW model, usually called as Loop Quantum Cosmology (LQC) and following the conventional quantisation ¯µ scheme [75] one introduces two new conjugate variables defined as follows: u ≡ 2 √ 3 sgn(p)/¯µ3, b ≡ ¯µc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='11) – 4 – where ¯µ = � ∆/|p| and ∆ = 4 √ 3 πγGℏ being the minimum nonzero eigenvalue of the area operator [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Finally, one can show that the new variables obey the following Poisson bracket: {b, u} = 2 ℏ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='12) and that in Hgrav there are two elementary operators, namely � eib/2 and ˆu related to the holonomy and the flux conjugate variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, it turns out that the eigenstates |u⟩ of ˆu form an orthonormal basis in Hgrav kin and the actions of these two operators in this basis can read as � eib/2 |u⟩ = |u + 1⟩ , ˆu |u⟩ = u |u⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='13) Letting now |φ⟩ being the orthonormal bases in Hmatter kin one can define |u, φ⟩ ≡ |u⟩ ⊗ |φ⟩ as the generalized basis of the whole kinematic Hilbert space Hkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, after defining the relevant Hilbert space and the associated to it orthonormal basis, one can promote the Hamiltonian to a quantum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, it is possible to define a quasi- classical sharped initial state living in Hkin, which can be viewed as wavepacket around a classical trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Consequently, expressing the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='5) in terms of fluxes and holonomies one can derive the expectation value of the Hamiltonian operator over the initial semi-classical sharped state which at the end will contain first order quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Finally, accounting only for the holonomy corrections (since flux or inverse volume corrections face the issue of a fiducial cell dependence [75]) one obtains the following modified Friedmann equation [75, 77, 78]: H2 = 8πG 3 ρ � 1 − ρ ρc � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='14) where ρc = 2 √ 3 M4 Pl γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' As it can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='14) for ρ > ρc there is no physical evo- lution since H2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' One then finds that the effect of holonomies leads to a non-singular evolution where the classical Big Bang singularity is replaced by a non-singular quantum bounce where ρ = ρc and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This bouncing point constitutes a transitioning point between a contracting (H < 0) and an expanding phase (H > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 3 The threshold of primordial black hole formation in loop quantum gravity Having introduced before the fundamentals of LQG, we estimate in this section the PBH formation threshold δc accounting for effects of loop quantum gravity at the level of the background cosmic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' To do so, we assume that the collapsing overdensity region is described by a homo- geneous core (closed Universe) described by the following fiducial metric: ds2 = −dt2 + a2(t) � dχ2 + sin2 χdΩ2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1) – 5 – where dΩ2 is the line element of a unit two-sphere and a(t) is the scale factor of the perturbed overdensity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' For this type of close homogeneous and isotropic spacetime foliations one can show that following the procedure as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 2 the modified Friedmann equation in k = 1 LQC accounting only for the holonomy corrections will read as [79, 80]: H2 = � ˙a a �2 = 1 3M2 Pl (ρ − ρ∗) � 1 − ρ − ρ∗ ρc � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2) where ρ is the energy density of the overdense region and ρ∗ = ρc � (1 + γ2)D2 + sin2 D � with D ≡ λ(2π2)1/3/v1/3, λ2 = 4 √ 3 πγℓ2 Pl [81], ℓPl being the Planck length and v = 2π2a3 being the physical volume of the unit sphere spatial manifold [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Since v increases with time one can expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2) in the limit u ≫ 1 [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, keeping terms up to O(1/v2/3) one can show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2) takes the following form: H2 = � ˙a a �2 = 1 3M2 Pl � ρ � 1 − ρ ρc � − 3M2 Pl a2 � 1 − 2 ρ ρc �� = F(ρ) 3M2 Pl − G(ρ) a2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='3) where F(ρ) = ρ(1 − ρ/ρc) and G(ρ) = 1 − 2ρ/ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In the limit where γ = 0, ρc → ∞ and one recovers the standard GR k = 1 Friedmann equation H2 = ρ 3M2 Pl − 1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' As regards now the background, the latter it will behave as the standard homoge- neous and isotropic FLRW background whose fiducial metric reads as ds2 = −dt2 + a2 b(t) � dr2 + r2dΩ2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='4) and whose modified Friedmann equation within LQC will read as H2 b = � ˙ab ab �2 = ρb 3M2 Pl � 1 − ρ ρc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='5) In this setup, the collapsing overdense region corresponds to the region where 0 ≤ χ ≤ χa and the areal radius at the edge of the ovedensity will read as Ra = a sin χa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='6) At this point, we need to stress that the characteristic size of the overdensity is initially super-horizon and will reenter the cosmological horizon when the areal radius of the overdensity becomes equal to the cosmological horizon H−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 1 Hhc = ahc sin χa, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='7) where the index “hc” denotes quantities at the horizon crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Writing now the energy density of the overdensity as ρ = ρb(1 + δ), where δ ≡ δρ ρb , one can plug ρ into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='3) and working within the uniform Hubble gauge where H = Hb they can recast Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='7) as sin2 χa = 1 Gδ hc �F δ hc Fhc − 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) – 6 – where F δ hc = F [ρb,hc(1 + δ)] and Gδ hc = G [ρb,hc(1 + δ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Once then the overdensity region crosses the cosmological horizon will initially follow the cosmic expansion and at some point it will detach from it starting to collapse to form a black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This basically happens at the time of maximum expansion of the overdensity, when the Hubble parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='5) becomes zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Hm = 0, or equivalently when am = 3M2 PlGm Fm , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='9) with the subscript “m” denoting quantities at the maximum expansion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Having derived above the horizon crossing time and the time at maximum expan- sion we establish below a criterion for PBH formation by investigating the necessary conditions for the triggering of the gravitational collapse process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Doing so, we confront the gravitational force which pushes matter inwards and enhances in this way the black hole gravitational collapse with the sound wave pressure force which pushes matter out- wards, thus disfavoring the collapse of the overdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, the criterion which we adopt is the requirement that the time at which the pressure sound wave crosses the radius of the overdensity region should be larger than the time at the maximum expansion, which is actually the time of the onset of the gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, the sound pressure force will not have time to disperse the collapsing fluid matter to the background medium and prevent in this way the collapsing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Equivalently, we require that the proper size of the overdensity χa is larger than the sound crossing distance by the time of maximum expansion χs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' χa > χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='10) To compute now the sound crossing distance by the time of maximum expansion we assume matter in terms of a perfect fluid characterized by a constant equation-of-state (EoS) parameter w, defined as the ratio between the pressure p and the energy density ρ of the fluid, w ≡ p/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Within this framework, the sound wave propagation equation reads as adχ dt = √w , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='11) where we used the fact that for a perfect fluid with a constant EoS parameter the square sound wave c2 s is equal to w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' c2 s = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, χs can be recast in following form: χs = √w � tm tini dt a = √w 3 � ρini ρb,m dρb (1 + w)ρb �� ρb,m ρb � 2 3(1+w) GmF(ρb) Fm − G(ρb) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='12) where we have assumed that for a perfect fluid ρb ∝ a−3(1+w) and used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='9) to express am in terms of Fm and Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) the criterion for PBH formation reads as 1 Gδ hc �F δ hc Fhc − 1 � > sin2 χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='13) – 7 – To determine therefore the PBH formation threshold, one can follow the following procedure: From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8), one should firstly determine the the ratio ρhc/ρm for a given value of χa and then solve numerically the inequality Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='13) in order to extract the value of the critical energy density contrast δc required for the overdensity region to collapse and form a PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Practically, one should compute χs from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='12) for a given value of ρm and equate χs with χa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Then, solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) they will extract the ratio ρhc/ρm and then plugging it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='13) they can extract numerically δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, given the fact that ρb,hc < ρc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' PBHs form after the quantum bounce, and that δ < 1 since we want to be within the perturbative regime, one can show from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) that δc ≃ sin2 χs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='14) with χs given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At this point, we should stress that the above expression for the value of δc is a lower bound estimate of its true value since it assumes the homogeneity of the collapsing overdensity region which in general is not the case when one is met with strong pressure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, it is strictly valid for regimes where w ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' However, PBH formation was never studied before in a rigorous way within the context of LQG through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' To that end, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='14) provides a reliable estimate for the value of δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 4 Results Following the procedure described above, we calculate here the PBH formation threshold δc within the framework of LQG and we compare it with its value in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 1 we show the PBH formation threshold as a function of the energy density at the time of maximum expansion ρm by fixing the EoS parameter w = 1/3 since we study PBH formation during the RD era and the value of the Barbero- Immirzi parameter γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2375 obtained from the computation of the entropy of black holes [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Interestingly, we see a deviation from GR for high energies at the time of maximum expansion which correspond to very small mass PBHs forming close to the quantum bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This behavior is somehow expected since in this high energy regime, one expects to see a quantifiable effect of the quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, we observe a drastic reduction of the value of δc in this region of high values of ρm up to 50% compared the GR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This reduction of δc should be related with a smaller cosmological/sound horizon in LQC compared to GR as it can be speculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' To see this, let us find the necessary conditions to get a cosmological horizon in LQC smaller than that in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Doing so, one should require that H2 LQC > H2 GR ⇔ ρ 3M2 Pl � ρ − ρ ρc � − 1 a2 � 1 − 2 ρ ρc � > ρ 3M2 Pl − 1 a2 ⇔ ρ < 6M2 Pl a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1) For ρ = ρm and a = am as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='9) one can verify that the inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1 is identically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, indeed the cosmological horizon in LQC is smaller than in that GR leading to a reduction of δc compared to its GR value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' – 8 – 10−13 10−11 10−9 10−7 10−5 10−3 10−1 101 ρm/M 4 Pl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='60 δc w = 1/3 LQG GR 10−5 10−3 10−1 101 103 105 107 MPBH in grams 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='60 δc w = 1/3 γ=10−1 γ=1 γ=101 γ=102 γ=103 Figure 1: Left Panel:The PBH formation threshold in the uniform Hubble gauge in the radiation-dominated era (w = 1/3) as a function of the energy density at the onset of the PBH gravitational collapse in LQG (green curve) and in GR (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1)) (black dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Right Panel: The PBH formation threshold in the uniform Hubble gauge in the radiation-dominated era (w = 1/3) as a function of the primordial black hole mass MPBH in LQG for different values of the Barbero-Immirzi parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' The vertical black dashed line corresponds to MPBH = MPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Then, we consider the Barbero-Immirzi parameter γ as a free parameter of the underlying quantum theory in the context of LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, despite the fact that the Bekenstein-Hawking entropy has been standardly used as a way to fix the value of γ, the dependence of the entropy calculation on γ is controversial, and the value γ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2375, calculated using thermodynamical arguments, is not broadly accepted [83–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In fact, the choice to vary this parameter is motivated by the fact that γ is actually a coupling constant with a topological term in the gravitational action, with no consequence at the level of the classical equations of motion [86–91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Thus, we vary the Barbero-Immirzi parameter within the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1 < γ < 1000 accounting for observational constraints for the duration of inflation after a quantum bounce [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At the end, we plot in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 1 the PBH formation threshold δc as a function of the PBH mass for different values of the parameter γ within the observationally allowed range γ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='1, 1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' We set the lower bound on the PBH mass equal to the Planck mass as predicted within the quantum gravity approach [94] (See vertical black dashed line in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In order to get the PBH mass, we account for the fact that the PBH mass is of the order of the cosmological horizon mass at horizon crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Solving at the end numerically Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='8) we found that ρhc/ρm ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' This corresponds to N = ln � am ahc � = 1 4 ln �ρhc ρm � ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='6 e − folds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content='2) passing from horizon crossing time up to the onset of the gravitational collapse process at – 9 – ρ = ρm, thus being in agreement with the results from PBH numerical simulations [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' As expected, when we increase the value of γ the overall mass range moves to higher masses given the fact that higher values of γ are equivalent with lower values of ρc, thus the quantum bounce happens at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Consequently, PBHs if formed will form at later times, thus will acquire larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Interestingly, independently on the value of the Barbero-Immirzi variable δc is re- duced on the low mass region, which for γ < 1000 corresponds to masses MPBH < 103g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, this reduction in δc in this very small PBH mass range will entail an enhancement in their abundances with tremendous consequences on the associated to them phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Indicatively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' we mention here that these ultra-light PBHs can trigger early PBH-matter dominated eras [20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 96] before BBN and reheat the Uni- verse through their evaporation [97] while at the same time they can account for the Hubble tension through the injection to the primordial plasma of light dark radiation degrees of freedom [98,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 99] while at the same time they can produce naturally the baryon assymetry through CP violating out-of-equilibrium decays of their Hawking evaporation products [100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Consequently, one can constrain the above mentioned observa- tional/phenomenological signatures by studying PBH formation within the context of LQG while vice-versa given the above mentioned phenomenology one can constrain the Barbero-Immirzi parameter γ which is the fundamental parameter within LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In this way, PBHs are promoted as a novel probe to constrain the potential quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' 5 Conclusions PBHs firstly introduced in ’70s are of great significance, since they can naturally account for a part or all of the dark matter sector, while at the same time they might seed the formation of large-scale structures through Poisson fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Moreover, they can also offer the seeds for the progenitors of the black-hole merging events recently detected by LIGO/VIRGO as well as for the supermassive black holes present in the galactic centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Their formation was mainly studied within the context of general relativity using both analytic and numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In this work, we studied PBH formation within the context of LQG by investigating the impact of the potential quantum character of spacetime on the critical PBH forma- tion threshold δc, whose value can crucially affect the abundance of PBHs, a quantity which is constrained by numerous observational probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In particular, by comparing the gravitational force with the sound wave pressure force during the process of the gravitational collapse we obtained a reliable estimate on the value of δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Interestingly, we found that for low mass PBHs formed close to the quantum bounce, the value of δc is drastically reduced up to 50% compared to the general rel- ativistic regime with tremendous consequences for the observational/phenomenological footprints of such small PBH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' In this way, we quantified for the first time to the best of our knowledge how quantum effects can influence PBH formation in the early Universe within a quantum gravity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' – 10 – Finally, by treating the Barbero-Immirzi parameter γ as the free parameter of LQG we varied its value by studying its effect on the value of the PBH formation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' As expected, we found an overall shift of the PBH masses affected by the choice of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Very interestingly, we showed as well that using the observational and phenomenological signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, one can constrain the quantum parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' At this point, we should highlight the fact that our formalism can be applied to any quantum theory of gravity giving an explicit form for the equations of the background cosmic evolution establishing in this way the PBH portal as a novel probe to constrain the potential quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' Acknowledgments The author acknowledges financial support from the Foundation for Education and European Culture in Greece as well the contribution of the COST Action CA18108 “Quantum Gravity Phenomenology in the multi-messenger approach”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' References [1] Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} +page_content=' – 16 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfEyy7/content/2301.11439v1.pdf'} diff --git a/HtE5T4oBgHgl3EQfWg9Q/content/tmp_files/2301.05559v1.pdf.txt b/HtE5T4oBgHgl3EQfWg9Q/content/tmp_files/2301.05559v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..853a9bc2686118f82b04d978ae3ee5ca4688636e --- /dev/null +++ b/HtE5T4oBgHgl3EQfWg9Q/content/tmp_files/2301.05559v1.pdf.txt @@ -0,0 +1,641 @@ +arXiv:2301.05559v1 [quant-ph] 11 Jan 2023 +Supercurrent and Electromotive force generations +by the Berry connection from many-body wave +functions +Hiroyasu Koizumi +Division of Quantum Condensed Matter Physics, Center for Computational Sciences, +University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan +E-mail: koizumi.hiroyasu.fn@u.tsukuba.ac.jp +January 2023 +Abstract. +The velocity field composed of the electromagnetic field vector potential +and the Berry connection from many-body wave functions explains supercurrent +generation, Faraday’s law for the electromotive force (EMF) generation, and other +EMF generations whose origins are not electromagnetism. An example calculation +for the EMF from the Berry connection is performed using a model for the cuprate +superconductivity. +1. Introduction +The Berry phase first discovered in the context of the adiabatic approximation now +prevails in various fields of physics [1, 2]. +In particular, it is now an indispensable +mathematical tool to detect topological defects in quantum wave functions [3]. Recently, +the Berry connection from many-body wave functions was defined and its usefulness to +calculate supercurrent is demonstrated [4]. A salient feature of such a formalism is that it +provides a vector potential directly related to the velocity field for electric current. In the +present work, we consider the supercurrent and electromotive force (EMF) generations +based on the same formalism [4, 5]. +The EMF is expressed using a non-irrotational ‘electric field’, Eirrot, whose origin +may not be a real electric field. It is defined as +E = +� +C +Enon−irrot · dr +(1) +where C is a closed electric circuit. This EMF appears due to various causes, such as +chemical reactions in batters or temperature differences in metals. +One of the important EMF generation mechanisms is the Faraday’s law of magnetic +induction. It is expressed as a total time-derivative of a magnetic flux of the magnetic +field B +E = − d +dt +� +S +B · dS +(2) + +Supercurrent and EMF by Berry connection +2 +where S is a surface whose circumference is C. This EMF formula is often called the +“flux rule”, since +� +S B · dS is the magnetic flux through the surface S; it has been +claimed curious since it is composed of two different fundamental equations in classical +theory [6], i.e., the Faraday’s law of induction and the Lorentz force. The curiosity +is increased by the fact that one of them is an equation for fields only, and the other +includes particles and is an equation for a force on a particle. +This peculiarity disappears in quantum theory using the vector potential A that +is more fundamental than the magnetic field B [7, 8, 9], and the wave function makes +the velocity of a particle a velocity field [10]. Then, the two contributions in the “flux +rule” are connected by the duality that a U(1) phase factor added on a wave function +describes a whole system motion, and also plays the role of the vector potential when +it is transferred into the Hamiltonian [11]. +In the present work, we extend the above vector potential and velocity field +approach for the electric current generation to cases where the vector potential of the +Berry connection from many-body wave functions appears [4]. We show that the EMF +generation other than the electromagnetic field origin, such as those due to chemical +reactions or temperature gradients can be expressed by it. +The organization of the present work is as follows: we explain the velocity field +appearing from the Berry connection from many-body wave functions in Section 2. +We reexamine the Faraday’s EMF generation formula using the velocity field from the +electromagnetic vector potential in Section 3. We examine the EMF generation by the +Berry connection in Section 4, and an example calculation is performed for the Nernst +effect in Section 5. Lastly, we conclude the present work by mentioning implications of +the present new theory in Section 6. +2. The velocity field from the Berry connection form many-body wave +functions and supercurrent generation +The key ingredient in the present work is the Berry connection from many-body wave +functions for electrons given by +AMB +Ψ (r)= +1 +ℏρ(r)Re +�� +dσ1dx2 · · ·dxNΨ∗(r, σ1, · · · , xN)(−iℏ∇)Ψ(r, σ1, · · · , xN) +� +(3) +where N is the total number of electrons in the system, ‘Re’ denotes the real part, Ψ +is the total wave function, xi collectively stands for the coordinate ri and the spin σi +of the ith electron, −iℏ∇ is the Schr¨odinger’s momentum operator for the coordinate +vector r, and ρ(r) is the number density calculated from Ψ. This Berry connection is +obtained by regarding r as the “adiabatic parameter”[1]. +Let us consider the electron system whose kinetic energy operator in the Schr¨odinger + +Supercurrent and EMF by Berry connection +3 +representation is given by +ˆT = − +N +� +j=1 +ℏ2 +2me +∇2 +j +(4) +where me is the electron mass. +For convenience, we also use the following χ defined as +χ(r) = −2 +� r +0 +AMB +Ψ (r′) · dr′ +(5) +and express the many-electron wave function Ψ as +Ψ(x1, · · · , xN) = exp +� +− i +2 +N +� +j=1 +χ(rj) +� +Ψ0(x1, · · · , xN) +(6) +Then, Ψ0 = Ψ exp +� +i +2 +�N +j=1 χ(rj) +� +is a currentless wave function for the current +operator associated with ˆT in Eq. (4) since the contribution from Ψ and that from +exp +� +i +2 +�N +j=1 χ(rj) +� +cancel out. In other words, a wave function is given as a product +of a currentless one, Ψ0, and the factor for the current exp +� +− i +2 +�N +j=1 χ(rj) +� +. The total +wave function Ψ must be a single-valued function of coordinates. This makes χ as an +angular variable that satisfies some periodicity. This periodicity gives rise to non-trivial +topological integer as will be explained, shortly. +When electromagnetic field is included, the kinetic energy operator becomes +ˆT ′ = +N +� +j=1 +1 +2me +(−iℏ∇j − qA)2 +(7) +where q = −e is the electron charge, and A is the electromagnetic field vector potential. +The magnetic field is given by B = ∇ × A. +In the following, we will use the same expression, Ψ, for the total wave function. +Then, the current density for Ψ is given by +j = −eρv +(8) +with the velocity field v given by +v = e +me +� +A − ℏ +2e∇χ +� += e +me +A + ℏ +me +AMB +Ψ +(9) +The current density in Eq. (8) is known to give rise to the Meissner effect if it is a +stable one due to the fact that it explicitly depends on A [10]. For the stable current +case, ∇χ compensates the gauge ambiguity in A and makes v in Eq. (9) gauge invariant. +If the Meissner effect is realize, the magnetic filed is expelled from the bulk of a +superconductor [10]. Then, the flux quantization is observed for magnetic flux through + +Supercurrent and EMF by Berry connection +4 +a loop C that goes through the bulk of a ring-shaped superconductor +� +S +B · dS = +� +C +A · dr += ℏ +2e +� +C +∇χ · dr += h +2ewC[χ] +(10) +where wC[χ] is the topological integer ‘winding number’ defined by +wC[χ] = 1 +2π +� +C +∇χ · dr +(11) +According to Eq. (9), the presence of non-zero wC[χ] means the existence of the +stable velocity field that satisfies +� +C +v · dr = +h +2me +wC[χ] +(12) +In superconductors, the quantized flux persists. This means that the condition +d +dtwC[χ] = 0 +(13) +is realized. +In normal metals, the time-derivative of the velocity field is often expressed as +dv +dt = −1 +τ v +(14) +using a relaxation time approximation, where τ is the relaxation time. +Combination of this with Eq. (12) yields +τ d +dtwC[χ] = −wC[χ] +(15) +If the condition in Eq. (13) with nonzero wC[χ] is realized, Eq. (15) means that τ +must be ∞, i.e., an infinite conductivity, or zero resistivity is realized. +3. The vorticity field from the vector potential A and Faraday’s flux rule +In this section, we consider the case where non-trivial AMB +Ψ +is absent. When AMB +Ψ +is +trivial, it satisfies +∇ × AMB +Ψ += 0 +(16) +Thus, by applying ∇× on the both sides of Eq. (9) +∇ × v = e +me +B +(17) +is obtained. +Taking the total time-derivative of the above yields +∇ × dv +dt = e +me +∂tB + e +me +(v · ∇)B +(18) + +Supercurrent and EMF by Berry connection +5 +where the total time-derivative of the field B is the Eulerian time-derivative given by +dB +dt = ∂tB + (v · ∇)B +(19) +Integrating Eq. (18) over the surface S, we have +� +C +dv +dt · dr = e +me +� +S +∂tB · dS + e +me +� +S +(v · ∇)B · dS +(20) +where the Stokes theorem is used to convert the surface integral to the line integral. +Noting that the electromotive force for an electron is given by +E = 1 +−e +� +C +d(mev) +dt +· dr +(21) +where −e is the electron charge and me is the electron mass, the following relation is +obtained +E = − +� +S +∂tB · dS − +� +S +(v · ∇)B · dS +(22) +This is equal to the Faraday’s formula in Eq. (2). +In the situation where the circuit C moves with a constant velocity v0, we have the +following relation +(v0 · ∇)B = ∇ × (B × v0) + v0(∇ · B) += ∇ × (B × v0) +(23) +due to the fact that B satisfies ∇ · B = 0 [12]. +As a consequence, the well-known EMF formula +E = − +� +S +∂tB · dS + +� +C +(v0 × B) · dr +(24) +is obtained. The first term in it is attributed to the Faraday’s law of induction, and +the second to the Lorentz force. This formula is composed of two different fundamental +equations in classical theory [6]. However, in the quantum mechanical formalism, two +contributions stem from a single relation in Eq. (9). +4. The EMF generation by the Berry connection +The velocity field in Eq. (9) contains the vector potential AMB +Ψ +in addition to the +electromagnetic vector potential A. Just like A, AMB +Ψ +will also give rise to the EMF. +We now consider a general case where the Berry connection arises from a set of +states {Ψj} and given by +AMB = +� +j +pjAMB +Ψj +(25) +where pj’s are probabilities satisfy +� +j +pj = 1 +(26) +and AMB +Ψj is obtained from Eq. (3) by replacing Ψ with Ψj. + +Supercurrent and EMF by Berry connection +6 +We express AMB using the following density matrix +ˆd = +� +j +pj|Ψj⟩⟨Ψj| +(27) +where the operator ˆAMB is defined through the relation +⟨Ψj| ˆAMB|Ψj⟩ = AMB +Ψj +(28) +From now on, we allow the time-dependence in Ψj. When Ψj is time-dependent, AMB +Ψj +is also time-dependent. The distribution probability pj can be also time and coordinate +dependent. +Using the density operator ˆd and the operator ˆAMB, the vector potential from the +Berry connection is given by +AMB = tr +� +ˆd ˆAMB� +(29) +We define BMB by +BMB = ∇ × AMB +(30) +Then, the EMF from the Berry connection is given by +EMB = −ℏ +e +� +S +∂tBMB · dS − ℏ +e +� +S +(v · ∇)BMB · dS +(31) +The first term in the right hand side can arise from the time-dependence of pj. This +means that if pj varies with time due to chemical reactions, photo excitations, or etc. +it will give rise to the EMF. The second term will arise if the temperature depends on +the coordinate, T(r), and pj contains the Boltzmann factor exp(− +Ej +kBT(r)), where Ej is +the energy for the state Ψj. It also arises when pj depends on the coordinate due, for +example, to the concentration gradient of chemical spices. +Now we consider the case where the circuit moves with a constant vector v0. The +circuit in this case should be regarded as a region of the system which flows due to the +flow existing in the system. Such a motion may arise from a temperature gradient or +concentration gradient in the system. In this case, we have the following relation, +(v · ∇)BMB = −∇ × (v0 × BMB) +(32) +due to the fact that ∇ · BMB = ∇ · (∇ × AMB) = 0. +The equation (31) can be cast into the following form +EMB = −ℏ +e +� +C +� +∂tAMB − v0 × (∇ × AMB) +� +· dr +(33) +that only contains AMB. However, the above formula may not be convenient to use due +to the fact that AMB contains topological singularities. A convenient one may be the +following +EMB = −ℏ +e +d +dt +� +S +BMB · dS +(34) +where B in the Faraday’s law in Eq. (2) is replaced by BMB. + +Supercurrent and EMF by Berry connection +7 +5. Nernst effect +In this section, we examine the Nernst effect observed in cuprate superconductors +[13, 14, 15]. We examine this phenomenon using Eq. (34). A theory of superconductivity +in the cuprate predicts the appearance of spin-vortices in the CuO2 plane around doped +holes that become small polarons [16, 17, 18]. The spin-vortices generate the vector +potential +AMB = −1 +2∇χ +(35) +where χ is an angular variable with period 2π. This angular variable appears due to +the requirement that the wave function to be a single-valued function of coordinates +in the situation where itinerant motion of electrons around the small polaron hole is a +spin-twisting one. +We can decompose χ as a sum over spin-vortices +χ = +Nh +� +j=1 +χj +(36) +where χj is a contribution form the jth small polaron hole, and Nh is the total number +of holes that become small polarons. +Each χj is characterized by its winding number +wj = 1 +2π +� +Cj +∇χj · dr +(37) +where Cj is a loop that only encircles the center of the jth spin-vortex. We can assume +wj to be +1 or −1; only odd integers are allowed due to the spin-twisting motion. The +numbers ±1 are favorable from the energetic point of view. +C(t) +v0 +x +y +Ly +C(t+Δt) +v0 +x0+ +Δt +x0 +Figure 1. A schematic picture for the EMF appearing from the Berry connection +generated by spin-vortices. +The Berry connection creates the vector potential +proportional to ∇χ, which creates vortices (loop currents) denoted by circles with +arrows. We consider two loops C(t) and C(t + ∆t), where t and t + ∆t denote two +times with interval ∆t. The loop moves with velocity v0 in the x-direction due to the +temperature gradient in that direction. A constant magnetic field is applied in the z- +direction. A voltage is generated across the y-direction. The sample exists 0 ≤ y ≤ Ly. +The left edge of the loop at time t is x0 and that at time t + ∆t is x0 + v0∆t. + +Supercurrent and EMF by Berry connection +8 +Let us consider the situation depicted in Fig. 1. We neglect the contribution from +A assuming that it is small. The EMF generated across the sample in the y-direction +is given by +EMB = − ℏ +e +1 +∆t +�� +S(t+∆t) +BMB · dS − +� +S(t) +BMB · dS +� += − ℏ +e +1 +∆t +�� +C(t+∆t) +AMB · dr − +� +C(t) +AMB · dr +� += ℏ +e +1 +∆t +� +∆C +AMB · dr +(38) +where S(t+∆t) and S(t) are surfaces in the xy-plane with circumferences C(t+∆t) and +C(t), respectively; ∆C is the loop encircling the area x0 ≤ x ≤ x0 + v0∆t, 0 ≤ y ≤ Ly, +with the counterclockwise direction. +We approximate +� +∆C AMB · dr by +� +∆C +AMB · dr = − 1 +2 +� +∆C +∇χ · dr +≈ − 1 +22π(nm − na)Lyv0∆t +(39) +where nm and na are average densities of wj = 1 (‘meron’) and wj = −1 (‘antimeron’) +vortices, respectively. Thus, nmLyv0∆t and naLyv0∆t are expected numbers of wj = 1 +and wj = −1 vortices within the loop ∆C, respectively. +From Eqs. (38) and (38), the approximate EMB is given by +EMB ≈ hv0 +2e (na − nm)Ly +(40) +Thus, the electric field generated by EMB in the y-direction is given by +Ey ≈ hv0 +2e (na − nm) +(41) +In our previous work, na is denoted as nd indicting that it yields a diamagnetic +current, and nm as np indicting that it yields a paramagnetic current [17, 18]. Using nd +and np, the Nernst signal is obtained as +eN = +Ey +|∂xT| = hv0(nd − np) +2e|∂xT| +(42) +The same formula was obtained previously for the situation where spin-vortices move +by the temperature gradient [17, 18]. Here, the situation is different; the spin-vortices +do not move, but the electron system affected by ∇χ moves. +Considering that the +small polaron movement is negligible at low temperature, the present situation is more +realistic than the previous one. The temperature dependence is the same as the one +that qualitatively explains the experimental result [18]. +Note that experiments indicating the presence of loop currents different from +ordinary Abrikosov vortices [19] in the cuprate [20, 21]. The present result indicates +that the observed Nernst can be explained by the presence of spin-vortex-induced loop +currents. + +Supercurrent and EMF by Berry connection +9 +6. Concluding remarks +Since the EMF by the Berry connection is not the electromagnetic field origin, it may +be more appropriate to call it the Berry-connection motive force (BCMF) given by +F BMF = −eEMB = ℏ d +dt +� +S +BMB · dS +(43) +The BCMF will arise from quantum mechanical dynamics of particles other than +electrons; for example, from proton dynamics, through chemical reactions. The non- +trivial Berry phase effect has been predicted [22], and observed in the hydrogen transfer +reactions [23]. Quantum mechanical effects are important in such reactions due to the +relatively light mass of protons [24, 25]. It is known that the EMF generated by the +proton pumps is a very important chemical process in biological systems, and the Berry- +connection motive force may play some roles in the working of the proton pumps. It +may be also useful to invent high performance batteries. +References +[1] Berry M V 1984 Proc. Roy. Soc. London Ser. A 391 45 +[2] Bohm A, Mostafazadeh A, Koizumi H, Niu Q and Zwanziger J 2003 The Geometric Phase in +Quantum Systems (Springer) +[3] Chiu C K, Teo J C Y, Schnyder A P and Ryu S 2016 Rev. Mod. Phys. 88(3) 035005 URL +https://link.aps.org/doi/10.1103/RevModPhys.88.035005 +[4] Koizumi +H +2022 +Physics +Letters +A +450 +128367 +ISSN +0375-9601 +URL +https://www.sciencedirect.com/science/article/pii/S0375960122004492 +[5] Koizumi H 2022 arXiv:2211.08759 [cond-mat.supr-con] +[6] Feynman R P, Leighton R B and Sands M 1963 The Feynman Lectures on Physics vol 2 (Addison- +Wesley) +[7] Aharonov Y and Bohm D 1959 Phys. Rev. 115 167 +[8] Tonomura A, Matsuda T, Suzuki R, Fukuhara A, Osakabe N, Umezaki H, Endo J, Shinagawa K, +Sugita Y and Fujiwara H 1982 Phys. Rev. Lett. 48 1443 +[9] Tonomura A, Osakabe N, Matsuda T, Kawasaki T, Endo J, Yano S and Yamada H 1986 Phys. +Rev. Lett. 56(8) 792–795 URL https://link.aps.org/doi/10.1103/PhysRevLett.56.792 +[10] London F 1950 Superfluids vol 1 (New York: Wiley) +[11] Koizumi H 2017 J. Supercond. Nov. Magn. 30 3345–3349 +[12] Jackson J D 1998 Classical Electrodynamics (Wiley) +[13] Xu Z A, Ong N P, Wang Y, Kakeshita T and Uchida S 2000 Nature 406 486 +[14] Wang Y, Li L, Naughton M J, Gu G D, Uchida S and Ong N P 2005 Phys. Rev. Lett. 95 247002 +[15] Daou R, Chang J, LeBoeuf D, Cyr-Choiniere O, Laliberte F, Doiron-Leyraud N, Ramshaw +B J, Liang R, Bonn D A, Hardy W N and Taillefer L 2010 Nature 463 519–522 URL +http://dx.doi.org/10.1038/nature08716 +[16] Koizumi H 2009 J. Phys. Chem. A 113 3997 +[17] Koizumi H 2011 J. Supercond. Nov. Magn. 24 1997 +[18] Hidekata R and Koizumi H 2011 J. Supercond. Nov. Magn. 24 2253 +[19] Abrikosov A A 1957 Sov. Phys. JETP 5 1174 +[20] Xia J, Schemm E, Deutscher G, Kivelson S A, Bonn D A, Hardy W H, Liang R, Siemons W, +Koster G, Fejer M M and Kapitulnik A 2008 Phys. Rev. Lett. 100 127002 +[21] He R H, Hashimoto M, Karapetyan H, Koralek J D, Hinton J P, Testaud J P, Nathan V, Yoshida + +Supercurrent and EMF by Berry connection +10 +Y, Yao H, Tanaka K, Meevasana W, Moore R G, Lu D H, Mo S K, Ishikado M, Eisaki H, andT +P Devereaux Z H, Kivelson S A, Orenstein J, Kapitulnik A and Shen Z X +[22] Mead C A and Truhlar D 1979 J. Chem. Phys. 70 2284 +[23] Yuan D, Guan Y, Chen W, Zhao H, Yu S, Luo C, Tan Y, Xie T, Wang X, Sun +Z, +Zhang +D +H +and +Yang +X +2018 +Science +362 +1289–1293 +ISSN +0036-8075 +URL +https://science.sciencemag.org/content/362/6420/1289 +[24] Kuppermann A and Schatz G C 1975 J. Chem. Phys. 62 62 2502 +[25] Elkowitz A B and Wyatt R E 1975 J. Chem. Phys. 62 2504 + diff --git a/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/2301.11823v1.pdf.txt b/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/2301.11823v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..36e95dfd5126ba9f4ed2725bb66cacb14a3b723a --- /dev/null +++ b/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/2301.11823v1.pdf.txt @@ -0,0 +1,983 @@ +HDPV-SLAM: Hybrid Depth-augmented Panoramic Visual SLAM for +Mobile Mapping System with Tilted LiDAR and Panoramic Visual +Camera +Mostafa Ahmadi1, Amin Alizadeh Naeini1, Zahra Arjmandi1, Yujia Zhang1, Mohammad Moein Sheikholeslami1, +and Gunho Sohn1† +Abstract— This paper proposes a novel visual simultane- +ous localization and mapping (SLAM), called Hybrid Depth- +augmented Panoramic Visual SLAM (HDPV-SLAM), generat- +ing accurate and metrically scaled vehicle trajectories using a +panoramic camera and a titled multi-beam LiDAR scanner. +RGB-D SLAM served as the design foundation for HDPV- +SLAM, adding depth information to visual features. It seeks +to overcome the two problems that limit the performance of +RGB-D SLAM systems. The first barrier is the sparseness of +LiDAR depth, which makes it challenging to connect it with +visual features extracted from the RGB image. We address this +issue by proposing a depth estimation module for iteratively +densifying sparse LiDAR depth based on deep learning (DL). +The second issue relates to the challenges in the depth associa- +tion caused by a significant deficiency of horizontal overlapping +coverage between the panoramic camera and the tilted LiDAR +sensor. To overcome this difficulty, we present a hybrid depth +association module that optimally combines depth information +estimated by two independent procedures, feature triangulation +and depth estimation. This hybrid depth association module +intends to maximize the use of more accurate depth information +between the triangulated depth with visual features tracked +and the DL-based corrected depth during a phase of feature +tracking. We assessed HDPV-SLAM’s performance using the +18.95 km-long York University and Teledyne Optech (YUTO) +MMS dataset. Experimental results demonstrate that the pro- +posed two modules significantly contribute to HDPV-SLAM’s +performance, which outperforms the state-of-the-art (SOTA) +SLAM systems. +I. INTRODUCTION +Recently, a vehicle-mounted mobile mapping system +(MMS)has become the principal spatial imaging system cap- +turing a high-fidelity map of built environments. Most com- +mercially available MMS combines mature geo-referencing +technology with precise, high-speed, long-range laser scan- +ning and high-resolution imaging sensors. A comprehen- +sive review of the modern MMSs is presented [7]. Latest +MMS enables rapidly collecting an enormous amount of +highly accurate and geo-referenced spatial data for different +applications such as producing high-definition maps for +autonomous vehicles and transforming them into 3D models +of large-scale cities [8]. However, MMS’s geo-referencing +†Corresponding author +1The authors are with the Department of Earth and Space Science +and Engineering, Lassonde School of Engineering, York University, 4700 +Keele Street, Toronto, Ontario M3J 1P3, Canada. �ahmadism@yorku.ca, +naeini@yorku.ca, +zahraarj@yorku.com, +zhang89@yorku.ca, +mmoein@yorku.ca, gsohn@yorku.ca +Mapping Module +Tracking Module +Tilted LiDAR: Panoramic Camera: +Mobile Mapping System +Sparse LiDAR Data +LiDAR Data +Projection +Loop Closing Module +Depth Association +t1 +t2 +Panoramic Image +Apply scale & bias +Pre-trained +Monodepth2 +RGB-D Image +Depth Estimation +Correction +Optimizer +Fig. 1: The mobile mapping system with a tilted LiDAR and +a panoramic camera. +capability heavily relies on high-end GNSS /IMU. The roles +of spatial imaging sensors such as cameras and LiDAR are +limited to generating photo-realistic textures and metrically- +scaled depths. Thus, MMS still shows its limitation in geo- +referencing spatial data in GNSS-unfavorable environments +such as urban canyons and tunnels. Also, we have recently +observed a high demand to design a compact and inexpensive +MMS to reduce system complexity, including the less or no +use of GNSS/IMU [7]. +To respond to this growing demand, some research efforts +have been made to exploit the potential of visual simultane- +ous localization and mapping (SLAM) to produce accurate +geo-referencing maps using only spatial imaging sensors [9]. +However, it is not trivial to directly adopt the SLAM pipeline +to the MMS’s spatial imaging sensors. A typical MMS is +equipped with a single multi-beam LiDAR tilted towards +the ground to increase the Vertical Field of View (FoV) +arXiv:2301.11823v1 [cs.RO] 27 Jan 2023 + +6 +Qand a panoramic camera to capture a 360o scene. However, +the tilted LiDAR is always considered unsuited for SLAM +because of its limited sensing coverage in the horizontal +direction. On the other hand, the full sensing coverage in +a panoramic camera’s horizontal and vertical directions is +advantageous for SLAM since it facilitates the extraction and +tracking of features in all directions. +To address these challenges, this paper proposes a novel +visual simultaneous localization and mapping (SLAM), +called Hybrid Depth-augmented Panoramic Visual SLAM +(HDPV-SLAM). Inspired by RPV-SLAM [9], HDPV-SLAM +has been developed based for metrically-scaled results using +a panoramic camera and a tilted LiDAR without using +additional sensors. HDPV-SLAM is facilitated by augment- +ing depth information after processing LiDAR point clouds +into visual features extracted from the panoramic camera to +produce scaled results. HDPV-SLAM’s major contributions +are as follows: +• We propose a novel depth estimation module for iter- +atively densifying sparse LiDAR depth based on deep +learning (DL) for increasing the quality of depth aug- +mentation on visual features. +• We propose a novel hybrid depth association module, +optimally combining depth information driven by trian- +gulating visual features and LiDAR depth association +for addressing a challenge caused by the significant lack +of spatial overlapping between the panoramic camera +and titled LIDAR sensor. +• Through extensive experiments in challenging outdoor +conditions on an 18.95 km long road, we demonstrate +that HDPV-SLAM can deliver superior metrically scaled +results compared to the state-of-the-art (SOTA) perfor- +mance. +We review related works in Section II. The proposed HDPV- +SLAM system is presented in Section III, and experimental +results are discussed in Section IV. Finally, we present the +concluding remarks on our work in Section V. +II. RELATED WORKS +In recent years visual SLAM (VS) and odometry have had +various remarkable works including ORB-SLAM [10], LSD- +SLAM [11], DSO [12], SVO [13] and RGB-D SLAM [14]. +The SOTA VS can be classified into filter-based SLAM and +keyframe-based SLAM. MonoSLAM [15] is one of the filter- +based method that using extended Kalman filter (EKF). In +comparison, keyframe-based VS uses selected frames instead +of all frames with bundle adjustment (BA) optimization is +more wildly used. The loop closing is a significant module +for keyframe-based VS, which detects the revisit keyframes +and improves the performance of BA optimization for large- +scale environment. +The camera’s FoV plays essential role in the VS. Most of +the VS systems apply pinhole model with limited FoV, which +will easily fail in the tracking process due to the insufficient +features that may caused by rapid motion, changing light +condition and texture-less scenes. In order to improve the +performance of the VS, the alternative solution is extending +the FoV by using fisheye or panoramic cameras. The SOTA +methods [16]–[18] have added fisheye camera module into +the SLAM systems. The works of [2]–[6] have presented +the advantages of using a panoramic camera into the SLAM +system. +Almost all real-world applications need metrically-scaled +trajectories. The other problem of monocular based VS is +scale ambiguity. Among all the current works only a small +number of works generate metrically-scaled results with the +help of auxiliary sensors. For example, [4] uses GPS data +and [5] uses GCPs; both have limited use cases. In contrast, +PIW-SLAM [23] generates metrically-scaled results with the +help of IMU and wheel encoder, which does not limit the +SLAM for real-world applications. Furthermore the works of +[19]–[22] use RGB-D camera like Kinect providing aligned +dense depth data for monocular images. RPV-SLAM [9] also +generates metrically scaled results using the help of a LiDAR +sensor. Although usage of LiDAR sensor with a pinhole +camera in the SLAM community has been addressed and +has proven its strengths [24], utilizing LiDAR sensor with a +panoramic camera has been an undervalued topic, and to the +best of our knowledge it is the only one. +In many research works, DL has demonstrated signif- +icant enhancements by substituting some modules of the +traditional VS. This end-to-end learning-based strategy was +still unable to outperform conventional methods. DL-based +approaches have been implemented in several SLAM compo- +nents, including feature extraction, posture extraction, depth +estimation, dynamic object elimination, and loop closure +detection. Feature extraction is one of the procedures that +DL can contribute to. NeuralBundler [26] presents a hy- +brid visual system that combines a monocular VO with a +pose graph optimization back-end. With the similar strategy, +LIFT-SLAM [27] developed a method for feature extraction +without the requirement of fine-tuning parameters. Dynamic +objects can significantly reduce the performance of the VS. +Their environment is thought to be static, contrary to actual +working conditions. DynaSLAM [28] attempts to detect +moving objects with DL’s assistance. DPC-net [29] propose +an approach that uses DL to learn corrections to the pose +estimation. We suggest two complimentary review articles +[30] and [31] for a more comprehensive examination of DL- +based SLAMs. Traditional VS still has advantages for posi- +tioning purposes. In our approach we use classical methods +for feature extraction and pose estimation. We leverage our +method with DL’s supports for depth estimation. +III. HYBRID DEPTH-AUGMENTED PANORAMIC +VISUAL SLAM +The main body of the SLAM system consists of depth +estimation, tracking, depth association, mapping, and loop +closing modules. Also, raw sensor inputs need processing +and feeding to the main body in a separate module. We will +explain each of the mentioned parts in this section. + +A. SLAM Input +As the first step, we create an RGB-D image made of a +panoramic RGB image and its corresponding LiDAR point +cloud. Channel D in the image is created by projecting the +LiDAR data into the image plane. This is achievable using +registeration parameters between the camera and the LiDAR +sensor following the previous work [9]. +Then, for each RGB-D image in the capturing order, frame +ft is created, where t ∈ {1, 2, 3, ..., n} and n is the total +number of RGB-D images. F is the array of all frames. +Each ft ∈ F contains RGB image It ∈ Rw×h×3 of the +width w and the height h and a corresponding projected +sparse LiDAR Ds +t ∈ Rw×h. In Section III-B, we describe a +method to densify Ds +t and get ˆDt ∈ Rw×h. Also, a frame ft +contains Ot which is the array of the ORB features extracted +from It using ORB feature extractor. For each oi +t ∈ Ot we +have oi +t ∈ {(u, v)|u, v ∈ W, 0 ≤ u < w, 0 ≤ v < h}where +i ∈ {1, 2, 3, ..., mt} and mt is the total number of ORB +features of It. +The output of the SLAM system is the array of all the +camera positions J and the set of all the map points ρ. Each +jt ∈ R3 × SO(3) in J is the position and the orientation of +the camera at time t in world coordinate, and each ϱ ∈ R3 +in ρ shows the position of a map point in world coordinate. +We define subsets of ρ, for each t as we create map points +using ft. The relation between ρt for different t is {} = +ρ0 ⊂ ρ1 ⊆ ρ2 ⊆ ...ρn−1 ⊆ ρn = ρ. Each ρt ⊂ ρ is updated +with the new map points created from ft each time a new +frame ft comes in. And when t = n, we have ρt = ρ +B. Depth Estimation Module +The overlapping regions (ORs) between LiDAR maps and +panoramic images can significantly contribute to the frame +matching through augmenting the RGB features with depth. +Although a tilted LiDAR has been used to increase the area +of ORs, the LiDAR data is still sparse. Therefore, taking +an input RGB image It at the time t and a corresponding +sparse LiDAR Ds +t, the aim of this module is to provide us +with a dense depth map ˆDt and resulting in bigger ORt. +However, depth estimation on panoramic images comes with +a number of challenges. First, as no standard dataset is +publicly available for panoramic depth estimation, supervised +models cannot be used. Second, self-supervised approaches +suffer from non-metricly scaled estimations and are con- +siderably complicated for panoramic geometry. Thus, the +proposed solution is to adapt a depth estimation baseline, +pre-trained on perspective images, to our panoramic dataset. +For this purpose, a double-stage adaptive refinement scheme +for panoramic images (PanoDARS), based on [43], has been +proposed (see Figure 2). +Guided by Ds +t, PanoDARS utilizes a depth estimation +baseline and iteratively refines the predicted depth map +Dt ∈ Rw×h. The selected baseline, Monodepth2 [34], was +originally trained on perspective images; however, using +PanoDARS, the predictions on panoramic images are sig- +nificantly refined in the inference time and without any need +for finetuning or re-training. +PanoDARS consists of two stages, correction and op- +timization. In [43], the depth maps are split into three +horizontal slices, and the correction values in each slice are +calculated independently. While in the proposed method, the +correction stage skips slicing, since the LiDAR sparsity pat- +terns are different in panoramic and perspective geometries. +Moreover, PanoDARS estimates depth only in the regions of +It that the sparse depth is available close to them (ORt, as +shown in a sample in the left image of Figure 3, rainbow +colored). +In the first stage, a correction value δds +t ∈ ∆Ds +t between +each valid pixel in the sparse depth map ds +t ∈ Ds +t and its +correspondence dt ∈ Dt is calculated using δds +t = dt − ds +t. +Then, an interpolation function Q : R2 �→ R based on +Delaunay triangulation [39] is leveraged to obtain a dense +correction map ∆Dt = Q(∆Ds +t). Finally, the corrected depth +map ˆDt = Dt + ∆Dt is calculated. ˆDt is a sufficiently +accurate initial value for the second stage. +In the second stage, while the pre-trained weights are +fixed, some learnable auxiliary parameters are applied on +intermediate features in the baseline. Optimizing those pa- +rameters, the predicted depth is refined. Therefore, the overall +performance of PanoDARS is as follows. +Given an input RGB image It ∈ Rw×h×3, PanoDARS +splits the pre-trained baseline M +: Rw×h×3 �→ Rw×h +into a body G : Rw×h×3 �→ Ra×b×c and a head H : +Ra×b×c �→ Rw×h, where a, b, and c are respectively width, +height and number of channels of the intermediate feature set +G(It) ∈ Ra×b×c. The auxiliary parameters, scales S ∈ Rc +and biases B ∈ Rc are applied on G(It), and the depth Dt = +H(S � G(It) � B) is predicted, where � and � represent +channel-wise multiplication and addition. Afterwards, the +correction module C : Rw×h �→ Rw×h carries out the first +refinement stage on Dt and returns ˆDt = C(Dt, Ds +t). +The auxiliary parameters X ∈ R2c, i.e., concatenated +channel-wise scales (S) and biases (B), are learnable. There- +fore, the following optimization problem can be formulated +as L(ˆDt(It, X + ∆X), Ds +t) → min +∆X . where ∆X is the correc- +tions applied on the parameters and L is the cost function +given to particle swarm optimizer (PSO) [38]. Hence, the +second stage of refinement is conducted by PSO. +C. Tracking Module +The tracking module runs in its own thread and has a close +collaboration with the mapping module described in Section +III-D. From the beginning of a SLAM run, the tracking +module reads ft for each t in sequential order. The matching +module inside the tracking module will find the matches +between ρt−1 and create the array of matched map points +Pt with Ot. The array Pt has the same size with Ot (mt) +and can contain Null values in some indices as well. For +each matched pair of oi +t and a map point from ρt−1, Pt has +a pi +t pointing to the map point. If oi +t do not match with any +map point from ρt−1, the corresponding index in Pt equals +to Null (pi +t = Null). +Using these local matches, the tracking module can esti- +mate jt. Then, in certain conditions that the system realizes + + +Apply +scale & bias +Pre-trained +Monodepth2 +Correction +Channel-wise +Scale (S) & Bias (B) +1.0 +… +0.7 +0.07 +… +0.8 +… +1.0 +… +0.9 +0.00 +… +1.2 +0.5 +… +0.7 +0.27 +… +0.6 +Particle 1 +Particle 2 +Particle n + + + + + +Optimizer +inference +input +optimization +Fig. 2: The proposed method for depth estimation module. +the tracking will perform weakly in the future with the +estimations of jt based on a small number of map points, +the system decides a frame ft to be a keyframe. Keyframes +are frames, the only difference being that the mapping and +loop closing modules are working with them. For example, +at time t = 1 the tracking module is not effective since ρ0 +is an empty set, and it will turn ft to a keyframe so that the +mapping module continues with it. +We can partition the set F based on them being keyframes +or not keyframes. We name the resulting subset containing +all keyframes KF and the other subset non-keyframes NF. +D. Mapping Module +The mapping module, like the tracking module, has its +own thread. When the tracking module decides ft to be a +keyframe, the mapping module creates new map points using +ft and puts them together with ρt−1 to save all of them in +ρt. The module attempts to create a map point for each oi +t +that its corresponding pi +t is Null. If oi +t ∈ ORt, it creates +a map point using oi +t, ˆDt(oi +t), and jt. If oi +t /∈ ORt, it will +triangulate ft with the nearest neighbouring keyframe fs to +estimate the depth for oi +t and create the map point with the +same set of information. If triangulation is not successful, +the module will not create a map point for oi +t. The mapping +module is also responsible for local bundle adjustment, which +is accomplished by following [9]. +E. Depth Association Module +In the tracking thread, the depth association module is +called after the tracking module tracks all the visible map +points in the current frame ft and estimates jt. This module +helps to make use ˆDt even if ft is not a keyframe. And +we call it hybrid depth association. Algorithm 1 shows the +process. +It inputs ft and the tracked array of map points Pt. For +each i, if the matching module has not assigned any map +points to oi +t or if oi +t is out of ORt, it skips the i and goes +for the next member of Oi +t. If oi +t passes these conditions, +then a new pi +t ∈ R3 will be created using oi +t and its depth +value and the position and orientation of the camera at time +t. But, sometimes the tracking module can make mistakes +in assigning ORB visual features to the corresponding map +points. So we have defined a rejection threshold θ to prevent +big mistakes and their effect on the whole trajectory. If the +distance between the current position of pi +t and the newly +calculated new pi +t is bigger than θ, we will not change +anything. If not, then we have defined two policies here: +• If pi +t has been created using triangulation and not +modified using any depth map later, the algorithm will +change the location of pi +t to new pi +t. +• Or, if the depth value of the visual feature that pre- +viously has modified (or created) the position of pi +t +(ˆDp(oi +p)) is larger the current depth (ˆDt(oi +t)), again the +algorithm will change the location of pi +t to new pi +t. +Figure 3 shows an example of this algorithm for the first +policy with a simple scenario where we only have one ORB +feature in each frame. And the second policy helps with the +precise estimation of the position of the map points. Each +time a map point is observed with a smaller depth value, the +algorithm recalculates its position based on it. +Algorithm 1 Depth Association Module Algorithm +Input: Current frame ft and the tracked array of map points Pt +1: θ ←The rejection threshold +2: for Each pi +t in Pt do +3: +if pi +t = Null or oi +t /∈ ORt then +4: +continue +5: +end if +6: +fp ←Last keyframe who created or modified pi +t +7: +new pi +t ←Estimated 3D position using oi +t, ˆDt(oi +t), and jt +8: +if distance(pi +t, new pi +t) <θ then +9: +if (pi +t created using triangulation and not modified after that) or +(ˆDt(oi +t)<ˆDp(oi +p)) then +10: +pi +t’s position ← new pi +t +11: +Update pi +t in ρt and all the previous subsets +12: +end if +13: +end if +14: end for +F. Loop Closing Module +Finally, after the loop closing module detects a loop, +it performs the global bundle adjustment on the trajectory +inside the loop. The global bundle adjustment performs a +pose-graph optimization on jt of the camera in each ft ∈ + +D一一cost function Lt - 4 +t +t + 3 +Frame +Keyframe +Visual feature +Visual feature and +matched map point +Triangulation +Trajectory: +Time: +o +1 +t-4 +o +1 +t +p +1 +t +o +1 +t& +p +1 +t+3 +o +1 +t+3& +The same map point in ρ +OR +t-4 +Fig. 3: A map point p1 +t is created at time t by triangulating visual features of times t−4 and t. This triangulation is because +these two visual features are not in ORt−4 and ORt. In ft+3, o1 +t+3 points to the middle of the parked car’s rooftop and +matches with the previously created p1 +t at time t which is now called p1 +t+3. At time t + 3, o1 +t+3 is inside ORt+3, which +means the depth data is available. This enables the system to estimate the map points’ 3D position more precisely through +the depth coming from depth estimation method. +{the subset of KF in the loop}. And also it optimizes all +of the map points created in those keyframes. For the pose- +graph optimization, we have used a similarity transformation. +IV. EXPERIMENTAL RESULTS AND DISCUSSION +In this section, initially, a brief overview of the dataset +is given. Afterwards, we demonstrate our experiments in +two parts. First, the effectiveness of each of our modules +is studied and the best setting for the proposed method +is chosen. Second, we the results of the proposed and +competing SLAM systems on all the sequences of YUTO +MMS dataset are discussed. +A. Dataset Characteristics +In this study, we used York University and Teledyne +Optech (YUTO) MMS dataset, and the acquisition details +can be found in a reference [9].The dataset has been acquired +by Teledyne Optech’s Maverick MMS with four sequences +in various outdoor environments with an 18.95 km long road +in York University’s Keele Campus. Table I shows the results +along with characteristics of each sequence. The results of +the percentage column conveys the fact that, on average, the +base work uses only one-fourth of the available depth maps +to create map points. +B. SLAM Trajectory Results +In this section, first an ablation study was conducted on +Sequence B of the dataset (see Table II). Then, the proposed +method was compared with the competing methods, i.e., +Google Cartographer [40] and RPV-SLAM [9] (see Table +III). +The ablation study aims to find the optimum value for θ in +Algorithm 1 and also to evaluate the effectiveness of depth +association and depth estimation modules. Bi-interpolation +described in [9] has been used as a rival to the depth +estimation module. +According to Table II, in the absence of depth association +module, i.e., first three rows, RPV-SLAM and the proposed +method show a similar performance in terms of both ATE +and RTE, yet significantly better than Cartographer. Further, +it proves that bi-interpolation and depth estimation module +have no superiority on each other, when there is no depth +association module. Considering different values of θ, bi- +interpolation and depth estimation module illustrate a similar +behaviour, where increasing θ leads to worse ATE and RTE. +Moreover, the best accuracy for both densification methods +was obtained using θ = 2. +As the Table II suggests, when depth association mod- +ule was utilized (with θ), more accurate results were ob- +tained. In addition, depth estimation module outperforms bi- +interpolation in both ATE and RTE, given identical values +for θ. Regardless of selected depth densification methods, +θ = 2 shows the best performance in both ATE and RTE. In +conclusion, we can attain the best performance when the +depth estimation module and θ = 2 was used. It means +the depth estimation and depth association module have +contributed to the improvement of the SLAM performance +in both ATE and RTE. +Table III shows the results of our best setting (depth +estimation module with θ = 2) in comparison with Google +Cartographer [40] and RPV-SLAM [9]. Google Cartographer +is a LiDAR-centric SLAM that is also equipped with IMU +and RPV-SLAM is our base work. Overall, relatively poor +results are obtained on residential areas due to unfavorable +illumination conditions such as shadows. As expected, the + +TABLE I: YUTO MMS Dataset Characteristics. +Region +Distance +travelled +Running +time +Total #frames +Average #keyframes +Percentage +Sequence A +Parking lot +324m +94 seconds +717 +192.2 +26.8% +Sequence B +Campus area +7035m +19 minutes +8382 +2306.4 +27.51% +Sequence C +Residential area +7965m +22 minutes +10778 +2576.2 +23.9% +Sequence D +Residential area +3634m +10 minutes +4500 +1117.8 +24.84% +TABLE II: SLAM trajectory results for ablation study on Sequence B. +SLAM +Densification method +θ +ATE (m) +RTE (%) +RRE (◦/m) +Cartographer +N/A +N/A +142.97 +16.57 +0.0093 +RPV-SLAM +bi-interpolation +N/A +12.91 +1.51 +0.0009 +Proposed method +depth estimation module +N/A +12.95 +1.64 +0.0010 +Proposed method +bi-interpolation +1 meter +11.99 +1.62 +0.0009 +Proposed method +bi-interpolation +2 meters +9.93 +1.43 +0.0010 +Proposed method +bi-interpolation +3 meters +10.19 +1.56 +0.0009 +Proposed method +bi-interpolation +4 meters +13.01 +1.63 +0.0010 +Proposed method +bi-interpolation +5 meters +13.70 +1.74 +0.0010 +Proposed method +depth estimation module +1 meter +11.28 +1.54 +0.0009 +Proposed method +depth estimation module +2 meters +9.58 +1.23 +0.0011 +Proposed method +depth estimation module +3 meters +11.96 +1.65 +0.0009 +Proposed method +depth estimation module +4 meters +12.24 +1.52 +0.0010 +Proposed method +depth estimation module +5 meters +14.16 +1.95 +0.0010 +TABLE III: SLAM trajectory results for all the sequences. Parameters of HDPV-SLAM has been set based on Table II. +SLAM +Sequence +ATE (m) +RTE (%) +RRE (◦/m) +Sequence +ATE (m) +RTE (%) +RRE (◦/m) +Cartographer +A +4.15 +6.79 +0.0507 +C +180.95 +5.78 +0.0133 +RPV-SLAM +A +1.62 +3.25 +0.0268 +C +30.66 +2.82 +0.0042 +Proposed method +A +1.40 +3.59 +0.0041 +C +11.93 +2.77 +0.0034 +Cartographer +B +142.97 +16.57 +0.0093 +D +57.74 +4.65 +0.0137 +RPV-SLAM +B +12.91 +1.51 +0.0009 +D +5.67 +1.49 +0.0016 +Proposed method +B +9.58 +1.23 +0.0011 +D +4.69 +0.9 +0.001 +parking lots sequence produced the best ATE performance +because of its shorter length and lower scene complexity. +Furthermore, the largest improvement in ATE was achieved +in Sequence C due to the relatively shorter LiDAR ranges in +residential areas. +To conclude, HDPV-SLAM produced the best results +compared to Cartographer and RPV-SLAM over all the +test sequences in terms of both ATE and RTE. Although +the performance of HDPV-SLAM varies depending on the +sequence, the other SLAM systems follow a similar pattern +in their performances as well. +C. Discussion +As seen in Table II, the proposed technique outperforms +the alternatives in terms of ATE. ATE is a metric that com- +pares the entire trajectory to the ground truth and handles the +form matching between them, indicating that the proposed +method preserves the shape more effectively than the other +methods. Furthermore, in three-quarters of the dataset, the +RTE and RRE of the proposed technique are superior to +those of the competing methods, and in one-quarter, they +are second best by a slight margin. All of these comparisons +demonstrate that the proposed strategy is superior. +V. CONCLUSIONS +In this study, we present a novel HDPV-SLAM system +with two modules, namely, depth estimation and depth +association. These two help bundle adjustment perform in a +better way. 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Sohn, “An Adaptive +Refinement Scheme for Depth Estimation Networks,” MDPI Sensors,. + diff --git a/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/load_file.txt b/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d4d90e4ca9a0e0b3a331740928a1e89a02a04da --- /dev/null +++ b/I9FKT4oBgHgl3EQfeC7T/content/tmp_files/load_file.txt @@ -0,0 +1,609 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf,len=608 +page_content='HDPV-SLAM: Hybrid Depth-augmented Panoramic Visual SLAM for Mobile Mapping System with Tilted LiDAR and Panoramic Visual Camera Mostafa Ahmadi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Amin Alizadeh Naeini1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Zahra Arjmandi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Yujia Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Mohammad Moein Sheikholeslami1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' and Gunho Sohn1† Abstract— This paper proposes a novel visual simultane- ous localization and mapping (SLAM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' called Hybrid Depth- augmented Panoramic Visual SLAM (HDPV-SLAM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' generat- ing accurate and metrically scaled vehicle trajectories using a panoramic camera and a titled multi-beam LiDAR scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' RGB-D SLAM served as the design foundation for HDPV- SLAM, adding depth information to visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' It seeks to overcome the two problems that limit the performance of RGB-D SLAM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The first barrier is the sparseness of LiDAR depth, which makes it challenging to connect it with visual features extracted from the RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We address this issue by proposing a depth estimation module for iteratively densifying sparse LiDAR depth based on deep learning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The second issue relates to the challenges in the depth associa- tion caused by a significant deficiency of horizontal overlapping coverage between the panoramic camera and the tilted LiDAR sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' To overcome this difficulty, we present a hybrid depth association module that optimally combines depth information estimated by two independent procedures, feature triangulation and depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This hybrid depth association module intends to maximize the use of more accurate depth information between the triangulated depth with visual features tracked and the DL-based corrected depth during a phase of feature tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We assessed HDPV-SLAM’s performance using the 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 km-long York University and Teledyne Optech (YUTO) MMS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Experimental results demonstrate that the pro- posed two modules significantly contribute to HDPV-SLAM’s performance, which outperforms the state-of-the-art (SOTA) SLAM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' INTRODUCTION Recently, a vehicle-mounted mobile mapping system (MMS)has become the principal spatial imaging system cap- turing a high-fidelity map of built environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Most com- mercially available MMS combines mature geo-referencing technology with precise, high-speed, long-range laser scan- ning and high-resolution imaging sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' A comprehen- sive review of the modern MMSs is presented [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Latest MMS enables rapidly collecting an enormous amount of highly accurate and geo-referenced spatial data for different applications such as producing high-definition maps for autonomous vehicles and transforming them into 3D models of large-scale cities [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' However, MMS’s geo-referencing †Corresponding author 1The authors are with the Department of Earth and Space Science and Engineering, Lassonde School of Engineering, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' �ahmadism@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ca, naeini@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ca, zahraarj@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='com, zhang89@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ca, mmoein@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ca, gsohn@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ca Mapping Module Tracking Module Tilted LiDAR: Panoramic Camera: Mobile Mapping System Sparse LiDAR Data LiDAR Data Projection Loop Closing Module Depth Association t1 t2 Panoramic Image Apply scale & bias Pre-trained Monodepth2 RGB-D Image Depth Estimation Correction Optimizer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' 1: The mobile mapping system with a tilted LiDAR and a panoramic camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' capability heavily relies on high-end GNSS /IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The roles of spatial imaging sensors such as cameras and LiDAR are limited to generating photo-realistic textures and metrically- scaled depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Thus, MMS still shows its limitation in geo- referencing spatial data in GNSS-unfavorable environments such as urban canyons and tunnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Also, we have recently observed a high demand to design a compact and inexpensive MMS to reduce system complexity, including the less or no use of GNSS/IMU [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' To respond to this growing demand, some research efforts have been made to exploit the potential of visual simultane- ous localization and mapping (SLAM) to produce accurate geo-referencing maps using only spatial imaging sensors [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' However, it is not trivial to directly adopt the SLAM pipeline to the MMS’s spatial imaging sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' A typical MMS is equipped with a single multi-beam LiDAR tilted towards the ground to increase the Vertical Field of View (FoV) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='11823v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='RO] 27 Jan 2023 6 Qand a panoramic camera to capture a 360o scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' However, the tilted LiDAR is always considered unsuited for SLAM because of its limited sensing coverage in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' On the other hand, the full sensing coverage in a panoramic camera’s horizontal and vertical directions is advantageous for SLAM since it facilitates the extraction and tracking of features in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' To address these challenges, this paper proposes a novel visual simultaneous localization and mapping (SLAM), called Hybrid Depth-augmented Panoramic Visual SLAM (HDPV-SLAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Inspired by RPV-SLAM [9], HDPV-SLAM has been developed based for metrically-scaled results using a panoramic camera and a tilted LiDAR without using additional sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' HDPV-SLAM is facilitated by augment- ing depth information after processing LiDAR point clouds into visual features extracted from the panoramic camera to produce scaled results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' HDPV-SLAM’s major contributions are as follows: We propose a novel depth estimation module for iter- atively densifying sparse LiDAR depth based on deep learning (DL) for increasing the quality of depth aug- mentation on visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We propose a novel hybrid depth association module, optimally combining depth information driven by trian- gulating visual features and LiDAR depth association for addressing a challenge caused by the significant lack of spatial overlapping between the panoramic camera and titled LIDAR sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Through extensive experiments in challenging outdoor conditions on an 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 km long road, we demonstrate that HDPV-SLAM can deliver superior metrically scaled results compared to the state-of-the-art (SOTA) perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We review related works in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The proposed HDPV- SLAM system is presented in Section III, and experimental results are discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Finally, we present the concluding remarks on our work in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' RELATED WORKS In recent years visual SLAM (VS) and odometry have had various remarkable works including ORB-SLAM [10], LSD- SLAM [11], DSO [12], SVO [13] and RGB-D SLAM [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The SOTA VS can be classified into filter-based SLAM and keyframe-based SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' MonoSLAM [15] is one of the filter- based method that using extended Kalman filter (EKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In comparison, keyframe-based VS uses selected frames instead of all frames with bundle adjustment (BA) optimization is more wildly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The loop closing is a significant module for keyframe-based VS, which detects the revisit keyframes and improves the performance of BA optimization for large- scale environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The camera’s FoV plays essential role in the VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Most of the VS systems apply pinhole model with limited FoV, which will easily fail in the tracking process due to the insufficient features that may caused by rapid motion, changing light condition and texture-less scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In order to improve the performance of the VS, the alternative solution is extending the FoV by using fisheye or panoramic cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The SOTA methods [16]–[18] have added fisheye camera module into the SLAM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The works of [2]–[6] have presented the advantages of using a panoramic camera into the SLAM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Almost all real-world applications need metrically-scaled trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The other problem of monocular based VS is scale ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Among all the current works only a small number of works generate metrically-scaled results with the help of auxiliary sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For example, [4] uses GPS data and [5] uses GCPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' both have limited use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In contrast, PIW-SLAM [23] generates metrically-scaled results with the help of IMU and wheel encoder, which does not limit the SLAM for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Furthermore the works of [19]–[22] use RGB-D camera like Kinect providing aligned dense depth data for monocular images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' RPV-SLAM [9] also generates metrically scaled results using the help of a LiDAR sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Although usage of LiDAR sensor with a pinhole camera in the SLAM community has been addressed and has proven its strengths [24], utilizing LiDAR sensor with a panoramic camera has been an undervalued topic, and to the best of our knowledge it is the only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In many research works, DL has demonstrated signif- icant enhancements by substituting some modules of the traditional VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This end-to-end learning-based strategy was still unable to outperform conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' DL-based approaches have been implemented in several SLAM compo- nents, including feature extraction, posture extraction, depth estimation, dynamic object elimination, and loop closure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Feature extraction is one of the procedures that DL can contribute to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' NeuralBundler [26] presents a hy- brid visual system that combines a monocular VO with a pose graph optimization back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' With the similar strategy, LIFT-SLAM [27] developed a method for feature extraction without the requirement of fine-tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Dynamic objects can significantly reduce the performance of the VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Their environment is thought to be static, contrary to actual working conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' DynaSLAM [28] attempts to detect moving objects with DL’s assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' DPC-net [29] propose an approach that uses DL to learn corrections to the pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We suggest two complimentary review articles [30] and [31] for a more comprehensive examination of DL- based SLAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Traditional VS still has advantages for posi- tioning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In our approach we use classical methods for feature extraction and pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We leverage our method with DL’s supports for depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' HYBRID DEPTH-AUGMENTED PANORAMIC VISUAL SLAM The main body of the SLAM system consists of depth estimation, tracking, depth association, mapping, and loop closing modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Also, raw sensor inputs need processing and feeding to the main body in a separate module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We will explain each of the mentioned parts in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' SLAM Input As the first step, we create an RGB-D image made of a panoramic RGB image and its corresponding LiDAR point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Channel D in the image is created by projecting the LiDAR data into the image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This is achievable using registeration parameters between the camera and the LiDAR sensor following the previous work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Then, for each RGB-D image in the capturing order, frame ft is created, where t ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=', n} and n is the total number of RGB-D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' F is the array of all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Each ft ∈ F contains RGB image It ∈ Rw×h×3 of the width w and the height h and a corresponding projected sparse LiDAR Ds t ∈ Rw×h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In Section III-B, we describe a method to densify Ds t and get ˆDt ∈ Rw×h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Also, a frame ft contains Ot which is the array of the ORB features extracted from It using ORB feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For each oi t ∈ Ot we have oi t ∈ {(u, v)|u, v ∈ W, 0 ≤ u < w, 0 ≤ v < h}where i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=', mt} and mt is the total number of ORB features of It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The output of the SLAM system is the array of all the camera positions J and the set of all the map points ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Each jt ∈ R3 × SO(3) in J is the position and the orientation of the camera at time t in world coordinate, and each ϱ ∈ R3 in ρ shows the position of a map point in world coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We define subsets of ρ, for each t as we create map points using ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The relation between ρt for different t is {} = ρ0 ⊂ ρ1 ⊆ ρ2 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='ρn−1 ⊆ ρn = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Each ρt ⊂ ρ is updated with the new map points created from ft each time a new frame ft comes in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' And when t = n, we have ρt = ρ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Depth Estimation Module The overlapping regions (ORs) between LiDAR maps and panoramic images can significantly contribute to the frame matching through augmenting the RGB features with depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Although a tilted LiDAR has been used to increase the area of ORs, the LiDAR data is still sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Therefore, taking an input RGB image It at the time t and a corresponding sparse LiDAR Ds t, the aim of this module is to provide us with a dense depth map ˆDt and resulting in bigger ORt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' However, depth estimation on panoramic images comes with a number of challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' First, as no standard dataset is publicly available for panoramic depth estimation, supervised models cannot be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Second, self-supervised approaches suffer from non-metricly scaled estimations and are con- siderably complicated for panoramic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Thus, the proposed solution is to adapt a depth estimation baseline, pre-trained on perspective images, to our panoramic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For this purpose, a double-stage adaptive refinement scheme for panoramic images (PanoDARS), based on [43], has been proposed (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Guided by Ds t, PanoDARS utilizes a depth estimation baseline and iteratively refines the predicted depth map Dt ∈ Rw×h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The selected baseline, Monodepth2 [34], was originally trained on perspective images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' however, using PanoDARS, the predictions on panoramic images are sig- nificantly refined in the inference time and without any need for finetuning or re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' PanoDARS consists of two stages, correction and op- timization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In [43], the depth maps are split into three horizontal slices, and the correction values in each slice are calculated independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' While in the proposed method, the correction stage skips slicing, since the LiDAR sparsity pat- terns are different in panoramic and perspective geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Moreover, PanoDARS estimates depth only in the regions of It that the sparse depth is available close to them (ORt, as shown in a sample in the left image of Figure 3, rainbow colored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In the first stage, a correction value δds t ∈ ∆Ds t between each valid pixel in the sparse depth map ds t ∈ Ds t and its correspondence dt ∈ Dt is calculated using δds t = dt − ds t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Then, an interpolation function Q : R2 �→ R based on Delaunay triangulation [39] is leveraged to obtain a dense correction map ∆Dt = Q(∆Ds t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Finally, the corrected depth map ˆDt = Dt + ∆Dt is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' ˆDt is a sufficiently accurate initial value for the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In the second stage, while the pre-trained weights are fixed, some learnable auxiliary parameters are applied on intermediate features in the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Optimizing those pa- rameters, the predicted depth is refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Therefore, the overall performance of PanoDARS is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Given an input RGB image It ∈ Rw×h×3, PanoDARS splits the pre-trained baseline M : Rw×h×3 �→ Rw×h into a body G : Rw×h×3 �→ Ra×b×c and a head H : Ra×b×c �→ Rw×h, where a, b, and c are respectively width, height and number of channels of the intermediate feature set G(It) ∈ Ra×b×c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The auxiliary parameters, scales S ∈ Rc and biases B ∈ Rc are applied on G(It), and the depth Dt = H(S � G(It) � B) is predicted, where � and � represent channel-wise multiplication and addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Afterwards, the correction module C : Rw×h �→ Rw×h carries out the first refinement stage on Dt and returns ˆDt = C(Dt, Ds t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The auxiliary parameters X ∈ R2c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=', concatenated channel-wise scales (S) and biases (B), are learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' There- fore, the following optimization problem can be formulated as L(ˆDt(It, X + ∆X), Ds t) → min ∆X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' where ∆X is the correc- tions applied on the parameters and L is the cost function given to particle swarm optimizer (PSO) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Hence, the second stage of refinement is conducted by PSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Tracking Module The tracking module runs in its own thread and has a close collaboration with the mapping module described in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' From the beginning of a SLAM run, the tracking module reads ft for each t in sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The matching module inside the tracking module will find the matches between ρt−1 and create the array of matched map points Pt with Ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The array Pt has the same size with Ot (mt) and can contain Null values in some indices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For each matched pair of oi t and a map point from ρt−1, Pt has a pi t pointing to the map point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If oi t do not match with any map point from ρt−1, the corresponding index in Pt equals to Null (pi t = Null).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Using these local matches, the tracking module can esti- mate jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Then, in certain conditions that the system realizes Apply scale & bias Pre-trained Monodepth2 Correction Channel-wise Scale (S) & Bias (B) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0 … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='07 … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='8 … 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0 … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='00 … 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='5 … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='27 … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='6 Particle 1 Particle 2 Particle n Optimizer inference input optimization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' 2: The proposed method for depth estimation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' the tracking will perform weakly in the future with the estimations of jt based on a small number of map points, the system decides a frame ft to be a keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Keyframes are frames, the only difference being that the mapping and loop closing modules are working with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For example, at time t = 1 the tracking module is not effective since ρ0 is an empty set, and it will turn ft to a keyframe so that the mapping module continues with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We can partition the set F based on them being keyframes or not keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' We name the resulting subset containing all keyframes KF and the other subset non-keyframes NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Mapping Module The mapping module, like the tracking module, has its own thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' When the tracking module decides ft to be a keyframe, the mapping module creates new map points using ft and puts them together with ρt−1 to save all of them in ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The module attempts to create a map point for each oi t that its corresponding pi t is Null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If oi t ∈ ORt, it creates a map point using oi t, ˆDt(oi t), and jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If oi t /∈ ORt, it will triangulate ft with the nearest neighbouring keyframe fs to estimate the depth for oi t and create the map point with the same set of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If triangulation is not successful, the module will not create a map point for oi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The mapping module is also responsible for local bundle adjustment, which is accomplished by following [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Depth Association Module In the tracking thread, the depth association module is called after the tracking module tracks all the visible map points in the current frame ft and estimates jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This module helps to make use ˆDt even if ft is not a keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' And we call it hybrid depth association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Algorithm 1 shows the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' It inputs ft and the tracked array of map points Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For each i, if the matching module has not assigned any map points to oi t or if oi t is out of ORt, it skips the i and goes for the next member of Oi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If oi t passes these conditions, then a new pi t ∈ R3 will be created using oi t and its depth value and the position and orientation of the camera at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' But, sometimes the tracking module can make mistakes in assigning ORB visual features to the corresponding map points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' So we have defined a rejection threshold θ to prevent big mistakes and their effect on the whole trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If the distance between the current position of pi t and the newly calculated new pi t is bigger than θ, we will not change anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' If not, then we have defined two policies here: If pi t has been created using triangulation and not modified using any depth map later, the algorithm will change the location of pi t to new pi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Or, if the depth value of the visual feature that pre- viously has modified (or created) the position of pi t (ˆDp(oi p)) is larger the current depth (ˆDt(oi t)), again the algorithm will change the location of pi t to new pi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Figure 3 shows an example of this algorithm for the first policy with a simple scenario where we only have one ORB feature in each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' And the second policy helps with the precise estimation of the position of the map points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Each time a map point is observed with a smaller depth value, the algorithm recalculates its position based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Algorithm 1 Depth Association Module Algorithm Input: Current frame ft and the tracked array of map points Pt 1: θ ←The rejection threshold 2: for Each pi t in Pt do 3: if pi t = Null or oi t /∈ ORt then 4: continue 5: end if 6: fp ←Last keyframe who created or modified pi t 7: new pi t ←Estimated 3D position using oi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' ˆDt(oi t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' and jt 8: if distance(pi t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' new pi t) <θ then 9: if (pi t created using triangulation and not modified after that) or (ˆDt(oi t)<ˆDp(oi p)) then 10: pi t’s position ← new pi t 11: Update pi t in ρt and all the previous subsets 12: end if 13: end if 14: end for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Loop Closing Module Finally, after the loop closing module detects a loop, it performs the global bundle adjustment on the trajectory inside the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The global bundle adjustment performs a pose-graph optimization on jt of the camera in each ft ∈ D一一cost function Lt - 4 t t + 3 Frame Keyframe Visual feature Visual feature and matched map point Triangulation Trajectory: Time: o 1 t-4 o 1 t p 1 t o 1 t& p 1 t+3 o 1 t+3& The same map point in ρ OR t-4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' 3: A map point p1 t is created at time t by triangulating visual features of times t−4 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This triangulation is because these two visual features are not in ORt−4 and ORt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In ft+3, o1 t+3 points to the middle of the parked car’s rooftop and matches with the previously created p1 t at time t which is now called p1 t+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' At time t + 3, o1 t+3 is inside ORt+3, which means the depth data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This enables the system to estimate the map points’ 3D position more precisely through the depth coming from depth estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' {the subset of KF in the loop}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' And also it optimizes all of the map points created in those keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' For the pose- graph optimization, we have used a similarity transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' EXPERIMENTAL RESULTS AND DISCUSSION In this section, initially, a brief overview of the dataset is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Afterwards, we demonstrate our experiments in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' First, the effectiveness of each of our modules is studied and the best setting for the proposed method is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Second, we the results of the proposed and competing SLAM systems on all the sequences of YUTO MMS dataset are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Dataset Characteristics In this study, we used York University and Teledyne Optech (YUTO) MMS dataset, and the acquisition details can be found in a reference [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='The dataset has been acquired by Teledyne Optech’s Maverick MMS with four sequences in various outdoor environments with an 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 km long road in York University’s Keele Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Table I shows the results along with characteristics of each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The results of the percentage column conveys the fact that, on average, the base work uses only one-fourth of the available depth maps to create map points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' SLAM Trajectory Results In this section, first an ablation study was conducted on Sequence B of the dataset (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Then, the proposed method was compared with the competing methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=', Google Cartographer [40] and RPV-SLAM [9] (see Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The ablation study aims to find the optimum value for θ in Algorithm 1 and also to evaluate the effectiveness of depth association and depth estimation modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Bi-interpolation described in [9] has been used as a rival to the depth estimation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' According to Table II, in the absence of depth association module, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=', first three rows, RPV-SLAM and the proposed method show a similar performance in terms of both ATE and RTE, yet significantly better than Cartographer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Further, it proves that bi-interpolation and depth estimation module have no superiority on each other, when there is no depth association module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Considering different values of θ, bi- interpolation and depth estimation module illustrate a similar behaviour, where increasing θ leads to worse ATE and RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Moreover, the best accuracy for both densification methods was obtained using θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' As the Table II suggests, when depth association mod- ule was utilized (with θ), more accurate results were ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In addition, depth estimation module outperforms bi- interpolation in both ATE and RTE, given identical values for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Regardless of selected depth densification methods, θ = 2 shows the best performance in both ATE and RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' In conclusion, we can attain the best performance when the depth estimation module and θ = 2 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' It means the depth estimation and depth association module have contributed to the improvement of the SLAM performance in both ATE and RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Table III shows the results of our best setting (depth estimation module with θ = 2) in comparison with Google Cartographer [40] and RPV-SLAM [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Google Cartographer is a LiDAR-centric SLAM that is also equipped with IMU and RPV-SLAM is our base work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Overall, relatively poor results are obtained on residential areas due to unfavorable illumination conditions such as shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' As expected, the TABLE I: YUTO MMS Dataset Characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Region Distance travelled Running time Total #frames Average #keyframes Percentage Sequence A Parking lot 324m 94 seconds 717 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='8% Sequence B Campus area 7035m 19 minutes 8382 2306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='51% Sequence C Residential area 7965m 22 minutes 10778 2576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='9% Sequence D Residential area 3634m 10 minutes 4500 1117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='84% TABLE II: SLAM trajectory results for ablation study on Sequence B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' SLAM Densification method θ ATE (m) RTE (%) RRE (◦/m) Cartographer N/A N/A 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='97 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0093 RPV-SLAM bi-interpolation N/A 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 Proposed method depth estimation module N/A 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 Proposed method bi-interpolation 1 meter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 Proposed method bi-interpolation 2 meters 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 Proposed method bi-interpolation 3 meters 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 Proposed method bi-interpolation 4 meters 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 Proposed method bi-interpolation 5 meters 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 Proposed method depth estimation module 1 meter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 Proposed method depth estimation module 2 meters 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0011 Proposed method depth estimation module 3 meters 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 Proposed method depth estimation module 4 meters 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 Proposed method depth estimation module 5 meters 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0010 TABLE III: SLAM trajectory results for all the sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Parameters of HDPV-SLAM has been set based on Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' SLAM Sequence ATE (m) RTE (%) RRE (◦/m) Sequence ATE (m) RTE (%) RRE (◦/m) Cartographer A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0507 C 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0133 RPV-SLAM A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0268 C 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0042 Proposed method A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0041 C 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0034 Cartographer B 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='97 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0093 D 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0137 RPV-SLAM B 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0009 D 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0016 Proposed method B 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='0011 D 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content='001 parking lots sequence produced the best ATE performance because of its shorter length and lower scene complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Furthermore, the largest improvement in ATE was achieved in Sequence C due to the relatively shorter LiDAR ranges in residential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' To conclude, HDPV-SLAM produced the best results compared to Cartographer and RPV-SLAM over all the test sequences in terms of both ATE and RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Although the performance of HDPV-SLAM varies depending on the sequence, the other SLAM systems follow a similar pattern in their performances as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Discussion As seen in Table II, the proposed technique outperforms the alternatives in terms of ATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' ATE is a metric that com- pares the entire trajectory to the ground truth and handles the form matching between them, indicating that the proposed method preserves the shape more effectively than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Furthermore, in three-quarters of the dataset, the RTE and RRE of the proposed technique are superior to those of the competing methods, and in one-quarter, they are second best by a slight margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' All of these comparisons demonstrate that the proposed strategy is superior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' CONCLUSIONS In this study, we present a novel HDPV-SLAM system with two modules, namely, depth estimation and depth association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' These two help bundle adjustment perform in a better way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This system generated superior metrically-scaled results compared to rival methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Our work can be extended using these improvements in the future: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' The depth association module can further contribute to the outcomes by extending depth estim ation to the entire panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' Using depth estimation and successive frames, identify the dynamic items in the scene and filter out the visual characteristics observed on these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' This can reduce errors, as conventional SLAM systems are susceptible to dynamic scene objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' ACKNOWLEDGEMENT This project is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Collab- orative Research Development (CRD) and Teledyne Optech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FKT4oBgHgl3EQfeC7T/content/2301.11823v1.pdf'} +page_content=' REFERENCES [1] B.' 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diff --git a/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/2301.02906v1.pdf.txt b/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/2301.02906v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d08c3044784dd914e55713fe6735211bf4825b15 --- /dev/null +++ b/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/2301.02906v1.pdf.txt @@ -0,0 +1,1903 @@ + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +1 + +Abstract— Continuous monitoring of inter-beat-interval +(IBI) and heart rate variability (HRV) provides insights in +cardiovascular, +neurological, +and +mental +health. +Photoplethysmography (PPG) from wearables assures +convenient +measurement +of +IBI. +However, +PPG +is +susceptible to motion artifacts, considerably deteriorating +the accuracy of IBIs estimation. Although a multi-channel +model in previous study improves accuracy, prevailing +compact commercial wearables would favor single-channel +sensors, causing benefits of multi-channel applications to +have restrictions. In this paper, a greedy-optimized +framework is proposed for measurement of IBI and HRV +featuring single-channel and multi-channel PPG signals +collected during daily activities. Utilizing the fact of +continuity in heartbeats, the IBI estimation problem is +converted into the shortest path problem in a directed +acyclic graph, where candidate heartbeats from the noisy +PPG are regarded as vertices. The framework exploits a +convex penalty function to optimize weight assignment in +the shortest path calculation and a greedy-optimized fusion +method to mitigate overly fluctuating patterns in estimated +IBIs. The results achieve correlation of 0.96 with percentage +error of 3.2% for IBI estimation using single-channel PPG +signals from the 2015 IEEE Signal Processing Cup dataset, +where percentage error is reduced by 58.4% and correlation +is improved by 11.6% in comparison to those without +greedy-optimized fusion. In the multi-channel model, it +achieves correlation of 0.98 with percentage error of 2.2%. +Estimated and true HRV parameters are also highly +correlated with low percentage errors. This paper further +validates these techniques on the PPG-DaLiA dataset, +indicating the robustness of the proposed framework. +Index Terms— Daily Activities, Heart Rate Variability, +Motion Artifacts, Wearable Physiological Sensing +I. INTRODUCTION +EMOTE healthcare monitoring is increasing in popularity +nowadays through more and more emerging wearable +devices. These commercialized wearables promote self-health +management, which benefits the engagement of patients and the +quality of medicine. Continuous cardiovascular activity + +Luffina C. Huang was with the Department of Computer Science and +Engineering, Texas A&M University and she is now with the Department of +monitoring is one important focus for self-health management. +Among various physiological parameters, average heart rate +(HR) and heart rate variability (HRV) are significant parameters +in continuous cardiovascular monitoring. Average HR, the +measurement of the number of heart beats per minute in a +certain time period, is one of the vital signs routinely monitored +by healthcare providers and serves as an important indicator of +cardiovascular health, such as hemodynamic stability and heart +rhythm. HRV, however, could provide integrated information +of the cardiovascular system and the autonomic nervous +system. Interbeat intervals, IBIs, are the time elapsed between +two successive heart beats. HRV quantifies the variability of +IBIs in a certain period of time and is widely used as a crucial +indicator in healthcare research and clinical practice. HRV +parameters could evaluate the sympathetic and parasympathetic +activity of the autonomic nervous system, which controls heart +rate and blood pressure in response to dynamic physiological +changes, such as respiration, exercise, physical stress and +mental load [1]. Further, low HRV, reduced level of beat-to- +beat heart rate fluctuations, is not only independently associated +with a 32–45 % increased risk of first fatal and non-fatal +cardiovascular disease (CVD) events but also a prognostic +factor with higher mortality in patients with a CVD event [2, 3]. +In addition, elevated HRV has a protective effect in reduction +of CVD events, which could be enhanced through increased +physical activity and aerobic exercise training. A study has +shown the 1% increase in a HRV parameter, standard deviation +of the normalized NN interval (SDNN), leads to a roughly 1 % +reduction of fatal or non-fatal CVD events [1, 2]. +IBI +and +HRV +could +be +derived +from +both +Electrocardiography (ECG) and Photoplethysmography (PPG) +[1, 2]. Traditionally, the gold standard of IBI and HRV +measurements is multi-lead ambulatory ECG. Although +ambulatory ECG provides the possibility of out-of-hospital +monitoring, it requires setup by specialized technicians and +needs to attach multiple electrodes to the chest skin, which is +not comfortable to wear for a long period of time [1, 3]. Given +the recent trend for integrating health assessments into wearable +technologies, more and more commercialized wearable devices +are equipped with single-lead PPG sensors. PPG-based +wearables with single contact point have become widely +Electrical and Computer Engineering, Rice University, Houston, TX 77005 +USA (e-mail: luffina.c.huang@rice.edu). +A Greedy-optimized Framework for Heart Rate +Variability Monitoring during Daily Activities +using Wearable Photoplethysmography +Luffina C. Huang +Department of Electrical and Computer Engineering, Rice University, Houston, Texas, USA +E-mail: luffina.c.huang@rice.edu +R + + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +2 +accessible due to the advantage of low-cost, non-invasive, and +easy to use, which makes them a convenient and practical tool +for continuous IBI and HRV monitoring in daily life, served as +an alternative to the standard ECG [1, 2, 4]. Through the +enhancement of continuous monitoring of HRV, wearable +technologies open a new window of remote healthcare +monitoring and the trend of self-health management. +PPG is an optical biomonitoring technique that emits single +or multi-wavelength light by LED to penetrate the skin and +blood vessels and captures the reflected light by photodiodes to +measure blood volumetric changes in microvascular tissue at +fingers and wrists [5]. Featured physiological parameters +related to the cardiopulmonary system could be estimated via +PPG, such as blood oxygen saturation (SpO2), average heart +rate (HR), respiratory rate (RR) [6], and blood pressure (BP) +[7]. Furthermore, PPG-based techniques can achieve highly +accurate HRV estimation in stationary conditions, such as +sitting, rest and supine. Studies have shown the IBIs and HRV +parameters derived from PPG wearables are highly associated +with those derived from ECG signals (correlation coefficient +ranged from 0.85 to 0.99) [4]. However, motion artifacts are an +inherent problem when applying PPG-related techniques to +healthcare monitoring in free movement condition. The motion +artifacts in PPG degrade the accuracy of IBIs/HRV estimation +as the level of physical activity increases. The frequency +spectrum of motion artifacts ranges from 0.01 to 10 Hz, which +overlaps with the normal frequency of PPG signal (0.5 - 5 Hz). +Therefore, it is not easy to denoise the motion-contaminated +PPG signal by applying general filtering techniques [8]. +Average HR exhibits more consistency over time compared to +noises and HRV. It is easier to attain average HR from noisy +PPG, whereas estimating HRV is challenging during intensive +physical activity [9]. Hence, there is an unmet need to develop +accurate algorithms for HRV estimation from PPG wearables. +This paper proposes a greedy-optimized framework to tackle +the aforementioned unmet need, which transforms IBI +estimation into a shortest path problem in directed acyclic graph +subsequently combined with a greedy-optimized fusion method +for +morphological +features +extracted +from +motion- +contaminated PPG. I use the physiological property of the +temporal continuity of heartbeats (i.e., the end of one heartbeat +is the start of the next heartbeat) and construct a directed acyclic +graph, where the vertices represent the feature candidates (i.e., +heartbeats) and the edges represent the candidate IBIs. Shortest +path algorithm is then used to remove noisy feature candidates +and calculate accurate IBIs. The proposed convex penalty +function for edge weight assignment is designed to augment the +power of the shortest path algorithm with increments of the +accuracy in IBI estimation. Subsequently, the greedy-optimized +fusion method is developed to optimize the process of selecting +estimated IBIs from the three morphological features, systolic +peaks, maximum slope, and onset points, which are extracted +from motion-corrupted PPG. Through this fusion strategy, it +mitigates the inherent overly fluctuating patterns of estimated +IBIs from noisy PPG signals and calculates accurate IBIs and +HRV. The advantage of the greedy-optimized framework is that +it could process both single-channel and multi-channel motion- +contaminated PPG signals with computational efficiency. +Hence, it could be adaptive with prevailing commercial +wearable +devices, +which +have, +in +general, +limited +computational capacity. In summary, the contributions of this +article are as the follow: +1) A greedy-optimized framework is developed to attain +high accuracy of IBI and HRV estimation with the +capacity of processing single-channel and multi-channel +motion-contaminated PPG. +2) The convex penalty function could optimize weight +assignment to augment the power of the shortest path +algorithm and the greedy-optimized fusion method is +developed for mitigating overly fluctuating patterns of +estimated consecutive IBIs. +3) Performance is validated on two public datasets, IEEE +Signal Processing Cup and PPG-DaLiA, which have +noisy PPG from wearables collected during intensive +exercise and daily activities, indicating the robustness of +the proposed framework. +II. RELATED WORKS +A. Average Heart Rate Estimation +In past decades, research works about HR estimation for +wearable PPG signals have quite matured. Several studies have +demonstrated that high accurate estimation of average HR +during intensive physical exercise can be achieved by +incorporating the accelerometer signals with a variety of motion +artifact reduction framework using single-channel PPG, such as +TROIKA [10], WFPV [11] and particle filtering [12]. Other +studies have attempted to leverage multi-channel PPG that used +truncated singular value decomposition (SVD) or template- +matching algorithm for accurate average HR estimation and +showed the multi-channel estimation outperformed the single- +channel estimation during intensive exercise [13, 14]. These +techniques, nevertheless, were not able to be adapted for IBIs +and HRV estimation. + +B. Heart Rate Variability Estimation +IBI/HRV have extensive physiological applications in +clinical practice, but it is more challenging to attain accurate +IBIs from wearable PPG sensors. Initially, some studies began +to demonstrate the feasibility and good performance of IBI and +HRV estimation using wrist-worn PPG sensors in post- +anesthesia patients and in healthy volunteers during sleep [15, +16]. Although these studies have shown satisfactory small +absolute errors of IBI and HRV parameters between wrist-worn +PPG sensors and ECG, most of their PPG signals do not have +motion artifacts distortion. One study has shown that although +the good association (correlation coefficient 0.74 - 0.88) +between wrist-worn PPG sensors and ECG in HRV parameters +was achieved in baseline rest condition, the correlation was +degraded to 0.42 - 0.67 when subjects were talking [17]. One +work has benchmarked the HRV parameters for different ages +and genders using a dataset of 8 million users, which is the +largest to date. It applied noise spikes cleaning algorithms and +achieved high correlation (0.97) in HRV parameters during a +randomly selected 24 hours period [3]. However, these above +studies eliminated all PPG signals that were corrupted with +motion artifacts from their HRV analysis. There are some +studies exploring accurate R peaks detection and IBI estimation +in noisy ECG signals using deep learning models. Vijayarangan + + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +3 +et al. has proposed a novel application of the IncResU-Net, a +fully convolutional Encoder-Decoder architecture, to detect R- +peak from the ECG. The model could provide good +performance in R-peak detection in ECG with noise level up to +0 dB [18]. However, deep learning-based methods require a +very large amount of data for computationally expensive +training with GPUs. Furthermore, these techniques have not +been investigated in noisy PPG signals yet. One study works on +single-channel motion-corrupted PPG using combinatorial +algorithms. The model leverages the shortest path algorithm +with exponential function, which presents a medium-high (0. 82 +to 0.86) correlation between the PPG sensors and ECG [19]. +However, this study only provided the average results. No +breakdown of subjects and no accuracy evaluation was reported +for both IBI and HRV estimation. +C. Fusion of Physiological Signals +Fusion approaches have been explored to improve the +accuracy of heartbeats detection by incorporating the +information across different physiological signal modalities or +multiple morphological features. One fusion approach is signal +switching, where the candidate fiducial points from a signal +modality with the best signal quality are selected as final +fiducial points in a certain segment. Singh et al. used the sample +entropy to assess the noise content in multiple signal modalities, +such as ECG and arterial blood pressure (ABP) signals, and +switch between them to enhance the accuracy of heartbeat +detection [20]. Aygun et al. obtained the best set of IBI arrays +from three PPG morphological features by selecting those +segments with minimal standard deviation of IBI subarray [19]. +Some studies explored voting method for fusion, where the +candidate fiducial points detected in each signal modality cast a +vote to select final fiducial points for a certain segment. In the +majority voting, the fiducial points that have most agreement +among different signal modalities are selected as the final +fiducial points [21]. Furthermore, the vote could be weighted +by the signal quality index or other evaluation metrics to select +fiducial points with best quality [22]. Some fusion methods are +based on sophisticated probabilistic models. For example, Zia +et al. decoded the waveform segments of ECG and ABP into +different hidden states in a Hidden Markov Model. Followed by +this, the authors employed a Bayesian Network to model the +relationship of the hidden states of the ECG, ABP and +classification, which indicates output of the QRS segment [23]. + +III. Method +In this work, a greedy-optimized framework is proposed for +IBI and HRV estimation with the capacity of processing +motion-contaminated PPG. The overview of the proposed +framework is shown in Fig. 1, which consists of two main +models, shortest path calculation and greedy-optimized fusion +method. In the preprocessing stage, PPG signals are upsampled +and filtered. Then morphological features of a cardiac cycle in +PPG signals are extracted as the fiducial points to represent the +heartbeats. Firstly, the heartbeat detection is modeled as the +shortest path problem that aims to differentiate the true +heartbeat from the noise spikes in noisy PPG. A graph is +constructed where the vertices represent all potential heartbeats, +and the edges represent all the candidate interbeat intervals. A +convex penalty function is proposed to optimize weight +assignment in the shortest path algorithm. The time difference +of two consecutive heartbeats selected by the shortest path +algorithm is regarded as an estimated IBI. Above processes +could be applied in single-channel and multi-channel PPG +signals, respectively. Secondly, a greedy-optimized fusion +method is introduced to utilize the complementary information +of the three IBI arrays estimated from the three different +morphological features to further improve accuracy of IBI +estimation. Note that both the shortest path calculation and the +greedy-optimized fusion method are generalized and could be +adopted into different signal modalities such as PPG, ECG or +other physiological signals. +A. Morphological Feature Extraction +In ECG signals, the QRS complex represents the +depolarization and contraction of ventricles, pumping blood +into vessels going to the body and lungs. The R-peak is the most +prominent positive peak of R-wave in the QRS complex. The +time elapsed between two consecutive R-peaks, called R-R +interval, is the most common and standard way to calculate the +IBIs in ECG. To strengthen the R-peak detection in ECG +signals, ECG signals are preprocessed by a continuous wavelet +transform (CWT) using Mexican hat with a center frequency of +0.25Hz [12]. The peaks detected from the wavelet are regarded +as ground truth R-peaks and the calculated IBI from those peaks + +Fig. 1. Overview of greedy-optimized framework for noisy PPG signals + +PPG1 +PPG2 +Upsampling +Band pass filter +IBls (Sys) +Fusion +IBis (Ms) +IBls (Onset) +Greedy +Convex Penalty Function +Average HR +Upsampling +Band pass filter Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +4 +are regarded as the ground truth for metric performance +evaluation. +The PPG waveform of a cardiac cycle is commonly divided +into two phases: The anacrotic phase is the rising edge of the +waveform, whereas the catacrotic phase is the falling edge of +the waveform as shown in Fig. 2. The anacrotic phase is +primarily associated with systole (heart contraction), which is +the most crucial physiological function of heart activity. +Systolic peaks, maximum slopes and onset points are important +morphological features that characterize the systolic waveform +of a cardiac cycle in PPG signals [24]. The time elapsed +between two consecutive systolic peaks in PPG signals is +referred to as Peak-Peak interval, whereas the time elapsed +between the onset and the end of the PPG waveform is referred +to as Pulse interval, as shown in Fig. 2. Some studies observe +that the Peak-Peak interval in PPG signal is highly correlated +with the R-R interval in the ECG signal [25]. Other studies +show that the HRV from the Pulse interval in PPG signals are +highly correlated with the HRV from R-R intervals in ECG +signals. Both Peak-Peak interval and Pulse interval have been +used to detect the heart rate and HRV in stationary condition +[26, 27]. Maximum slope, which is centered between systolic +peak and onset in PPG signals, has also been applied to HRV +estimation in intensive physical activity [19]. Therefore, these +three features are used for IBI estimation in this study. +Following steps are the process of extracting these features, +which are regarded as candidate fiducial points in this study. +First, the filtered PPG signals are smoothed by a 5th order +smoothing spline. Then, a general peak detection algorithm in +SciPy detects the local maxima of PPG and ECG signals to +obtain the systolic peak candidates of PPG and R-peak +candidates of ECG, respectively [28]. Then, I use the same +strategy in a previous study to extract the local maximum of the +first derivative and second derivative of PPG signals as the +maximum slope candidates and onset candidates, respectively +[9]. All pairs of fiducial points in the same morphological +features compose the set of candidate IBIs. However, many of +the candidate fiducial points are induced by motion artifacts, +which leads to false IBIs and therefore need to be eliminated. +B. Graph Modeling and Shortest Path Calculation +To improve IBI estimation accuracy, the false fiducial points +need to be removed whereas fiducial points representing true +heartbeats need to be selected. Based on aforementioned +features, a weighted graph is constructed, where vertices +represent candidate heartbeats and edges represent candidate +IBIs. The shortest path calculation is then used to find the real +IBIs and filter out those induced by motion artifacts. According +to a previous study, the multi-channel PPG model outperforms +the single-channel PPG model in IBI estimation during +intensive physical activity [9]. However, prevailing commercial +wearable devices favor a single-channel PPG sensor due to the +need for portable and compact [3]. Hence, this study +investigates single-channel and multi-channel models for PPG +signals as below. +1) Graph Construction with Single- and Multi-Channel +Firstly, three directed acyclic graphs are constructed using +candidate fiducial points extracted from three morphological +features, respectively. In the graph, the vertices are marked as +the candidate fiducial points while edges are designed as +candidate IBIs. Vertices are denoted as 𝑣!, 𝑖 = 1, 2, … , 𝑁, where +𝑁 is the total number of vertices in the graph, and their values +are equal to their timestamp. A time interval 𝑡! prior to each +vertex 𝑣! with the range of 1.5 folds of its average IBI is +considered to identify neighbors of 𝑣! , where *𝐼𝐵𝐼"#$-! = +6000 (𝐻𝑅"#$)! +⁄ + [𝑚𝑠] . Vertices within this interval are +considered as neighbors of the vertex 𝑣!. Edges are formed +between vertex 𝑣! and its neighbors and denoted as 𝑒!%, where +vertices 𝑣% are 𝑣!’s neighbors. The value of an edge is assigned +as the time difference of the two connected vertices and herein +represents the candidate IBI. (𝐻𝑅"#$)!, the average heart rate +of vertex 𝑣!, is equal to the average heart rate of the 8-second +PPG window which is closest to 𝑣!. Average heart rates are +estimated from PPG1 using the WFPV algorithm by every 8- +second window with 6-second overlap [11]. Following the +above steps, the graph construction of the single-channel model +is completed. +To model the multi-channel PPG in the graph, I exploit the +same observation in a previous study that false fiducial points +included by noise would have a bigger time gap between PPG1 +and PPG2 than the true fiducial points [9]. Based on this +observation, the vertices from different channels are labeled +with different colors to differentiate its origin. For example, +vertices from PPG1 are labeled blue whereas vertices from +PPG2 are red, as shown in Fig. 1. Then, vertices from two +channels are concatenated and sorted by timestamps. The multi- +channel graph model is constructed by the above steps. +2) Weight Assignment by Convex Penalty Function +The shortest path algorithm is used as the graph search +algorithm to attain the correct IBI path. The weight of each edge +is assigned based on their deviation from the average IBI and +the weight of each vertex is accumulated from the start vertex +to the current one. Therefore, an effective penalty function +would be crucial for assigning edge weights of the graph [9]. +There are many directions to design customized penalty +functions, which might bring different debates. One approach is +a tolerance-based penalty function, where edge weight is not +penalized until certain criteria meet. Another approach is to +penalize the edge weight by a convex function. In this study, I +examine the concept by proposing a sigmoid penalty function +as a good representative example, shown in formula (1). +𝑤!" = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑 )min )-.𝐼𝐵𝐼#$%1! − 𝜀 − 𝑒!"- , -.𝐼𝐵𝐼#$%1! + 𝜀 − 𝑒!"-77 +(1) + +Fig. 2. Standard PPG waveforms +Systolic +Peak +Peak-Peak interval +Pulse interval +Onset +Anacrotic +Phase +Catacrotic +Phase + + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +5 +where e&' is the time difference between i’ th and j’ th vertex. +The sigmoid penalty function penalizes the edges which are out +of the range of *𝐼𝐵𝐼"#$-! ± 𝜀 with the sigmoid function. +Otherwise, it will assign zero to w&' if 𝑒!% is within the range of +*𝐼𝐵𝐼"#$-! ± 𝜀 . This approach adapts an intuition that the +average IBI with a tolerance of 𝜀 could be a good indicator of +the time difference between any pair of true fiducial points [19]. +However, the sigmoid penalty function in this example has a +deficiency of assigning weights. There is no weight discrepancy +for those edges inside the tolerance interval, even though there +are time differences among them, leading to the difficulty of +reaching the optimum during the vertex selecting process for +the shortest path algorithm. For those edges outside of the +tolerance interval, the edge weights are prone to be +underestimated with a plateau curve. Moreover, the empirical +results of tolerance-based penalty functions are highly +dependent on the epsilon 𝜀. Furthermore, an exponential +penalty function proposed in a previous study has similar +situations for the edges inside the tolerance interval [19], but for +those edges outside of the tolerance interval, the edge weights +are prone to be overestimated with enormous scale [9]. + An effective penalty function needs to guarantee discrepancy +for those edges inside the tolerance interval which have time +differences and also reflect cardiac physiology that true IBIs +would not have drastic changes in such a short time and should +be close to their average IBIs. To provide a solution, a convex +penalty function is proposed and calculated through (2)-(4) as +shown in a previous study [9]. +𝑣! +" = 𝑣! − $𝐼𝐵𝐼#$%'!, 𝑖 = 1,2. . , 𝑁 +(2) +𝑑!& = 1𝑣& − 𝑣! +"1, 𝑗 = 𝑖 − 1, … , 𝑖 − 𝑚! +(3) +𝑤!& = 𝜆 𝑑!& +', 𝑥 ∈ ℕ +(4) +Firstly, 𝑣’! is marked as the expected previous vertex of 𝑣!, +which is calculated by subtracting the average IBI from each +vertex 𝑣!. Secondly, the distance 𝑑!% is calculated between 𝑣! +( +and 𝑣%, where 𝑣% are 𝑣!’s neighbors and 𝑚! is the total number +of neighbors of vertex 𝑣!. Finally, 𝑤!%, the weight of the edge +that connect 𝑣! to neighbor 𝑣% is measured from the power +function. The power function raises the 𝑑!% to the power of 𝑥 +with a constant parameter 𝜆, where the power can be 1 or any +even positive integers. For example, 𝑥 is assigned as 2 in this +study. The edge weights assigned by the convex penalty +function grow smoothly compared to the exponential penalty +function and sigmoid penalty function. Furthermore, the strictly +convex property helps the shortest path algorithm approximate +optimal solutions and avoids potential overflow error in +numerical computation. +3) Shortest Path Calculation +Since the end of each heartbeat is the beginning of its next +heartbeat and they are continuous in time domain, the shortest +path is then used to select the fiducial points that correspond to +true heartbeats. After constructing the weighted graph, the +shortest path algorithm is applied on this graph and then the +path with the least total weight is chosen [19]. The weight of +vertex 𝑣! is assigned by finding the minimum of the weight of +previous neighbors plus the weight of edges connected between +them. The previous neighbor that contributes to the minimum +is selected as the previous vertex of 𝑣! and denoted as 𝑝𝑛! , as +shown in Algorithm 1. +C. Greedy-optimized Fusion for Various Shortest Path +One IBIs array is produced from one of three morphological +features. Consecutive IBIs, however, would not be estimated +precisely from motion-corrupted PPG during intensive daily +activities. The beat-to-beat IBI plots depict that the estimated +IBIs have overly fluctuating patterns, as compared to the true +IBIs from ECG. Further, estimated IBIs arrays derived from +different morphological features have different estimated time +lengths, even if they come from the same heartbeat, due to the +difficulty of extracting the true fiducial points from highly- +distorted PPG signals. Although the multi-channel model could +improve the accuracy of estimated IBIs, most prevailing +commercial wearable devices usually favor practical single +channel PPG sensors. A greedy-optimized fusion technique for + +Fig. 3. Greedy-optimized fusion method for various shortest path +utilizing morphological features + + +Systolic +Peaks +Maximum +slope +Onset +Fusion of +3 features +11误122误Algorithm 1 Shortest Path Detection for PPG Signals +Input: Candidate fiducial points from PPG signals +Output: the set of chosen vertices Vchosen +l: <> +2: Concatenate all fiducial points from multiple-channel PPGs and sort them +by timestamp to form the vertex set V = [Ui, U2, ...,Un] // N is the total +number of vertices +3: // Neighbor selection process for vertex Vi +4: for Vi E V do +5: +j=i-l +6: +while Vi - Vj < 1.5 * (IBIavg)i do +7: +eii=Vi-Vi +8: +Ni ← Vi / /N; is the set of neighbor vertices of Vi +9: +j=j-l +10: +end while +11: +mi = i - j + 1 // mi is the number of neighbors of vi +12: +end for +13: +<< Shortest Path Detection >> +14: f +for i = 2 to N do +15: +for j = i- l to i-mi do +16: +Calculate the edge weight wii by convex penalty function (Eq.2 - Eq.4) +17: +end for +18: +Wi = min(wij +wi), for j = i- l, ...,i- mi / /wi is the weight of vertex +19: +pni = argmin(wij + wi), for j = i - l,..,i - mi / /pni is the chosen +VjENi +previous vertex that contributes the minimal weight for Vi +20: end for +21: Udst = +argmin (wN, wN-1, .., WN-mn) / /choose the vertex that has the +UjE[NN,UN] +minimal vertice weight within the t window as the destination of the short- +est path, Udst +22: Backward search from Udst to select the vertices on the shortest path to form +Vchosen +23: return +Vchosen Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +6 +IBI arrays from various morphological features is proposed in +this study to provide a solution for those challenges. +Due to the fact that the onset feature represents the beginning +of a cardiac activity, IBIs array from the onset feature is selected +as the baseline for segmentation. Firstly, the IBIs array of the +onset feature is divided into q segments where each segment +contains three consecutive IBIs. The timestamps of this +segmentation are used as references to guide the segmentation +of IBI arrays for maximum slope and systolic peak features. For +each segment, if the starting points of IBIs from maximum +slope and systolic features are within this segment, these IBIs +are included as the candidate IBIs. Secondly, based on a +physiological phenomenon that IBIs would not have drastic +changes in a short period, true IBIs are expected to be close to +their average IBIs. Hence, an objective function as equation (5) +is proposed for the greedy-optimized fusion method to find +local optimum IBIs in each step, or each segment. +𝚤̂, 𝚥̂, 𝑘A = argmin +!(&() $1𝑖*+ +! +− 𝑖,+ +- 1 + 1𝑖*+ +& − 𝑖,+ +. 1 + 1𝑖*+ +) − 𝑖,+ +/ 1' +(5) +The following is the process of greedy-optimized fusion on +three morphological features. The IBIs set generated from onset +are denoted as 𝐼), IBIs set from systolic peak as 𝐼*, and IBIs set +from maximum slope as 𝐼+. The pth segment of 𝐼), 𝐼*, and 𝐼+ +are denoted as 𝐼),, 𝐼*,, and 𝐼+,. The individual IBIs in 𝐼), are +denoted as 𝑖), +- , … , 𝑖), +. , where k must be equal to 3 for segments +in 𝐼), but k can be any integer number close to 3 for segments +in 𝐼* and 𝐼+, for example k can be 2 or 4. The set of candidate +IBIs in the pth segment are denoted as 𝐼/,. The set of average +IBIs in the pth segment are denoted as 𝐼0,. Take the pth segment +in Fig. 3 as example, the starting points of four IBIs from +𝐼* (𝑖*, +- , 𝑖*, +1 , 𝑖*, +2 , 𝑖*, +3 ) and three IBIs from 𝐼+ *𝑖+, +- , 𝑖+, +1 , 𝑖+, +2 - are +within the pth segment. Hence, candidate IBIs in the pth segment, +𝐼/,, is composed of F𝑖*, +- , 𝑖*, +1 , 𝑖*, +2 , 𝑖*, +3 , 𝑖+, +- , 𝑖+, +1 , 𝑖+, +2 , 𝑖), +- , 𝑖), +1 , 𝑖), +2 G. +The three IBIs in 𝐼/, that minimize the absolute error are +chosen as shown in equation (5) and are concatenated into the +final IBIs set 𝐼4. This process iterates through the q segments to +obtain the complete final IBIs set 𝐼4. +D. Algorithms and Complexity +For the shortest path detection (Algorithm 1), the main loop +in the algorithm runs 𝑁 ∗ 𝑚! times, where the outer loop runs +𝑁 times for 𝑁 vertices and the inner loop runs 𝑚! times for 𝑚! +neighbors of any vertex, 𝑣!. Since neighbors are selected from +a bounded window, which is 1.5-fold of average IBI (around +0.45-1.5 seconds), 𝑚! is assumed to be constant. Hence, the +complexity of shortest path detection is 𝑂(𝑁). For the greedy- +optimized fusion method (Algorithm 2), the main loop in the +algorithm runs q times, where q is the number of segments and +less than N/3. Since the number of candidate IBIs for each +segment is around 9, the time complexity to find the least +absolute error between the three IBIs and average IBIs is +regarded as constant. Hence, the complexity of greedy- +optimized fusion method is 𝑂(𝑁/3) = O(N). + +IV. RESULTS +A. Dataset and Data Preprocessing +I test the greedy-optimized framework on two datasets, the +2015 IEEE Signal Processing Cup training dataset (referred to +as IEEE_Training) and the PPG-DaLiA dataset to have a +comprehensive evaluation of performance during intensive +exercises and daily activities [10, 29]. The IEEE_Training +dataset emphasizes the lab-based controlled conditions whereas +the PPG-DaLiA dataset puts focus on daily life activities +naturally, close to real-life conditions. +1) IEEE_Training Dataset +Two-channel PPG signals (PPG1 and PPG2) from wrist- +worn sensors and one-channel ECG signal were collected +synchronously from 12 healthy individuals aged 18 to 35 while +they were running on the treadmill [10]. The running program +was set up as Rest 30s → Jogging 1 min → Running 1 min → +Jogging 1 min → Running 1 min → Rest 30s. Both ECG and +PPG signals are at a sampling rate of 125 Hz and upsampled to +500 Hz to attain higher frequency resolution. The up-sampling +could provide precise timestamps when extracting features in +ECG and PPG signals. Then, to eliminate the low frequency +trending and high frequency noises, the single-channel PPG +signals are preprocessed with a band-pass Butterworth filter +with a cutoff frequency of 0.5 Hz and 15Hz whereas multi- +channel PPG signals are preprocessed with a band-pass +Butterworth filter with a cutoff frequency of 0.7 Hz and 15Hz. +ECG signals are filtered with a high-pass Butterworth filter with +a 0.5 Hz cutoff frequency. +2) PPG-DaLiA Dataset +This dataset includes synchronized PPG and ECG signals +recorded from wrist-worn devices (Empatica E4) and chest- +worn devices (RespiBAN Professional), respectively [29, 30]. +Data was recorded from 15 subjects while performing different +kinds of daily activities as naturally as possible for 2.5 hours, +such as sitting, ascending/descending stairs, cycling, lunch +break and working. I use two intense physical activities, +ascending/descending stairs (5 mins) and cycling (8 mins), to +evaluate the performance of the greedy-optimized framework. +The PPG signals from the PPG-DaLiA dataset are upsampled +from 64 Hz to 500 Hz and filtered with a band-pass filter with +a cutoff frequency of 0.5 Hz and 15Hz. The true R-peaks of +ECG provided in this dataset are used to calculate the ground- +truth IBIs for performance evaluation. + + +Algorithm 2 Greedy-optimized Fusion for Various Shortest Paths +Input: IBIs set from systolic peaks : Is +IBIs set from maximum slope: IM +IBIs set from onset : Io +Average IBIs set : IA +Output: The final IBIs set IF +1: Divide Io into q segments such that each segment has 3 IBIs. +2: Divide Is and Im based on Io segmentation. +3: for p ← 1 to q do +4: +Candidate IBIs set Icp = {Isp, IMp, Iop] +5: +Find 3 IBIs from Icp such that: +z,3, = argmin(licp -iapl +[icp -ipl +ip -ipl) +6: +iik +7: +where iAp, iAp and iAp are three average IBIs at pth segment +IFp =[icp,icp,ic, ] +8: +9: +IF← IFp +10: end for +ll: return IF Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +7 +B. Interbeat Intervals Evaluation +I evaluate the agreement between true and estimated IBIs +using Pearson Correlation Coefficient (Corr) for each subject. +As for accuracy performance metric, I rely on Mean Absolute +Percentage Errors (MAPE) for each subject, defined as (6): +𝑀𝐴𝑃𝐸 = 1 +𝑛 OP|𝑡𝑟𝑢𝑒𝐼𝐵𝐼! − 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝐼𝐵𝐼!| +𝑡𝑟𝑢𝑒𝐼𝐵𝐼! +× 100Z +0 +!1- +(6) +where n is the total number of IBIs in one subject, 𝑡𝑟𝑢𝑒𝐼𝐵𝐼! +denotes the i’th true IBI from ECG and 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝐼𝐵𝐼! denotes +the i’th estimated IBI from PPG. +1) Evaluation on IEEE_training Dataset +Table I. depicts the overall performance evaluation of IBI +estimation on IEEE_Training dataset, which compares different +penalty functions and shows results from single-channel and +two-channel models using each morphological feature +individually and fusion of them. The results are the average of +12 subjects in the IEEE_Training. Several observations from +Table I. are stretched out below. First, the two-channel model +outperforms both the single-channel model PPG1 and the +single-channel model PPG2. The two-channel model achieves +a MAPE of 5.9%, 4.8% and 4.5% for systolic peak (SP), +maximum slope (MS), and onset, respectively, which has +30.6%, 40.7%, and 41.6% improvement, respectively, as +compared to single-channel PPG1. Further, analysis on the +channel usage shows that the PPG1 and PPG2 accounts for +53.3% and 46.7% fiducial points in the two-channel model +using onset feature. +I implement the convex penalty function and other different +penalty +functions +using +Python +and +compare +them +comprehensively. Results show that the convex penalty +function outperforms exponential and sigmoid penalty +functions for all three morphological features either in the +single-channel model or two-channel model. I evaluated results +with sensitivity analysis of the convex penalty function with the +power of 1, 2, 4, and 6. Since they provide similar performance +and the 2nd power is the best, 2nd power is chosen for the +convex penalty function. Note that ε is a parameter that controls +the tolerance of assigning zero edge weight in exponential +penalty function and sigmoid penalty function. It is set as 0.1 +for the single-channel model as it is described in [19]. For the +two-channel model, I empirically tested the 𝜀 parameter and +found the performance is best when ε is set as 0.06, suggesting +that the exponential penalty function and sigmoid penalty +function are sensitive to the ε parameter. Interestingly, the +performance of exponential penalty function and sigmoid +penalty function are identical in all experiments. Overall, it +shows that an effective penalty function is critical in IBI +estimation using the shortest path algorithm where the convex +function is preferable. +Last but most importantly, results demonstrate the +effectiveness of the greedy-optimized fusion method. In the +two-channel model, the fusion method achieves correlation of +0.98 and MAPE of 2.2%, where the MAPE is improved by +51.1% (reducing from 4.5% to 2.2%) as compared to the case +using the onset feature individually. Further, IBI estimation in +the single-channel model has significant improvements using +the greedy-optimized fusion method. The correlation of single- +channel PPG1 from onset feature without fusion is 0.86, +whereas the correlation reaches to 0.96 after applying the +greedy-optimized fusion, which is improved by 11.6%. +TABLE I. +COMPARISON OF SINGLE- AND TWO-CHANNEL MODEL WITH DIFFERENT PENALTY FUNCTIONS IN IBI ESTIMATION +PERFORMANCE USING THREE MORPHOLOGICAL FEATURES AND FUSION OF THEM +Penalty Functions +SP +MS +Onset +Fusion** +Corr +MAPE +Corr +MAPE +Corr +MAPE +Corr +MAPE +Single-channel (PPG1) + + + + + + + + +Convex Penalty +0.83 +8.5% +0.83 +8.1% +0.86 +7.7% +0.96 +3.2% +Expo. (e = 0.1)* +0.82 +8.8% +0.82 +8.7% +0.82 +8.9% +n/a +n/a +Sigmoid (e = 0.1) +0.82 +8.8% +0.82 +8.7% +0.82 +8.9% +n/a +n/a +Single-channel (PPG2) + + + + + + + + +Convex Penalty +0.78 +10.6% +0.82 +9.3% +0.84 +8.5% +0.95 +3.7% +Expo. (e = 0.1)* +0.77 +10.8% +0.80 +10.3% +0.81 +10.1% +n/a +n/a +Sigmoid (e = 0.1) +0.77 +10.8% +0.80 +10.3% +0.81 +10.1% +n/a +n/a +Two-channel (PPG 1&2) + + + + + + + + +Convex Penalty +0.90 +5.9% +0.92 +4.8% +0.94 +4.5% +0.98 +2.2% +Expo. (e = 0.1)* +0.84 +8.5% +0.84 +8.5% +0.84 +9.5% +n/a +n/a +Expo. (e = 0.06)* +0.87 +7.1% +0.90 +6.2% +0.91 +5.9% +n/a +n/a +Sigmoid (e = 0.1) +0.84 +8.5% +0.84 +8.5% +0.84 +9.5% +n/a +n/a +Sigmoid (e = 0.06) +0.87 +7.1% +0.90 +6.2% +0.91 +5.9% +n/a +n/a +Best result (PPG 1&2) +0.90 +5.9% +0.92 +4.8% +0.94 +4.5% +0.98 +2.2% +Aygun et al. [19] (PPG1) +0.82 +n/a +0.85 +n/a +0.86 +n/a +0.89 +n/a +This table shows average metric over the first 12 subjects in IEEE_Training. n/a = not available. +* My implementation in python using the exponential penalty function [19]. +** Greedy-optimized fusion of three morphological features. + + + + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +8 +Similarly for the single-channel PPG2, the correlation jumps +greatly from 0.84 to 0.95, which is improved by 13.1%. In +addition, the MAPE is reduced to 3.2% and 3.7% in the single- +channel PP1 and PPG2, respectively, which is 58.4% and +56.5% improvement as compared to the case using the onset +feature individually. Table II. presents the breakdown result of +12 subjects after the greedy-optimized fusion method for single- +channel models and two-channel models. +Fig. 4 demonstrates the effectiveness of the greedy- +optimized fusion method for tackling the challenge of estimated +IBIs from PPG which have overly fluctuating patterns. The +challenge could be explicitly seen in Fig. 4 (a), which is the IBIs +calculated from the single-channel PPG1 onset feature of +subject 2 in IEEE_Training. This difficulty could also be +viewed in the estimated IBIs from the single-channel PPG1 +onset feature of subject 6, shown as Fig. 4 (c). The greedy- +optimized fusion method mitigates the overly fluctuating +pattern and improves the correlation by 10.3% and 8.8% and +reduces the MAPE by 54.3% and 69.2% in subject 2 and subject +6, respectively as shown in Fig. 4 (b, d). Furthermore, even +though the two-channel PPG1 + PPG2 model already provided +very high correlations of 0.94 and 0.98 and low percentage +errors of 5.5% and 3.9% for subject 2 and subject 6, respectively, +the fluctuation challenge remains, shown as Fig. 4 (e, g). The +effectiveness of the greedy-optimized fusion method is well +demonstrated in Fig. 4 (f, h), where the estimated IBIs are much +closer to the true IBIs with low fluctuation. The correlations +achieve 0.97 and 0.99 for subject 2 and subject 6, respectively, +and MAPEs reduce to 3.0% and 1.8%, which indicates the +greedy-optimized fusion provides a solution for the challenge. +2) Evaluation on PPG-DaLiA Dataset +To further evaluate the robustness of the greedy-optimized +framework in detecting IBIs on PPG signals in daily real-life +conditions, I also apply my techniques on the PPG-DaLiA +dataset. This dataset only provides one channel PPG from +commercial wearable wristbands [29]. I extract segments of two +intensive activities, ascending/descending stairs and cycling, +from PPG signals, which have duration of 5 mins and 8 mins, +respectively. Note that there are certain amounts of abnormal +high spikes in the true IBI of subject 6 and subject 10, so I +correct the true IBI annotation of subject 10 for stairs activity +and subject 6 for cycling activity. I use the average heart rate +(window length: 8 s, window shift: 2 s) provided in this dataset +to calculate the average IBIs. The single-channel model with +fusion of three morphological features achieves high correlation +of 0.91 ± 0.04 and low MAPE of 3.8% ± 0.8% for +ascending/descending stairs activity and high correlation of +0.95 ± 0.04 and low MAPE of 2.4% ± 0.7% for cycling +activity, shown in Table III. +C. Heart Rate Variability Analysis +HRV can be described using time-domain and frequency- +domain measurements. The time-domain measurements +quantify the amount of variability in measurements of the IBI +TABLE II. IBI ESTIMATION PERFORMANCE OF GREEDY-OPTIMIZED +FRAMEWORK AND COMPARISON OF SINGLE CHANNEL AND TWO +CHANEEL PPG SIGNALS FROM 12 SUBJECTS IN IEEE_TRAINING +Subject +ID +PPG1 +(Fusion) +PPG2 +(Fusion) +PPG1&2 +(Fusion) +1 +0.98 | 3.4% +0.98 | 4.4% +0.99 | 2.5% +2 +0.96 | 4.2% +0.94 | 4.9% +0.97 | 3.0% +3 +0.96 | 3.5% +0.95 | 3.9% +0.99 | 2.0% +4 +0.97 | 3.1% +0.96 | 3.7% +0.98 | 2.0% +5 +0.98 | 2.0% +0.97 | 2.5% +0.99 | 1.5% +6 +0.99 | 2.4% +0.99 | 2.8% +0.99 | 1.8% +7 +0.98 | 2.0% +0.97 | 2.9% +0.99 | 1.6% +8 +0.98 | 2.7% +0.96 | 4.0% +0.99 | 1.9% +9 +0.99 | 2.4% +0.98 | 3.1% +0.99 | 1.7% +10 +0.83 | 4.6% +0.86 | 4.0% +0.93 | 2.9% +11 +0.91 | 3.2% +0.86 | 5.1% +0.93 | 2.8% +12 +0.95 | 4.5% +0.97 | 3.2% +0.98 | 2.5% +Average +0.96 | 3.2% +0.95 | 3.7% +0.98 | 2.2% +SD +0.04 | 0.9% +0.04 | 0.8% +0.02 | 0.5% +The first column in each signal modality reports the correlation and the +second column reports the MAPE. SD = standard deviation + +Fig. 4. IBIs plot over time for subject 2 and subject 6 in IEEE_Training Dataset (Fusion of three features v.s. No fusion) + + +1000 +1000 +1000 +1000 +Subject 2 - PPG1 Onset (No Fusion) +Subject 2 - PPG1 (Fusion of 3 features) +Subject 6 - PPG1 Onset (No fusion) +Subject 6 - PPG1 (Fusion of 3 features) +900 +Correlation: 0.87, MAPE : 9.2% +900 +Correlation: 0.96, MAPE : 4.2% +900 +Correlation: 0.91, MAPE : 7.8% +900 +Correlation: 0.99, MAPE : 2.4% +800 +800 +800 +800 + 600 +B +500 +500 +500 +500 +400 +400 +400 +400 +Estimated IBI +Estimated IBI +Estimated IBl +Estimated IBI +300 +True IBI +300 +True IBI +300 +True IBI +300 +True IBI +50 +100 +150 +200 +50 +100 +200 +250 +300 +0 +50 +100 +200 +300 +0 +0 +50 +100 +200 +250 +300 +Time (sec) +Time (sec) +Time (sec) +Time (sec) +1000 +1000 +1000 +1000 +Subject 2 - PPG1 + PPG2 Onset (No Fusion) +Subject 2 - PPG1 + PPG2 (Fusion of 3 features) +Subject 6 - PPG1 + PPG2 Onset (No fusion) +Subject 6 - PPG1 + PPG2 (Fusion of 3 features) +006 +Correlation: 0.94, MAPE : 5.5% +006 +Correlation: 0.97, MAPE : 3.0% +900 +Correlation: 0.98, MAPE : 3.9% +900 +Correlation: 0.99, MAPE : 1.8% +800 +800 +800 +800 +700 +B +500 +500 +500 +500 +400 +400 +400 +400 +Estimated IBI +Estimated IBI +Estimated IBI +Estimated IBI +300 +True IBI +300 +True IBI +300 +True IBI +300 +True IBI +0 +50 +100 +150 +200 +250 +300 +0 +50 +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +Time (sec) +Time (sec) +Time (sec) +Time (sec) Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +9 +during monitoring periods. These metrics include the standard +deviation of heart rate (STD HR), standard deviation of the IBI +of normal sinus beats (SDNN), and so on. SDNN has been used +as the medical stratification of cardiac risk of morbidity and +mortality for heart attack survivors [31]. The frequency-domain +measurements estimate the distribution of absolute or relative +power into three frequency bands: very-low-frequency (VLF), +low-frequency (LF), and high-frequency (HF) bands. VLF +Power has been reported to be associated with all-cause +mortality, arrhythmic death, and post-traumatic stress disorder +(PTSD) [31]. HF band reflects parasympathetic activity and is +related to the respiratory cycle and is correlated with mental +health, such as stress, panic and anxiety [31]. +To comprehensively evaluate the performance of the greedy- +optimized framework in estimating HRV, I apply the HRV +analysis to two datasets, IEEE_Training (12 subjects) and PPG- +DaLiA (15 subjects). The estimated and true IBIs are used to +calculate four time-domain HRV parameters and four +frequency-domain HRV parameters using pyHRV [32]. The +four time-domain HRV parameters investigated in this paper +include Mean RR (ms), SDNN (ms), Mean HR(1/min) and STD +HR (1/min). The frequency domain HRV parameters +investigated in this paper are computed using the autoregressive +method to separate HRV into its component frequency band, +including the VLF Power (absolute power of the VLF band of +0.00 – 0.04 Hz), LF Power (absolute power of the LF band of +0.04–0.15 Hz), HF Power (absolute power of the HF band of +0.15–0.4 Hz), and the Total Power. The HRV results are +evaluated by Pearson Correlation Coefficient (Corr) and the +accuracy are evaluated by Mean Absolute Percentage Errors +(MAPE), defined as (7): +𝑀𝐴𝑃𝐸 = 1 +𝑛 @ A|𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟! − 𝑒𝑠𝑡𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!| +𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟! +× 100M +& +!'( +(7) + +where n is the total number of subjects in the dataset, +𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟! denotes the HRV parameter derived from +true IBIs and 𝑒𝑠𝑡𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟! denotes the HRV parameter +derived from estimated IBIs of the i’th subject. The Table IV +shows HRV analysis results of IEEE_training Dataset. HRV +analysis of this study is based on IBI estimation results of two- +channel (PPG1&2) after the greedy-optimized fusion method is +applied on three features, SP, MS and Onset. HRV analysis of +Aygun et al. is based on IBI estimation results of single-channel +(PPG1) from IEEE_Training using their fusion method of three +morphological features [19]. Results demonstrate that the +estimated and true HRV parameters are highly correlated with +low percentage errors in Table IV. The Pearson correlation +coefficients are above 0.9 significantly with all p-values less +than 0.001, except that the coefficient of HF Power is 0.858. +This study provides the MAPEs which are less than 1.7% for +all eight HRV parameters. +HRV analysis of PPG-DaLiA Dataset is shown in Table V, +which is based on IBI estimation results of single-channel PPG +from Empatica E4 of 15 subjects in stairs and cycling activities +after the greedy-optimized fusion method is applied on three +features, SP, MS and Onset. The results show that the estimated +and true HRV parameters are highly correlated with low +absolute percentage errors for both the stairs and cycling +activities and the performance of the cycling activity is better +TABLE III. IBI ESTIMATION PERFORMANCE OF GREEDY- +OPTIMIZED FRAMEWORK USING SINGLE-CHANNEL PPG FROM 15 +SUBJECTS IN THE PPG-DALIA +Subject +ID +Ascending/Descending +Stairs (5 mins) +Cycling +(8 mins) +1 +0.96 | 3.7 % +0.97 | 1.9 % +2 +0.94 | 3.9 % +0.94 | 3.1 % +3 +0.92 | 2.8 % +0.97 | 2.3 % +4 +0.91 | 4.4 % +0.94 | 3.5 % +5 +0.85 | 4.3 % +0.87 | 2.6 % +6 +0.89 | 2.8 % +0.97* | 1.6 %* +7 +0.89 | 4.2 % +0.97 | 2.2 % +8 +0.85 | 5.3 % +0.89 | 2.7 % +9 +0.96 | 3.8 % +0.98 | 2.9 % +10 +0.87* | 4.6 %* +0.92 | 3.2 % +11 +0.91 | 3.8 % +0.99 | 1.5 % +12 +0.92 | 4.0 % +0.98 | 1.8 % +13 +0.98 | 2.4 % +0.99 | 1.5 % +14 +0.96 | 2.6 % +0.98 | 2.2 % +15 +0.90 | 4.2 % +0.91 | 3.5 % +Average +0.91 | 3.8 % +0.95 | 2.4 % +SD +0.04 | 0.8 % +0.04 | 0.7 % +* I found amounts of abnormal high spikes for subject 6 and subject 10’s +true IBI. Hence, I corrected the true IBI annotation of subject 10 for stair +activities and subject 6 for cycling activities, respectively. + + +TABLE IV. HRV PARAMETERS PERFORMANCE OF IEEE_TRAINING +HRV +Parameters +Greedy-optimized +Framework* +Aygun et al.[19] +Corr** +MAPE +Corr +MAPE +Mean RR (ms) +0.999 +0.2 % +0.986 +n/a +SDNN (ms) +0.998 +1.3 % +0.956 +n/a +Mean HR (1/min) +0.999 +0.2 % +0.987 +n/a +STD HR (1/min) +0.990 +1.7 % +0.860 +n/a +VLF Power (ms2) +0.995 +0.2 % +0.981 +n/a +LF Power (ms2) +0.971 +0.8% +0.898 +n/a +HF Power (ms2) +0.858 +1.7% +0.828 +n/a +Total Power (ms2) +0.932 +1.0 % +0.974 +n/a +* HRV analysis is based on IBI estimation results of two-channel +(PPG1&2) from 12 subjects in treadmill activity on IEEE_Training using +the greedy-optimized framework. +** All p-value of Pearson correlation coefficient are less than 0.001. + + +TABLE V. HRV PARAMETERS PERFORMANCE OF PPG-DALIA +HRV +Parameters +Ascending/Descending +Stairs (8 mins) +Cycling (5 mins) +Corr* +MAPE +Corr* +MAPE +Mean RR (ms) +1 +0.2 % +1 +0.1 % +SDNN (ms) +0.981 +5.0 % +0.998 +2.4 % +Mean HR (1/min) +1 +0.2 % +1 +0.1 % +STD HR (1/min) +0.935 +6.5 % +0.996 +2.7 % +VLF Power (ms2) +0.996 +0.2 % +0.998 +0.2 % +LF Power (ms2) +0.905 +1.1 % +0.965 +0.8 % +HF Power (ms2) +0.786 +4.4 % +0.784 +3.5 % +Total Power (ms2) +0.843 +2.7 % +0.888 +1.9 % +HRV analysis is based on IBI estimation results of single-channel PPG +from Empatica E4 of 15 subjects in stairs and cycling activities on the +PPG-DaLiA using the greedy-optimized framework. +* All p-value of Pearson correlation coefficient are less than 0.001. + + + + + Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +10 +than the stairs activity. Note that HF Power has the lowest +correlation among the eight HRV parameters in the stairs and +cycling activities of the PPG-DaLiA and in the IEEE_Training. +Fig. 5 provides the scatterplots that compare the true and +estimated SDNN, STD HR, VLF Power, LF Power, HF Power +and Total Power derived from PPG signals during intensive +treadmill activities on IEEE_Training dataset and in the stairs +and cycling activities of PPG-DaLiA Dataset. These plotted +points in Fig. 5 (a, b, c, d, e, f) are distributed along with the +identity line closely, showing that the true and estimated HRV +results have high correlations and small absolute errors in all +three intensive activities. HF Power, however, has the highest +absolute errors among the four frequency-domain parameters +and is often overestimated in the PPG-DaLiA. +V. DISCUSSION +In the application of healthcare monitoring through wearable +sensors, IBI and HRV estimation from PPG are challenging +because motion-artifacts induced by daily or exercise activities +significantly deteriorates the accuracy. The most common +strategy in analyzing IBI and HRV from noisy PPG is discarding +motion-contained signal segments, which loses the opportunity +of discovering potential health information which is triggered +during exercise. In this study, I show that the greedy-optimized +framework, which leverages convex penalty function in shortest +path calculation and greedy-optimized fusion method, could +provide high accuracy in estimating IBI and HRV from whole +PPG signals obtained during daily and intensive exercise +activities. The set of IBIs selected by the short path algorithm in +a directed acyclic graph is regarded as the optimal among +candidate IBIs from one shortest path in terms of resembling the +true IBIs. Although this guarantees estimated IBIs to have high +correlation with true IBIs, the observation in IBI plots shows a +big challenge that those estimated IBIs are over-fluctuating and +cause large absolute errors as compared to true IBIs, causing +those estimated IBIs are not ideal representatives. To tackle this +challenge, the greedy-optimization fusion method for various +shortest paths is proposed in this study. By leveraging a +physiological phenomenon that true IBIs are close to their +average IBIs, I develop an objective function for the greedy- +optimized fusion method to find local optimum IBIs in each step, +or each segment. Through the process, the greedy-optimization +fusion method selects optimal IBIs that have the least absolute +error with the average IBI set. Results show that the greedy- +optimized fusion reduces the MAPE by at least 50% in both the +single-channel and two-channel models and enormously +mitigates inherently over-fluctuating beat-to-beat IBIs estimated +from noisy PPG. +IBI estimation from the multi-channel PPG signals outperform +the single-channel PPG signal in a previous study [9]. +Nevertheless, practical direction in healthcare remote monitoring +is to develop a compact and portable wearable sensor. Prevailing +commercial wearables are embedded with only one PPG sensor. +Hence, it is crucial to develop a model which is capable of +achieving high accuracy even if the wearable has a single-channel +signal. Results from PPG-DaLiA indicate that my techniques +have the ability to accurately estimate the IBI and HRV from +PPG on a commercial wearable, Empatica E4, which has one +channel of PPG sensor with low sampling rate of 64Hz. +Furthermore, the greedy framework proposed in this study has +efficient time complexity of O(n). Given the computational +efficient nature of greedy, the framework could be implemented +with edge computing for commercial wearables and could be +applied in real world healthcare remote monitoring applications. +Although the proposed greedy-optimized framework has nice +performance in IBI and HRV estimation from noisy PPG signals +under daily intensive activities, there is a crucial material needed +to know for applying the model. Average HR is an important +input in this optimization framework. The accuracy of IBI and +HRV estimation would be limited when the accuracy of average +HR decreases. Favorably, despite the above limitation, this +optimization framework for IBI and HRV estimation is +independent of average HR. Users could use any algorithm that +generates accurate average HR from noise-contaminated PPG, +such as WFPV [11], particle filtering [12] and Deep PPG [29], +which have been matured for decades. Another thing which +needs attention is that currently I evaluate this optimization +framework in the dataset of 5-8 minutes duration. It has not been +investigated in the dataset with longer duration. For example, +PPG signals which are collected overnight during sleep or over +one day period (above 24 hours). Further, some studies have +shown that long-term HRV parameters (24 hours) are a more + +Fig. 5. Scatterplot comparison of true/estimated HRV parameters +(a) SDNN (b) STD HR (c) VLF Power (d) LF Power (e) HF Power +and (f) Total Power from noisy PPG signals on IEEE_Training +(treadmill) and PPG-DaLiA (stairs and cycling) + +140 +IEEE_ Training +27.5 +IEEE Training +★ +★ +★ +DaLiA_cycling +DaLiA_cycling +25.0 +120 +DaLiA stairs +DaLiA stairs +(sw) i +SDNN( +100 +*+ +F 20.0 +** +工 +80 +17.5 +Estimated +60 +12.5 +40 +10.0 +20 +7.5 +20 +40 +60 +80 +100 +120 +140 +7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 +True SDNN (ms) +True STD HR (1/min) +2400 +IEEE Training +IEEE_Training +★ +5000 +DaLiA_cycling +DaLiA_cycling +DaLiA_ stairs +DaLiA_stairs +2250 +媒 +2100 +4000 +2100 +215022002250 +2300 +2350 +2400 +4000 +4200 +4400 +4600 +4800 +5000 +True VLF Power (ms2) +True LF Power (ms2) +10500 +18000 +IEEE_Training +IEEE_Training +10000 +DaLiA_cycling +DaLiA_cycling +(zsu) +DaLiA stairs +DaLiA stairs +9500 +Power +9000 +8500 +Estimated H +15000 +8000 +7500 +7000 +13000 +005010000T0056 0006 0058 0008 005L 000L +1300014000150001600017000 +18000 +True HF Power (ms2) +True Total Power (ms2) Luffina C. Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG +11 +powerful predictor of mortality than short-term HRV parameters +for patients with chronic heart failure and acute myocardial +infarction [33, 34]. A future work could be extended into +evaluating this greedy-optimization framework for IBI and HRV +estimation in long period wearable PPG signals. +VI. CONCLUSION +This paper proposes a greedy-optimized framework for IBI +and HRV estimation on single-channel and multi-channel PPG +signals collected during intensive daily activities. Two proposed +techniques, convex penalty function and greedy-optimized fusion +method, equip the framework with the capability of improving +the accuracy of the IBI and HRV estimation. The convex penalty +function is introduced to optimize edge weights assignment in the +shortest path calculation. The greedy-optimized fusion method +mitigates highly fluctuating patterns in estimated IBIs, achieving +the better approximation of true IBIs. On 2015 IEEE Signal +Processing Cup, the greedy-optimized framework achieves low +average percentage errors of 2.2% and 3.2% with high average +correlations of 0.98 and 0.96 for IBI estimation through two- +channel PPGs and single-channel PPG1, respectively, with O(n) +complexity. Results also demonstrate the convex penalty +function outperforms the exponential and sigmoid penalty +function in the shortest path algorithm. The proposed greedy- +optimized fusion successfully reduces the MAPE by 58.4% and +improves the correlation by 11.6% in the single-channel PPG1 +for IBI estimation. I further validate the proposed framework on +two daily activities from the PPG-DaLiA Dataset, which uses +single-channel PPG commercial wearables. The estimated IBIs +achieve high average correlations of 0.92 and 0.95 with low +percentage error of 3.8% and 2.4% for the ascending/descending +stairs and cycling activities, respectively, indicating that this +framework could be adaptive to single sensor PPG wearables on +the market. The estimated and true HRV parameters (Mean RR, +SDNN, Mean HR, STD HR, VLF Power, LF Power and Total +Power) are also highly correlated with low percentage errors. +Since the accuracy of IBI and HRV estimation is consistently +favorable across three activities from those two datasets with low +standard deviations among subjects, suggesting the robustness of +the greedy-optimized framework. +REFERENCES +[1] +N. Singh, K. J. Moneghetti, J. W. Christle, D. Hadley, D. Plews, and V. +Froelicher, "Heart Rate Variability: An Old Metric with New Meaning in +the Era of using mHealth Technologies for Health and Exercise Training +Guidance. 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Nolan et al., "Prospective study of heart rate variability and mortality +in chronic heart failure: results of the United Kingdom heart failure +evaluation and assessment of risk trial (UK-heart)," Circulation, vol. 98, +no. 15, pp. 1510-6, Oct 13 1998, doi: 10.1161/01.cir.98.15.1510. + + + diff --git a/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/load_file.txt b/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb59184984552cf111aed01775747283eb8e377c --- /dev/null +++ b/IdE1T4oBgHgl3EQfFwOO/content/tmp_files/load_file.txt @@ -0,0 +1,1340 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf,len=1339 +page_content='Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 1 Abstract— Continuous monitoring of inter-beat-interval (IBI) and heart rate variability (HRV) provides insights in cardiovascular, neurological, and mental health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Photoplethysmography (PPG) from wearables assures convenient measurement of IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, PPG is susceptible to motion artifacts, considerably deteriorating the accuracy of IBIs estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although a multi-channel model in previous study improves accuracy, prevailing compact commercial wearables would favor single-channel sensors, causing benefits of multi-channel applications to have restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In this paper, a greedy-optimized framework is proposed for measurement of IBI and HRV featuring single-channel and multi-channel PPG signals collected during daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Utilizing the fact of continuity in heartbeats, the IBI estimation problem is converted into the shortest path problem in a directed acyclic graph, where candidate heartbeats from the noisy PPG are regarded as vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The framework exploits a convex penalty function to optimize weight assignment in the shortest path calculation and a greedy-optimized fusion method to mitigate overly fluctuating patterns in estimated IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The results achieve correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 with percentage error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% for IBI estimation using single-channel PPG signals from the 2015 IEEE Signal Processing Cup dataset, where percentage error is reduced by 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% and correlation is improved by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6% in comparison to those without greedy-optimized fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In the multi-channel model, it achieves correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 with percentage error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Estimated and true HRV parameters are also highly correlated with low percentage errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This paper further validates these techniques on the PPG-DaLiA dataset, indicating the robustness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Index Terms— Daily Activities, Heart Rate Variability, Motion Artifacts, Wearable Physiological Sensing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' INTRODUCTION EMOTE healthcare monitoring is increasing in popularity nowadays through more and more emerging wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' These commercialized wearables promote self-health management, which benefits the engagement of patients and the quality of medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Continuous cardiovascular activity Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang was with the Department of Computer Science and Engineering, Texas A&M University and she is now with the Department of monitoring is one important focus for self-health management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Among various physiological parameters, average heart rate (HR) and heart rate variability (HRV) are significant parameters in continuous cardiovascular monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Average HR, the measurement of the number of heart beats per minute in a certain time period, is one of the vital signs routinely monitored by healthcare providers and serves as an important indicator of cardiovascular health, such as hemodynamic stability and heart rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV, however, could provide integrated information of the cardiovascular system and the autonomic nervous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Interbeat intervals, IBIs, are the time elapsed between two successive heart beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV quantifies the variability of IBIs in a certain period of time and is widely used as a crucial indicator in healthcare research and clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV parameters could evaluate the sympathetic and parasympathetic activity of the autonomic nervous system, which controls heart rate and blood pressure in response to dynamic physiological changes, such as respiration, exercise, physical stress and mental load [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Further, low HRV, reduced level of beat-to- beat heart rate fluctuations, is not only independently associated with a 32–45 % increased risk of first fatal and non-fatal cardiovascular disease (CVD) events but also a prognostic factor with higher mortality in patients with a CVD event [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In addition, elevated HRV has a protective effect in reduction of CVD events, which could be enhanced through increased physical activity and aerobic exercise training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A study has shown the 1% increase in a HRV parameter, standard deviation of the normalized NN interval (SDNN), leads to a roughly 1 % reduction of fatal or non-fatal CVD events [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' IBI and HRV could be derived from both Electrocardiography (ECG) and Photoplethysmography (PPG) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Traditionally, the gold standard of IBI and HRV measurements is multi-lead ambulatory ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although ambulatory ECG provides the possibility of out-of-hospital monitoring, it requires setup by specialized technicians and needs to attach multiple electrodes to the chest skin, which is not comfortable to wear for a long period of time [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Given the recent trend for integrating health assessments into wearable technologies, more and more commercialized wearable devices are equipped with single-lead PPG sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' PPG-based wearables with single contact point have become widely Electrical and Computer Engineering, Rice University, Houston, TX 77005 USA (e-mail: luffina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='huang@rice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A Greedy-optimized Framework for Heart Rate Variability Monitoring during Daily Activities using Wearable Photoplethysmography Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang Department of Electrical and Computer Engineering, Rice University, Houston, Texas, USA E-mail: luffina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='huang@rice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='edu R Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 2 accessible due to the advantage of low-cost, non-invasive, and easy to use, which makes them a convenient and practical tool for continuous IBI and HRV monitoring in daily life, served as an alternative to the standard ECG [1, 2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Through the enhancement of continuous monitoring of HRV, wearable technologies open a new window of remote healthcare monitoring and the trend of self-health management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' PPG is an optical biomonitoring technique that emits single or multi-wavelength light by LED to penetrate the skin and blood vessels and captures the reflected light by photodiodes to measure blood volumetric changes in microvascular tissue at fingers and wrists [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Featured physiological parameters related to the cardiopulmonary system could be estimated via PPG, such as blood oxygen saturation (SpO2), average heart rate (HR), respiratory rate (RR) [6], and blood pressure (BP) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, PPG-based techniques can achieve highly accurate HRV estimation in stationary conditions, such as sitting, rest and supine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Studies have shown the IBIs and HRV parameters derived from PPG wearables are highly associated with those derived from ECG signals (correlation coefficient ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='85 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, motion artifacts are an inherent problem when applying PPG-related techniques to healthcare monitoring in free movement condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The motion artifacts in PPG degrade the accuracy of IBIs/HRV estimation as the level of physical activity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The frequency spectrum of motion artifacts ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='01 to 10 Hz, which overlaps with the normal frequency of PPG signal (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 - 5 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Therefore, it is not easy to denoise the motion-contaminated PPG signal by applying general filtering techniques [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Average HR exhibits more consistency over time compared to noises and HRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' It is easier to attain average HR from noisy PPG, whereas estimating HRV is challenging during intensive physical activity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, there is an unmet need to develop accurate algorithms for HRV estimation from PPG wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This paper proposes a greedy-optimized framework to tackle the aforementioned unmet need, which transforms IBI estimation into a shortest path problem in directed acyclic graph subsequently combined with a greedy-optimized fusion method for morphological features extracted from motion- contaminated PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I use the physiological property of the temporal continuity of heartbeats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', the end of one heartbeat is the start of the next heartbeat) and construct a directed acyclic graph, where the vertices represent the feature candidates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', heartbeats) and the edges represent the candidate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Shortest path algorithm is then used to remove noisy feature candidates and calculate accurate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The proposed convex penalty function for edge weight assignment is designed to augment the power of the shortest path algorithm with increments of the accuracy in IBI estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Subsequently, the greedy-optimized fusion method is developed to optimize the process of selecting estimated IBIs from the three morphological features, systolic peaks, maximum slope, and onset points, which are extracted from motion-corrupted PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Through this fusion strategy, it mitigates the inherent overly fluctuating patterns of estimated IBIs from noisy PPG signals and calculates accurate IBIs and HRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The advantage of the greedy-optimized framework is that it could process both single-channel and multi-channel motion- contaminated PPG signals with computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, it could be adaptive with prevailing commercial wearable devices, which have, in general, limited computational capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In summary, the contributions of this article are as the follow: 1) A greedy-optimized framework is developed to attain high accuracy of IBI and HRV estimation with the capacity of processing single-channel and multi-channel motion-contaminated PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2) The convex penalty function could optimize weight assignment to augment the power of the shortest path algorithm and the greedy-optimized fusion method is developed for mitigating overly fluctuating patterns of estimated consecutive IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 3) Performance is validated on two public datasets, IEEE Signal Processing Cup and PPG-DaLiA, which have noisy PPG from wearables collected during intensive exercise and daily activities, indicating the robustness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Average Heart Rate Estimation In past decades, research works about HR estimation for wearable PPG signals have quite matured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Several studies have demonstrated that high accurate estimation of average HR during intensive physical exercise can be achieved by incorporating the accelerometer signals with a variety of motion artifact reduction framework using single-channel PPG, such as TROIKA [10], WFPV [11] and particle filtering [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Other studies have attempted to leverage multi-channel PPG that used truncated singular value decomposition (SVD) or template- matching algorithm for accurate average HR estimation and showed the multi-channel estimation outperformed the single- channel estimation during intensive exercise [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' These techniques, nevertheless, were not able to be adapted for IBIs and HRV estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Heart Rate Variability Estimation IBI/HRV have extensive physiological applications in clinical practice, but it is more challenging to attain accurate IBIs from wearable PPG sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Initially, some studies began to demonstrate the feasibility and good performance of IBI and HRV estimation using wrist-worn PPG sensors in post- anesthesia patients and in healthy volunteers during sleep [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although these studies have shown satisfactory small absolute errors of IBI and HRV parameters between wrist-worn PPG sensors and ECG, most of their PPG signals do not have motion artifacts distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' One study has shown that although the good association (correlation coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='74 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='88) between wrist-worn PPG sensors and ECG in HRV parameters was achieved in baseline rest condition, the correlation was degraded to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='42 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='67 when subjects were talking [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' One work has benchmarked the HRV parameters for different ages and genders using a dataset of 8 million users, which is the largest to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' It applied noise spikes cleaning algorithms and achieved high correlation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97) in HRV parameters during a randomly selected 24 hours period [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, these above studies eliminated all PPG signals that were corrupted with motion artifacts from their HRV analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' There are some studies exploring accurate R peaks detection and IBI estimation in noisy ECG signals using deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Vijayarangan Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 3 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' has proposed a novel application of the IncResU-Net, a fully convolutional Encoder-Decoder architecture, to detect R- peak from the ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The model could provide good performance in R-peak detection in ECG with noise level up to 0 dB [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, deep learning-based methods require a very large amount of data for computationally expensive training with GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, these techniques have not been investigated in noisy PPG signals yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' One study works on single-channel motion-corrupted PPG using combinatorial algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The model leverages the shortest path algorithm with exponential function, which presents a medium-high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 82 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='86) correlation between the PPG sensors and ECG [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, this study only provided the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' No breakdown of subjects and no accuracy evaluation was reported for both IBI and HRV estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Fusion of Physiological Signals Fusion approaches have been explored to improve the accuracy of heartbeats detection by incorporating the information across different physiological signal modalities or multiple morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' One fusion approach is signal switching, where the candidate fiducial points from a signal modality with the best signal quality are selected as final fiducial points in a certain segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' used the sample entropy to assess the noise content in multiple signal modalities, such as ECG and arterial blood pressure (ABP) signals, and switch between them to enhance the accuracy of heartbeat detection [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Aygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' obtained the best set of IBI arrays from three PPG morphological features by selecting those segments with minimal standard deviation of IBI subarray [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Some studies explored voting method for fusion, where the candidate fiducial points detected in each signal modality cast a vote to select final fiducial points for a certain segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In the majority voting, the fiducial points that have most agreement among different signal modalities are selected as the final fiducial points [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, the vote could be weighted by the signal quality index or other evaluation metrics to select fiducial points with best quality [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Some fusion methods are based on sophisticated probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For example, Zia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' decoded the waveform segments of ECG and ABP into different hidden states in a Hidden Markov Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Followed by this, the authors employed a Bayesian Network to model the relationship of the hidden states of the ECG, ABP and classification, which indicates output of the QRS segment [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Method In this work, a greedy-optimized framework is proposed for IBI and HRV estimation with the capacity of processing motion-contaminated PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The overview of the proposed framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1, which consists of two main models, shortest path calculation and greedy-optimized fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In the preprocessing stage, PPG signals are upsampled and filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Then morphological features of a cardiac cycle in PPG signals are extracted as the fiducial points to represent the heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Firstly, the heartbeat detection is modeled as the shortest path problem that aims to differentiate the true heartbeat from the noise spikes in noisy PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A graph is constructed where the vertices represent all potential heartbeats, and the edges represent all the candidate interbeat intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A convex penalty function is proposed to optimize weight assignment in the shortest path algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The time difference of two consecutive heartbeats selected by the shortest path algorithm is regarded as an estimated IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Above processes could be applied in single-channel and multi-channel PPG signals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Secondly, a greedy-optimized fusion method is introduced to utilize the complementary information of the three IBI arrays estimated from the three different morphological features to further improve accuracy of IBI estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Note that both the shortest path calculation and the greedy-optimized fusion method are generalized and could be adopted into different signal modalities such as PPG, ECG or other physiological signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Morphological Feature Extraction In ECG signals, the QRS complex represents the depolarization and contraction of ventricles, pumping blood into vessels going to the body and lungs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The R-peak is the most prominent positive peak of R-wave in the QRS complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The time elapsed between two consecutive R-peaks, called R-R interval, is the most common and standard way to calculate the IBIs in ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' To strengthen the R-peak detection in ECG signals, ECG signals are preprocessed by a continuous wavelet transform (CWT) using Mexican hat with a center frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='25Hz [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The peaks detected from the wavelet are regarded as ground truth R-peaks and the calculated IBI from those peaks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Overview of greedy-optimized framework for noisy PPG signals PPG1 PPG2 Upsampling Band pass filter IBls (Sys) Fusion IBis (Ms) IBls (Onset) Greedy Convex Penalty Function Average HR Upsampling Band pass filter Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 4 are regarded as the ground truth for metric performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The PPG waveform of a cardiac cycle is commonly divided into two phases: The anacrotic phase is the rising edge of the waveform, whereas the catacrotic phase is the falling edge of the waveform as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The anacrotic phase is primarily associated with systole (heart contraction), which is the most crucial physiological function of heart activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Systolic peaks, maximum slopes and onset points are important morphological features that characterize the systolic waveform of a cardiac cycle in PPG signals [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The time elapsed between two consecutive systolic peaks in PPG signals is referred to as Peak-Peak interval, whereas the time elapsed between the onset and the end of the PPG waveform is referred to as Pulse interval, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Some studies observe that the Peak-Peak interval in PPG signal is highly correlated with the R-R interval in the ECG signal [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Other studies show that the HRV from the Pulse interval in PPG signals are highly correlated with the HRV from R-R intervals in ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Both Peak-Peak interval and Pulse interval have been used to detect the heart rate and HRV in stationary condition [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Maximum slope, which is centered between systolic peak and onset in PPG signals, has also been applied to HRV estimation in intensive physical activity [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Therefore, these three features are used for IBI estimation in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Following steps are the process of extracting these features, which are regarded as candidate fiducial points in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' First, the filtered PPG signals are smoothed by a 5th order smoothing spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Then, a general peak detection algorithm in SciPy detects the local maxima of PPG and ECG signals to obtain the systolic peak candidates of PPG and R-peak candidates of ECG, respectively [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Then, I use the same strategy in a previous study to extract the local maximum of the first derivative and second derivative of PPG signals as the maximum slope candidates and onset candidates, respectively [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' All pairs of fiducial points in the same morphological features compose the set of candidate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, many of the candidate fiducial points are induced by motion artifacts, which leads to false IBIs and therefore need to be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Graph Modeling and Shortest Path Calculation To improve IBI estimation accuracy, the false fiducial points need to be removed whereas fiducial points representing true heartbeats need to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Based on aforementioned features, a weighted graph is constructed, where vertices represent candidate heartbeats and edges represent candidate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The shortest path calculation is then used to find the real IBIs and filter out those induced by motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' According to a previous study, the multi-channel PPG model outperforms the single-channel PPG model in IBI estimation during intensive physical activity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, prevailing commercial wearable devices favor a single-channel PPG sensor due to the need for portable and compact [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, this study investigates single-channel and multi-channel models for PPG signals as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1) Graph Construction with Single- and Multi-Channel Firstly, three directed acyclic graphs are constructed using candidate fiducial points extracted from three morphological features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In the graph, the vertices are marked as the candidate fiducial points while edges are designed as candidate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Vertices are denoted as 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', 𝑖 = 1, 2, … , 𝑁, where 𝑁 is the total number of vertices in the graph, and their values are equal to their timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A time interval 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' prior to each vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' with the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 folds of its average IBI is considered to identify neighbors of 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' , where *𝐼𝐵𝐼"#$-!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' = 6000 (𝐻𝑅"#$)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ⁄ [𝑚𝑠] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Vertices within this interval are considered as neighbors of the vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='. Edges are formed between vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' and its neighbors and denoted as 𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='%, where vertices 𝑣% are 𝑣!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='s neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The value of an edge is assigned as the time difference of the two connected vertices and herein represents the candidate IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' (𝐻𝑅"#$)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', the average heart rate of vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', is equal to the average heart rate of the 8-second PPG window which is closest to 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='. Average heart rates are estimated from PPG1 using the WFPV algorithm by every 8- second window with 6-second overlap [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Following the above steps, the graph construction of the single-channel model is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' To model the multi-channel PPG in the graph, I exploit the same observation in a previous study that false fiducial points included by noise would have a bigger time gap between PPG1 and PPG2 than the true fiducial points [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Based on this observation, the vertices from different channels are labeled with different colors to differentiate its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For example, vertices from PPG1 are labeled blue whereas vertices from PPG2 are red, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Then, vertices from two channels are concatenated and sorted by timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The multi- channel graph model is constructed by the above steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2) Weight Assignment by Convex Penalty Function The shortest path algorithm is used as the graph search algorithm to attain the correct IBI path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The weight of each edge is assigned based on their deviation from the average IBI and the weight of each vertex is accumulated from the start vertex to the current one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Therefore, an effective penalty function would be crucial for assigning edge weights of the graph [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' There are many directions to design customized penalty functions, which might bring different debates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' One approach is a tolerance-based penalty function, where edge weight is not penalized until certain criteria meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Another approach is to penalize the edge weight by a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In this study, I examine the concept by proposing a sigmoid penalty function as a good representative example, shown in formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 𝑤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='" = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑 )min )-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='𝐼𝐵𝐼#$%1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' − 𝜀 − 𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' "- , -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='𝐼𝐵𝐼#$%1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' + 𝜀 − 𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' "-77 (1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Standard PPG waveforms Systolic Peak Peak-Peak interval Pulse interval Onset Anacrotic Phase Catacrotic Phase Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=" Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 5 where e&' is the time difference between i’ th and j’ th vertex." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The sigmoid penalty function penalizes the edges which are out of the range of *𝐼𝐵𝐼"#$-!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ± 𝜀 with the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=" Otherwise, it will assign zero to w&' if 𝑒!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='% is within the range of 𝐼𝐵𝐼"#$-!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ± 𝜀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This approach adapts an intuition that the average IBI with a tolerance of 𝜀 could be a good indicator of the time difference between any pair of true fiducial points [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' However, the sigmoid penalty function in this example has a deficiency of assigning weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' There is no weight discrepancy for those edges inside the tolerance interval, even though there are time differences among them, leading to the difficulty of reaching the optimum during the vertex selecting process for the shortest path algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For those edges outside of the tolerance interval, the edge weights are prone to be underestimated with a plateau curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Moreover, the empirical results of tolerance-based penalty functions are highly dependent on the epsilon 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, an exponential penalty function proposed in a previous study has similar situations for the edges inside the tolerance interval [19], but for those edges outside of the tolerance interval, the edge weights are prone to be overestimated with enormous scale [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' An effective penalty function needs to guarantee discrepancy for those edges inside the tolerance interval which have time differences and also reflect cardiac physiology that true IBIs would not have drastic changes in such a short time and should be close to their average IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' To provide a solution, a convex penalty function is proposed and calculated through (2)-(4) as shown in a previous study [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' " = 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=" − $𝐼𝐵𝐼#$%'!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', 𝑖 = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' , 𝑁 (2) 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='& = 1𝑣& − 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' "1, 𝑗 = 𝑖 − 1, … , 𝑖 − 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' (3) 𝑤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='& = 𝜆 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content="& ', 𝑥 ∈ ℕ (4) Firstly, 𝑣’!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is marked as the expected previous vertex of 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=', which is calculated by subtracting the average IBI from each vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='. Secondly, the distance 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='% is calculated between 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ( and 𝑣%, where 𝑣% are 𝑣!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='s neighbors and 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is the total number of neighbors of vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='. Finally, 𝑤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='%, the weight of the edge that connect 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' to neighbor 𝑣% is measured from the power function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The power function raises the 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='% to the power of 𝑥 with a constant parameter 𝜆, where the power can be 1 or any even positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For example, 𝑥 is assigned as 2 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The edge weights assigned by the convex penalty function grow smoothly compared to the exponential penalty function and sigmoid penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, the strictly convex property helps the shortest path algorithm approximate optimal solutions and avoids potential overflow error in numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 3) Shortest Path Calculation Since the end of each heartbeat is the beginning of its next heartbeat and they are continuous in time domain, the shortest path is then used to select the fiducial points that correspond to true heartbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' After constructing the weighted graph, the shortest path algorithm is applied on this graph and then the path with the least total weight is chosen [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The weight of vertex 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is assigned by finding the minimum of the weight of previous neighbors plus the weight of edges connected between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The previous neighbor that contributes to the minimum is selected as the previous vertex of 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' and denoted as 𝑝𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' , as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Greedy-optimized Fusion for Various Shortest Path One IBIs array is produced from one of three morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Consecutive IBIs, however, would not be estimated precisely from motion-corrupted PPG during intensive daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The beat-to-beat IBI plots depict that the estimated IBIs have overly fluctuating patterns, as compared to the true IBIs from ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Further, estimated IBIs arrays derived from different morphological features have different estimated time lengths, even if they come from the same heartbeat, due to the difficulty of extracting the true fiducial points from highly- distorted PPG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although the multi-channel model could improve the accuracy of estimated IBIs, most prevailing commercial wearable devices usually favor practical single channel PPG sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A greedy-optimized fusion technique for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Greedy-optimized fusion method for various shortest path utilizing morphological features Systolic Peaks Maximum slope Onset Fusion of 3 features 11误122误Algorithm 1 Shortest Path Detection for PPG Signals Input: Candidate fiducial points from PPG signals Output: the set of chosen vertices Vchosen l: <> 2: Concatenate all fiducial points from multiple-channel PPGs and sort them by timestamp to form the vertex set V = [Ui, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=',Un] // N is the total number of vertices 3: // Neighbor selection process for vertex Vi 4: for Vi E V do 5: j=i-l 6: while Vi - Vj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 * (IBIavg)i do 7: eii=Vi-Vi 8: Ni ← Vi / /N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is the set of neighbor vertices of Vi 9: j=j-l 10: end while 11: mi = i - j + 1 // mi is the number of neighbors of vi 12: end for 13: << Shortest Path Detection >> 14: f for i = 2 to N do 15: for j = i- l to i-mi do 16: Calculate the edge weight wii by convex penalty function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4) 17: end for 18: Wi = min(wij +wi), for j = i- l, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=',i- mi / /wi is the weight of vertex 19: pni = argmin(wij + wi), for j = i - l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='.,i - mi / /pni is the chosen VjENi previous vertex that contributes the minimal weight for Vi 20: end for 21: Udst = argmin (wN, wN-1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='., WN-mn) / /choose the vertex that has the UjE[NN,UN] minimal vertice weight within the t window as the destination of the short- est path, Udst 22: Backward search from Udst to select the vertices on the shortest path to form Vchosen 23: return Vchosen Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 6 IBI arrays from various morphological features is proposed in this study to provide a solution for those challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Due to the fact that the onset feature represents the beginning of a cardiac activity, IBIs array from the onset feature is selected as the baseline for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Firstly, the IBIs array of the onset feature is divided into q segments where each segment contains three consecutive IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The timestamps of this segmentation are used as references to guide the segmentation of IBI arrays for maximum slope and systolic peak features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For each segment, if the starting points of IBIs from maximum slope and systolic features are within this segment, these IBIs are included as the candidate IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Secondly, based on a physiological phenomenon that IBIs would not have drastic changes in a short period, true IBIs are expected to be close to their average IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, an objective function as equation (5) is proposed for the greedy-optimized fusion method to find local optimum IBIs in each step, or each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 𝚤̂, 𝚥̂, 𝑘A = argmin !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' (&() $1𝑖*+ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' − 𝑖,+ 1 + 1𝑖*+ & − 𝑖,+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=" 1 + 1𝑖*+ ) − 𝑖,+ / 1' (5) The following is the process of greedy-optimized fusion on three morphological features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The IBIs set generated from onset are denoted as 𝐼), IBIs set from systolic peak as 𝐼*, and IBIs set from maximum slope as 𝐼+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The pth segment of 𝐼), 𝐼*, and 𝐼+ are denoted as 𝐼),, 𝐼*,, and 𝐼+,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The individual IBIs in 𝐼), are denoted as 𝑖), , … , 𝑖), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' , where k must be equal to 3 for segments in 𝐼), but k can be any integer number close to 3 for segments in 𝐼* and 𝐼+, for example k can be 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The set of candidate IBIs in the pth segment are denoted as 𝐼/,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The set of average IBIs in the pth segment are denoted as 𝐼0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Take the pth segment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 3 as example, the starting points of four IBIs from 𝐼* (𝑖*, , 𝑖*, 1 , 𝑖*, 2 , 𝑖*, 3 ) and three IBIs from 𝐼+ *𝑖+, , 𝑖+, 1 , 𝑖+, 2 - are within the pth segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, candidate IBIs in the pth segment, 𝐼/,, is composed of F𝑖*, , 𝑖*, 1 , 𝑖*, 2 , 𝑖*, 3 , 𝑖+, , 𝑖+, 1 , 𝑖+, 2 , 𝑖), , 𝑖), 1 , 𝑖), 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The three IBIs in 𝐼/, that minimize the absolute error are chosen as shown in equation (5) and are concatenated into the final IBIs set 𝐼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This process iterates through the q segments to obtain the complete final IBIs set 𝐼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Algorithms and Complexity For the shortest path detection (Algorithm 1), the main loop in the algorithm runs 𝑁 ∗ 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' times, where the outer loop runs 𝑁 times for 𝑁 vertices and the inner loop runs 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' times for 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' neighbors of any vertex, 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='. Since neighbors are selected from a bounded window, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5-fold of average IBI (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='45-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 seconds), 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, the complexity of shortest path detection is 𝑂(𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For the greedy- optimized fusion method (Algorithm 2), the main loop in the algorithm runs q times, where q is the number of segments and less than N/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Since the number of candidate IBIs for each segment is around 9, the time complexity to find the least absolute error between the three IBIs and average IBIs is regarded as constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, the complexity of greedy- optimized fusion method is 𝑂(𝑁/3) = O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Dataset and Data Preprocessing I test the greedy-optimized framework on two datasets, the 2015 IEEE Signal Processing Cup training dataset (referred to as IEEE_Training) and the PPG-DaLiA dataset to have a comprehensive evaluation of performance during intensive exercises and daily activities [10, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The IEEE_Training dataset emphasizes the lab-based controlled conditions whereas the PPG-DaLiA dataset puts focus on daily life activities naturally, close to real-life conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1) IEEE_Training Dataset Two-channel PPG signals (PPG1 and PPG2) from wrist- worn sensors and one-channel ECG signal were collected synchronously from 12 healthy individuals aged 18 to 35 while they were running on the treadmill [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The running program was set up as Rest 30s → Jogging 1 min → Running 1 min → Jogging 1 min → Running 1 min → Rest 30s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Both ECG and PPG signals are at a sampling rate of 125 Hz and upsampled to 500 Hz to attain higher frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The up-sampling could provide precise timestamps when extracting features in ECG and PPG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Then, to eliminate the low frequency trending and high frequency noises, the single-channel PPG signals are preprocessed with a band-pass Butterworth filter with a cutoff frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 Hz and 15Hz whereas multi- channel PPG signals are preprocessed with a band-pass Butterworth filter with a cutoff frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 Hz and 15Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ECG signals are filtered with a high-pass Butterworth filter with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 Hz cutoff frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2) PPG-DaLiA Dataset This dataset includes synchronized PPG and ECG signals recorded from wrist-worn devices (Empatica E4) and chest- worn devices (RespiBAN Professional), respectively [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Data was recorded from 15 subjects while performing different kinds of daily activities as naturally as possible for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 hours, such as sitting, ascending/descending stairs, cycling, lunch break and working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I use two intense physical activities, ascending/descending stairs (5 mins) and cycling (8 mins), to evaluate the performance of the greedy-optimized framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The PPG signals from the PPG-DaLiA dataset are upsampled from 64 Hz to 500 Hz and filtered with a band-pass filter with a cutoff frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 Hz and 15Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The true R-peaks of ECG provided in this dataset are used to calculate the ground- truth IBIs for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Algorithm 2 Greedy-optimized Fusion for Various Shortest Paths Input: IBIs set from systolic peaks : Is IBIs set from maximum slope: IM IBIs set from onset : Io Average IBIs set : IA Output: The final IBIs set IF 1: Divide Io into q segments such that each segment has 3 IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2: Divide Is and Im based on Io segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 3: for p ← 1 to q do 4: Candidate IBIs set Icp = {Isp, IMp, Iop] 5: Find 3 IBIs from Icp such that: z,3, = argmin(licp -iapl +[icp -ipl +ip -ipl) 6: iik 7: where iAp, iAp and iAp are three average IBIs at pth segment IFp =[icp,icp,ic, ] 8: 9: IF← IFp 10: end for ll: return IF Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Interbeat Intervals Evaluation I evaluate the agreement between true and estimated IBIs using Pearson Correlation Coefficient (Corr) for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' As for accuracy performance metric, I rely on Mean Absolute Percentage Errors (MAPE) for each subject, defined as (6): 𝑀𝐴𝑃𝐸 = 1 𝑛 OP|𝑡𝑟𝑢𝑒𝐼𝐵𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' − 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝐼𝐵𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='| 𝑡𝑟𝑢𝑒𝐼𝐵𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' × 100Z 0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1- (6) where n is the total number of IBIs in one subject, 𝑡𝑟𝑢𝑒𝐼𝐵𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' denotes the i’th true IBI from ECG and 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝐼𝐵𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' denotes the i’th estimated IBI from PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 1) Evaluation on IEEE_training Dataset Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' depicts the overall performance evaluation of IBI estimation on IEEE_Training dataset, which compares different penalty functions and shows results from single-channel and two-channel models using each morphological feature individually and fusion of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The results are the average of 12 subjects in the IEEE_Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Several observations from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' are stretched out below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' First, the two-channel model outperforms both the single-channel model PPG1 and the single-channel model PPG2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The two-channel model achieves a MAPE of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% for systolic peak (SP), maximum slope (MS), and onset, respectively, which has 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6%, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7%, and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6% improvement, respectively, as compared to single-channel PPG1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Further, analysis on the channel usage shows that the PPG1 and PPG2 accounts for 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3% and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7% fiducial points in the two-channel model using onset feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I implement the convex penalty function and other different penalty functions using Python and compare them comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Results show that the convex penalty function outperforms exponential and sigmoid penalty functions for all three morphological features either in the single-channel model or two-channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I evaluated results with sensitivity analysis of the convex penalty function with the power of 1, 2, 4, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Since they provide similar performance and the 2nd power is the best, 2nd power is chosen for the convex penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Note that ε is a parameter that controls the tolerance of assigning zero edge weight in exponential penalty function and sigmoid penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' It is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1 for the single-channel model as it is described in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For the two-channel model, I empirically tested the 𝜀 parameter and found the performance is best when ε is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='06, suggesting that the exponential penalty function and sigmoid penalty function are sensitive to the ε parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Interestingly, the performance of exponential penalty function and sigmoid penalty function are identical in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Overall, it shows that an effective penalty function is critical in IBI estimation using the shortest path algorithm where the convex function is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Last but most importantly, results demonstrate the effectiveness of the greedy-optimized fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In the two-channel model, the fusion method achieves correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 and MAPE of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2%, where the MAPE is improved by 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1% (reducing from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2%) as compared to the case using the onset feature individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Further, IBI estimation in the single-channel model has significant improvements using the greedy-optimized fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The correlation of single- channel PPG1 from onset feature without fusion is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='86, whereas the correlation reaches to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 after applying the greedy-optimized fusion, which is improved by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' COMPARISON OF SINGLE- AND TWO-CHANNEL MODEL WITH DIFFERENT PENALTY FUNCTIONS IN IBI ESTIMATION PERFORMANCE USING THREE MORPHOLOGICAL FEATURES AND FUSION OF THEM Penalty Functions SP MS Onset Fusion** Corr MAPE Corr MAPE Corr MAPE Corr MAPE Single-channel (PPG1) Convex Penalty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1% 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9% n/a n/a Best result (PPG 1&2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% Aygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' [19] (PPG1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='82 n/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='85 n/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='86 n/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='89 n/a This table shows average metric over the first 12 subjects in IEEE_Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' n/a = not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' My implementation in python using the exponential penalty function [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ** Greedy-optimized fusion of three morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 8 Similarly for the single-channel PPG2, the correlation jumps greatly from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='84 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='95, which is improved by 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In addition, the MAPE is reduced to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7% in the single- channel PP1 and PPG2, respectively, which is 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% improvement as compared to the case using the onset feature individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' presents the breakdown result of 12 subjects after the greedy-optimized fusion method for single- channel models and two-channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 demonstrates the effectiveness of the greedy- optimized fusion method for tackling the challenge of estimated IBIs from PPG which have overly fluctuating patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The challenge could be explicitly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 (a), which is the IBIs calculated from the single-channel PPG1 onset feature of subject 2 in IEEE_Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This difficulty could also be viewed in the estimated IBIs from the single-channel PPG1 onset feature of subject 6, shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The greedy- optimized fusion method mitigates the overly fluctuating pattern and improves the correlation by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% and reduces the MAPE by 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3% and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% in subject 2 and subject 6, respectively as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 (b, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, even though the two-channel PPG1 + PPG2 model already provided very high correlations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 and low percentage errors of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9% for subject 2 and subject 6, respectively, the fluctuation challenge remains, shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 (e, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The effectiveness of the greedy-optimized fusion method is well demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 4 (f, h), where the estimated IBIs are much closer to the true IBIs with low fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The correlations achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99 for subject 2 and subject 6, respectively, and MAPEs reduce to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8%, which indicates the greedy-optimized fusion provides a solution for the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 2) Evaluation on PPG-DaLiA Dataset To further evaluate the robustness of the greedy-optimized framework in detecting IBIs on PPG signals in daily real-life conditions, I also apply my techniques on the PPG-DaLiA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This dataset only provides one channel PPG from commercial wearable wristbands [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I extract segments of two intensive activities, ascending/descending stairs and cycling, from PPG signals, which have duration of 5 mins and 8 mins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Note that there are certain amounts of abnormal high spikes in the true IBI of subject 6 and subject 10, so I correct the true IBI annotation of subject 10 for stairs activity and subject 6 for cycling activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I use the average heart rate (window length: 8 s, window shift: 2 s) provided in this dataset to calculate the average IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The single-channel model with fusion of three morphological features achieves high correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04 and low MAPE of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% for ascending/descending stairs activity and high correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04 and low MAPE of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7% for cycling activity, shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Heart Rate Variability Analysis HRV can be described using time-domain and frequency- domain measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The time-domain measurements quantify the amount of variability in measurements of the IBI TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' IBI ESTIMATION PERFORMANCE OF GREEDY-OPTIMIZED FRAMEWORK AND COMPARISON OF SINGLE CHANNEL AND TWO CHANEEL PPG SIGNALS FROM 12 SUBJECTS IN IEEE_TRAINING Subject ID PPG1 (Fusion) PPG2 (Fusion) PPG1&2 (Fusion) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% 2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Time (sec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Time (sec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Time (sec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Time (sec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Subject 2 - PPG1 + PPG2 Onset (No Fusion) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Subject 2 - PPG1 + PPG2 (Fusion of 3 features) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Subject 6 - PPG1 + PPG2 Onset (No fusion) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Subject 6 - PPG1 + PPG2 (Fusion of 3 features) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='006 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94, MAPE : 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5% 006 Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97, MAPE : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0% 900 Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98, MAPE : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9% 900 Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99, MAPE : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% 800 800 800 800 700 B 500 500 500 500 400 400 400 400 Estimated IBI Estimated IBI Estimated IBI Estimated IBI 300 True IBI 300 True IBI 300 True IBI 300 True IBI 0 50 100 150 200 250 300 0 50 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 Time (sec) Time (sec) Time (sec) Time (sec) Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 9 during monitoring periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' These metrics include the standard deviation of heart rate (STD HR), standard deviation of the IBI of normal sinus beats (SDNN), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' SDNN has been used as the medical stratification of cardiac risk of morbidity and mortality for heart attack survivors [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The frequency-domain measurements estimate the distribution of absolute or relative power into three frequency bands: very-low-frequency (VLF), low-frequency (LF), and high-frequency (HF) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' VLF Power has been reported to be associated with all-cause mortality, arrhythmic death, and post-traumatic stress disorder (PTSD) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HF band reflects parasympathetic activity and is related to the respiratory cycle and is correlated with mental health, such as stress, panic and anxiety [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' To comprehensively evaluate the performance of the greedy- optimized framework in estimating HRV, I apply the HRV analysis to two datasets, IEEE_Training (12 subjects) and PPG- DaLiA (15 subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The estimated and true IBIs are used to calculate four time-domain HRV parameters and four frequency-domain HRV parameters using pyHRV [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The four time-domain HRV parameters investigated in this paper include Mean RR (ms), SDNN (ms), Mean HR(1/min) and STD HR (1/min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The frequency domain HRV parameters investigated in this paper are computed using the autoregressive method to separate HRV into its component frequency band, including the VLF Power (absolute power of the VLF band of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='00 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04 Hz), LF Power (absolute power of the LF band of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='15 Hz), HF Power (absolute power of the HF band of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='15–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 Hz), and the Total Power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The HRV results are evaluated by Pearson Correlation Coefficient (Corr) and the accuracy are evaluated by Mean Absolute Percentage Errors (MAPE), defined as (7): 𝑀𝐴𝑃𝐸 = 1 𝑛 @ A|𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' − 𝑒𝑠𝑡𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='| 𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' × 100M & !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=" '( (7) where n is the total number of subjects in the dataset, 𝑡𝑟𝑢𝑒𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' denotes the HRV parameter derived from true IBIs and 𝑒𝑠𝑡𝐻𝑅𝑉𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' denotes the HRV parameter derived from estimated IBIs of the i’th subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The Table IV shows HRV analysis results of IEEE_training Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV analysis of this study is based on IBI estimation results of two- channel (PPG1&2) after the greedy-optimized fusion method is applied on three features, SP, MS and Onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV analysis of Aygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' is based on IBI estimation results of single-channel (PPG1) from IEEE_Training using their fusion method of three morphological features [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Results demonstrate that the estimated and true HRV parameters are highly correlated with low percentage errors in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The Pearson correlation coefficients are above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9 significantly with all p-values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='001, except that the coefficient of HF Power is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' This study provides the MAPEs which are less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7% for all eight HRV parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV analysis of PPG-DaLiA Dataset is shown in Table V, which is based on IBI estimation results of single-channel PPG from Empatica E4 of 15 subjects in stairs and cycling activities after the greedy-optimized fusion method is applied on three features, SP, MS and Onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The results show that the estimated and true HRV parameters are highly correlated with low absolute percentage errors for both the stairs and cycling activities and the performance of the cycling activity is better TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' IBI ESTIMATION PERFORMANCE OF GREEDY- OPTIMIZED FRAMEWORK USING SINGLE-CHANNEL PPG FROM 15 SUBJECTS IN THE PPG-DALIA Subject ID Ascending/Descending Stairs (5 mins) Cycling (8 mins) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97 | 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9 % 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1 % 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='92 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3 % 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='94 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='85 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='87 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6 % 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='89 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97* | 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6 %* 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='89 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='97 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='85 | 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='89 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9 % 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='87* | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6 %* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='92 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99 | 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='92 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='99 | 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='90 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='91 | 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='95 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 % SD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04 | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='04 | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % I found amounts of abnormal high spikes for subject 6 and subject 10’s true IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, I corrected the true IBI annotation of subject 10 for stair activities and subject 6 for cycling activities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV PARAMETERS PERFORMANCE OF IEEE_TRAINING HRV Parameters Greedy-optimized Framework* Aygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' [19] Corr** MAPE Corr MAPE Mean RR (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='986 n/a SDNN (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='3 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='956 n/a Mean HR (1/min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='987 n/a STD HR (1/min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='990 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='860 n/a VLF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='981 n/a LF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='898 n/a HF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='858 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='828 n/a Total Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='932 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='974 n/a HRV analysis is based on IBI estimation results of two-channel (PPG1&2) from 12 subjects in treadmill activity on IEEE_Training using the greedy-optimized framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' ** All p-value of Pearson correlation coefficient are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HRV PARAMETERS PERFORMANCE OF PPG-DALIA HRV Parameters Ascending/Descending Stairs (8 mins) Cycling (5 mins) Corr* MAPE Corr* MAPE Mean RR (ms) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1 % SDNN (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='981 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='998 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 % Mean HR (1/min) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1 % STD HR (1/min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='935 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='996 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % VLF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2 % LF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='905 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='1 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8 % HF Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='786 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='784 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 % Total Power (ms2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='843 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='7 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='888 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='9 % HRV analysis is based on IBI estimation results of single-channel PPG from Empatica E4 of 15 subjects in stairs and cycling activities on the PPG-DaLiA using the greedy-optimized framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' All p-value of Pearson correlation coefficient are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 10 than the stairs activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Note that HF Power has the lowest correlation among the eight HRV parameters in the stairs and cycling activities of the PPG-DaLiA and in the IEEE_Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 5 provides the scatterplots that compare the true and estimated SDNN, STD HR, VLF Power, LF Power, HF Power and Total Power derived from PPG signals during intensive treadmill activities on IEEE_Training dataset and in the stairs and cycling activities of PPG-DaLiA Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' These plotted points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 5 (a, b, c, d, e, f) are distributed along with the identity line closely, showing that the true and estimated HRV results have high correlations and small absolute errors in all three intensive activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' HF Power, however, has the highest absolute errors among the four frequency-domain parameters and is often overestimated in the PPG-DaLiA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' DISCUSSION In the application of healthcare monitoring through wearable sensors, IBI and HRV estimation from PPG are challenging because motion-artifacts induced by daily or exercise activities significantly deteriorates the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The most common strategy in analyzing IBI and HRV from noisy PPG is discarding motion-contained signal segments, which loses the opportunity of discovering potential health information which is triggered during exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' In this study, I show that the greedy-optimized framework, which leverages convex penalty function in shortest path calculation and greedy-optimized fusion method, could provide high accuracy in estimating IBI and HRV from whole PPG signals obtained during daily and intensive exercise activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The set of IBIs selected by the short path algorithm in a directed acyclic graph is regarded as the optimal among candidate IBIs from one shortest path in terms of resembling the true IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although this guarantees estimated IBIs to have high correlation with true IBIs, the observation in IBI plots shows a big challenge that those estimated IBIs are over-fluctuating and cause large absolute errors as compared to true IBIs, causing those estimated IBIs are not ideal representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' To tackle this challenge, the greedy-optimization fusion method for various shortest paths is proposed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' By leveraging a physiological phenomenon that true IBIs are close to their average IBIs, I develop an objective function for the greedy- optimized fusion method to find local optimum IBIs in each step, or each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Through the process, the greedy-optimization fusion method selects optimal IBIs that have the least absolute error with the average IBI set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Results show that the greedy- optimized fusion reduces the MAPE by at least 50% in both the single-channel and two-channel models and enormously mitigates inherently over-fluctuating beat-to-beat IBIs estimated from noisy PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' IBI estimation from the multi-channel PPG signals outperform the single-channel PPG signal in a previous study [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Nevertheless, practical direction in healthcare remote monitoring is to develop a compact and portable wearable sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Prevailing commercial wearables are embedded with only one PPG sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Hence, it is crucial to develop a model which is capable of achieving high accuracy even if the wearable has a single-channel signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Results from PPG-DaLiA indicate that my techniques have the ability to accurately estimate the IBI and HRV from PPG on a commercial wearable, Empatica E4, which has one channel of PPG sensor with low sampling rate of 64Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Furthermore, the greedy framework proposed in this study has efficient time complexity of O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Given the computational efficient nature of greedy, the framework could be implemented with edge computing for commercial wearables and could be applied in real world healthcare remote monitoring applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Although the proposed greedy-optimized framework has nice performance in IBI and HRV estimation from noisy PPG signals under daily intensive activities, there is a crucial material needed to know for applying the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Average HR is an important input in this optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The accuracy of IBI and HRV estimation would be limited when the accuracy of average HR decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Favorably, despite the above limitation, this optimization framework for IBI and HRV estimation is independent of average HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Users could use any algorithm that generates accurate average HR from noise-contaminated PPG, such as WFPV [11], particle filtering [12] and Deep PPG [29], which have been matured for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Another thing which needs attention is that currently I evaluate this optimization framework in the dataset of 5-8 minutes duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' It has not been investigated in the dataset with longer duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' For example, PPG signals which are collected overnight during sleep or over one day period (above 24 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Further, some studies have shown that long-term HRV parameters (24 hours) are a more Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Scatterplot comparison of true/estimated HRV parameters (a) SDNN (b) STD HR (c) VLF Power (d) LF Power (e) HF Power and (f) Total Power from noisy PPG signals on IEEE_Training (treadmill) and PPG-DaLiA (stairs and cycling) 140 IEEE_ Training 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 IEEE Training ★ ★ ★ DaLiA_cycling DaLiA_cycling 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 120 DaLiA stairs DaLiA stairs (sw) i SDNN( 100 + F 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 ** 工 80 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 Estimated 60 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='5 40 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='0 20 7.' metadata={'source': 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+page_content='True HF Power (ms2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='True Total Power (ms2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='Luffina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Huang: A Greedy-optimized Framework for HRV Monitoring during Daily Activities using Wearable PPG 11 powerful predictor of mortality than short-term HRV parameters for patients with chronic heart failure and acute myocardial infarction [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' A future work could be extended into evaluating this greedy-optimization framework for IBI and HRV estimation in long period wearable PPG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' CONCLUSION This paper proposes a greedy-optimized framework for IBI and HRV estimation on single-channel and multi-channel PPG signals collected during intensive daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Two proposed techniques, convex penalty function and greedy-optimized fusion method, equip the framework with the capability of improving the accuracy of the IBI and HRV estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The convex penalty function is introduced to optimize edge weights assignment in the shortest path calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The greedy-optimized fusion method mitigates highly fluctuating patterns in estimated IBIs, achieving the better approximation of true IBIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' On 2015 IEEE Signal Processing Cup, the greedy-optimized framework achieves low average percentage errors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='2% with high average correlations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='98 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='96 for IBI estimation through two- channel PPGs and single-channel PPG1, respectively, with O(n) complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Results also demonstrate the convex penalty function outperforms the exponential and sigmoid penalty function in the shortest path algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The proposed greedy- optimized fusion successfully reduces the MAPE by 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% and improves the correlation by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='6% in the single-channel PPG1 for IBI estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' I further validate the proposed framework on two daily activities from the PPG-DaLiA Dataset, which uses single-channel PPG commercial wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The estimated IBIs achieve high average correlations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='92 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='95 with low percentage error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='8% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content='4% for the ascending/descending stairs and cycling activities, respectively, indicating that this framework could be adaptive to single sensor PPG wearables on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' The estimated and true HRV parameters (Mean RR, SDNN, Mean HR, STD HR, VLF Power, LF Power and Total Power) are also highly correlated with low percentage errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfFwOO/content/2301.02906v1.pdf'} +page_content=' Since the accuracy of IBI and HRV estimation is consistently favorable across three activities from those two datasets with low standard 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a/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/2301.12009v1.pdf.txt b/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/2301.12009v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6dc6553e7158fbe29f2d1b01a7a42f6dc932849 --- /dev/null +++ b/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/2301.12009v1.pdf.txt @@ -0,0 +1,17722 @@ +Inference for all variants of the multivariate coefficient of variation +in factorial designs +Marc Ditzhaus1 and �Lukasz Smaga2,∗ +1Faculty of Mathematics, Otto von Guericke University Magdeburg, Germany +2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland +January 31, 2023 +Abstract +The multivariate coefficient of variation (MCV) is an attractive and easy-to-interpret effect size +for the dispersion in multivariate data. Recently, the first inference methods for the MCV were +proposed by Ditzhaus and Smaga (2022) for general factorial designs covering k-sample settings +but also complex higher-way layouts. However, two questions are still pending: (1) The theory +on inference methods for MCV is primarily derived for one special MCV variant while there are +several reasonable proposals. (2) When rejecting a global null hypothesis in factorial designs, a +more in-depth analysis is typically of high interest to find the specific contrasts of MCV leading to +the aforementioned rejection. In this paper, we tackle both by, first, extending the aforementioned +nonparametric permutation procedure to the other MCV variants and, second, by proposing a +max-type test for post hoc analysis. To improve the small sample performance of the latter, we +suggest a novel studentized bootstrap strategy and prove its asymptotic validity. +The actual +performance of all proposed tests and post hoc procedures are compared in an extensive simulation +study and illustrated by a real data analysis. +Keywords: Coefficient of variation, factorial designs, multiple testing, bootstrap and permutation +procedures, multivariate analysis, standardized mean +1 +Introduction +The coefficient of variation (CV) is defined as the standard deviation divided by the population mean. +By this, it becomes a powerful, unit-free measure of dispersion and is used in diverse areas, e.g. in +medicine for reliability and reproducibility of measurements (Neumann et al., 2021), for risk evaluation +in finance (Ferri and Jones, 1979) or in psychology (Weber et al., 2004). Furthermore, it is used in +control charts for monitoring (Jalilibal et al., 2021). However, as pointed out by Yeong et al. (2016) +in the context of control charts: “There are many situations where multiple characteristics need to +be monitored simultaneously.” This is certainly true apart from control charts, e.g. when various +medical measurements are taken from the same patients. For such scenarios, the CV can be extended +to the multivariate setting in various ways (Reyment, 1960; Van Valen, 1974; Voinov and Nikulin, 1996; +Albert and Zhang, 2010): +CRR = +� +(det Σ)1/d +µ⊤µ +, CVV = +� +trΣ +µ⊤µ, CVN = +� +1 +µ⊤Σ−1µ, CAZ = +� +µ⊤Σµ +(µ⊤µ)2 , +(1) +which all reduces to CV in the univariate case (d = 1). Here µ ̸= 0 denotes the mean vector of a +d-dimensional random variable and Σ is the corresponding covariance matrix. The standardized means +as the reciprocal of Cv are of their own interest: +Bv = 1 +Cv +(v = RR, VV, VN, AZ). +(2) +∗ Corresponding author. Email address: ls@amu.edu.pl +1 +arXiv:2301.12009v1 [stat.ME] 27 Jan 2023 + +The differences between the four variants are discussed in great detail by Albert and Zhang (2010). One +remarkable difference is that CRR and CVN require a regular matrix Σ while the other two variants do +not. This regularity assumptions becomes rather restrictive for high-dimensional scenarios, such as +microarray data. Moreover, the variant CVV is the only one which does not take the covariance of the +different measurements into account. +When we turn to the inference problem, the literature regarding the MCV becomes scare while for +the univariate CV various two- and k-sample testing proposal can be found in the literature, see Aerts +and Haesbroeck (2017) and Pauly and Smaga (2020). Recently, Ditzhaus and Smaga (2022) addressed +the remaining question for more general CV testing procedures, namely in complex factorial designs. +The latter are highly relevant for various fields, e.g. in biomedicine or psychology (GISSI-2, 1990; +Baigent et al., 1998; Cassidy et al., 2008; Kurz et al., 2015), where the k-sample set-up is often too +narrow. Methods for factorial designs allow to discuss main effects (e.g. of the gender, measurement, +site) and even interaction effects: ’it is desirable for reports of factorial trials to include estimates +of the interaction between the treatments’ (Lubsen and Pocock, 1994). In addition to the extension +towards factorial designs, Ditzhaus and Smaga (2022) proposed their nonparametric methods directly +for CVN in the multivariate set-up. This complemented the prior proposal of Aerts and Haesbroeck +(2017), which, in contrast, relied on (semi-)parametric model assumptions and whose convergence +rate was rather slow, see the simulation results of Ditzhaus and Smaga (2022). To the best of our +knowledge, these two proposals are the only ones discussing the inference problem for MCV and, +moreover, both restricted their study to CVN. Thus, the natural question arises whether the inference +strategies can be transferred to the other MCV variants from (1), especially to be able to study +also settings with non-regular covariance matrices Σ. For this purpose, we will adopt the results of +Ditzhaus and Smaga (2022) and, in particular, derive permutation versions with a better performance +under small sample sizes, see Section 5. In this way, we add a further chapter to the success story of +studentized permutation tests in complex factorial designs (Pauly et al., 2015; Friedrich et al., 2017; +Harrar et al., 2019; Ditzhaus et al., 2021b). While classical permutation tests for exchangeable data +settings are well-known, it is less known that the studentized permutation versions are also valid +beyond exchangeability. +Beside global null hypothesis testing in k-sample settings or inferring main/interaction effects +in factorial designs, often a more in-depth analysis is wanted, e.g. multiple pairwise comparisons, +to get a better picture of the underlying effects. Since Bonferroni correction leads partially to a +significant power loss, strategies incorporating the concrete dependence structure are preferred. Therefor, +multiple contrast tests and corresponding simultaneous confidence intervals are well established for +means (Mukerjee et al., 1987; Bretz et al., 2001), the relative treatment effect (Umlauft et al., 2019; +Gunawardana and Konietschke, 2019), and the area under the receiving operating curve (Konietschke +et al., 2018; Wechsung and Konietschke, 2021). A respective multiple strategy for the MCV is missing. +The remainder of this paper is organized as follows. In Section 2, we derive a central limit theorem +for all MCV variants and their reciprocals in the one-sample setting. Moreover, we discuss assumptions +such that the limit is not degenerated and how the limiting variances can be estimated consistently. +All these results lie the foundation for the Wald-type statistics to infer main and interaction effects +in terms of MCVs and standardized means in general factorial designs, see Section 3. Moreover, we +develop respective permutation and bootstrap counterparts of these Wald-type statistics and prove +their asymptotic validity. In Section 4, we discuss the issue of simultaneous inference and present +max-type multiple contrast tests. Again, we complement this by an asymptotically valid resampling +procedure. An exhaustive simulation study is presented in Section 5. The tests’ applicability are +illustrated by analyzing data of external quality assessment in Section 6. Finally, Section 7 summarizes +the major results of the paper and discusses further research possibilities. All proofs and additional +simulation results are in the supplementary materials. +2 + +2 +The nonparametric framework +Consider n1 independent, identically distributed d-dimensional random variables +Xj = (Xj1, . . . , Xjd)⊤ +(j = 1, . . . , n1). +Hereby, we suppose no specific conditions on the distributions of Xj except the following assumptions +on the moments to ensure the well-definedness of Cj and Bj: +Assumption 1. Let µ ̸= 0 and E(X4 +jℓ) < ∞ for all j and ℓ. Moreover, we suppose: +(a) For CRR, CVN, BRR, and BVN, we consider only regular matrices Σ. +(b) For CVV and BVV, we assume Σ ̸= 0d×d. +(c) For CAZ and BAZ, we suppose µ⊤Σµ > 0. +Here and below, we denote by 0d×d a d × d-dimensional matrix consisting of zeros only. +Assumption 1 ensures that Cv, Bv are well-defined. Clearly, a ⇒ c ⇒ b. Thus, CVV seems to be +the most general variant. But, as mentioned in the introduction, CVV does not take the covariance +structure into account and only combines the marginal variability into one standardized effect size. For +statistical inference, we estimate the MCVs or their reciprocals, respectively, by plugging-in the sample +mean �µ and covariance matrix �Σ: +�CRR = +� +(det �Σ)1/d +�µ⊤ �µ +, �CVV = +� +tr�Σ +�µ⊤ �µ +, �CVN = +� +1 +�µ⊤ �Σ +−1 �µ +, �CAZ = +� +�µ⊤ �Σ�µ +(�µ⊤ �µ)2 , +(3) +and �Bv = 1/ �Cv. By extending the results of Ditzhaus and Smaga (2022) for CVN and BVN, we are +able to derive central limit theorems for all these estimators. The respective asymptotic variances have +a rather complex structure and depend on several quantities: +ARR(µ, Σ) = +� +−2d det(Σ) +µ⊤ +(µ⊤µ)d+1 + det(Σ) +� +vec(Σ−1) +�⊤ +(µ⊤µ)d +� +D(µ), det(Σ) +� +vec(Σ−1) +�⊤ +(µ⊤µ)d +� +, +AVV(µ, Σ) = +� +−2 tr(Σ) +µ⊤ +(µ⊤µ)2 + +1 +µ⊤µ (vec(Id))⊤ � +D(µ), +1 +µ⊤µ (vec(Id))⊤ +� +, +AVN(µ, Σ) = +� +2 µ⊤Σ−1 − [(µ⊤Σ−1) ⊗ (µ⊤Σ−1)] � +D(µ), −(µ⊤Σ−1) ⊗ (µ⊤Σ−1) +� +, +AAZ(µ, Σ) = +� +−4 µ⊤Σµ +µ⊤ +(µ⊤µ)3 + 2 µ⊤Σ +(µ⊤µ)2 + µ⊤ ⊗ µ⊤ +(µ⊤µ)2 � +D(µ), µ⊤ ⊗ µ⊤ +(µ⊤µ)2 +� +where ⊗ is the Kronecker product, and the matrices � +D(x) ∈ Rd2×d for x = (x1, . . . , xd)⊤ ∈ Rd, +Ψi3 ∈ Rd2×d as well as Ψi4 ∈ Rd2×d2 are given by their entries +[ � +D(x)]ad−d+r,s = −xrI{s = a ̸= r} − 2xsI{s = r = a} − xaI{r = s ̸= a} +[Ψ3]ad−d+r,s = E(X1aX1rX1s) − E(X1aX1r)E(X1s) +(4) +[Ψ4]ad−d+r,bd−d+s = E(X1aX1rX1bX1s) − E(X1aX1r)E(X1bX1s) +for a, b, r, s ∈ {1, . . . , d}. Now, we are able to formulate the central limit theorems. Here and subsequent, +all limits are meant as n1 → ∞ unless stated explicitly otherwise. +Theorem 1 (Central limit theorem). Let v ∈ {RR, VV, VN, AZ} and Assumption 1 be fulfilled. The +estimators �Cv and �Bv +i are asymptotically normal: +n1/2 +1 +� +�Cv − Cv� +d +−→ ZCv ∼ N(0, σ2 +Cv), and n1/2 +1 +� +�Bv − Bv� +d +−→ ZBv ∼ N(0, σ2 +Bv) +3 + +with asymptotic variances σ2 +Bv = (Cv)−4σ2 +Cv and +σ2 +Cv = Sv +4 Av(µ, Σ) +� +Σ +Ψ⊤ +3 +Ψ3 +Ψ4 +� +Av(µ, Σ)⊤, +where SRR = d−2(CRR)2−4d, SVV = (CVV)−2, SVN = (CVN)6, SAZ = (CAZ)−2. +The typically unknown variances, σ2 +Bv and σ2 +Cv, can be naturally estimated by replacing the +expectations and covariances by their empirical counterparts, for instance: +[ �Ψ3]ad−d+r,s = +� +n−1 +1 +n1 +� +j=1 +XjaXjrXjs +� +− +� +n−1 +1 +n1 +� +j=1 +XjaXjr +�� +n−1 +1 +n1 +� +j=1 +Xjs +� +. +In this way, we obtain +�σ2 +Cv = +�Sv +4 Av(�µ, �Σ) +� +�Σ +�Ψ +⊤ +3 +�Ψ3 +�Ψ4 +� +Av(�µ, �Σ)⊤, +�σ2 +Bv = ( �Cv)−4�σ2 +Cv, +(5) +where �SRR = d−2( �CRR)2−4d, �SVV = ( �CVV)−2, �SVN = ( �CVN)6, �SAZ = ( �CAZ)−2. A direct consequence +of the continuous mapping theorem and the strong law of large numbers is +Lemma 1. Under Assumption 1, �σ2 +Cv +p→ σ2 +Cv and �σ2 +Bv +p→ σ2 +Bv. +In general, there is no guarantee that the limits from Theorem 1 are not degenerated, i.e. σ2 +Cv = 0 +and, equivalently, σ2 +Bv = 0 might be possible. As in the univariate case (Pauly and Smaga, 2020) and +for v = VN in the multivariate setting (Ditzhaus and Smaga, 2022), degeneracy can just appear in +rather unusual scenarios of the following kind: +Definition 1 (Conditional two-point distribution). Let Y = (Y1, . . . , Yd)⊤ ∈ Rd be a multivariate +random variable. We call the rth coordinate Yr conditionally two-point distributed if it is (conditionally) +degenerated or it just takes (conditionally) two different values with positive probability, both given the +remaining components (Ys)s=1,...,d;s̸=r. +For example, the coordinates of (Y1, Y2, Y1 + Y 2 +2 ) for arbitrarily distributed Y1, Y2 are conditionally +two-point distributed. The same is true for the coordinates of (Y1, . . . , Yd) with binomial distributed +Yj. These extreme cases need to be excluded: +Assumption 2. No coordinate of X1 is conditionally two-point distributed. +In fact, a weaker assumption is also enough as illustrated in the proofs. In detail, it is sufficient to +suppose that the ℓth coordinate of X1 is not conditionally two-point distributed for some ℓ = 1, . . . , d. +However, we then additionally require [µ]ℓ ̸= 0 in case of v = AZ and [µ⊤Σ−1]ℓ ̸= 0 for v = VN. From +our point of view, Assumption 2 is easier to check, in particular later for the resampling procedures, +and we do not loose much of generality. +Lemma 2. Under Assumptions 1 and 2 we have σ2 +Cv > 0 and, thus, σ2 +Bv > 0. +By Theorem 1 and Lemma 1 [ �Cv ± n−1/2 +1 +�σCvz1−α/2] and [ �Bv ± n−1/2 +1 +�σBvz1−α/2] are asymptotically +valid confidence intervals for Cv and Bv, respectively, where z1−α/2 is the (1 − α/2)-quantile of N(0, 1). +Moreover, these results serve as the foundation for inference methods in more complex models as +discussed below. +4 + +3 +Global testing for factorial designs +3.1 +Factorial designs with respective null hypotheses +From the easy one-sample scenario from the previous section, we immediately turn to general factorial +designs covering two- and k-sample settings as special cases. Notationally, factorial designs can be +incorporated in a k-sample framework by interpreting the groups as subgroups for different factor +combinations. We explain this concept below in more detail but first start with introducing the concrete +model. For this purpose, we add another index i = 1, . . . , k, k ∈ N, to all quantities from Section 2. In +particular, let +Xij = (Xij1, . . . , Xijd)⊤ +(i = 1, . . . , k; j = 1, . . . , ni), +where Xi1, . . . , Xini are identically distributed for each i = 1, . . . , k and all observations X11, . . . , Xknk +are mutually independent. Depending on the research question, we choose a contrast matrix H ∈ Rr×k, +i.e. H1k×1 = 0r×1. We now like to infer the following null hypothesis in terms of the chosen H and +by this cover a huge variety of testing problems: +H0,Cv : HCv = 0r×1, +H0,Bv : HBv = 0r×1. +(6) +Here, Cv = (Cv +1, . . . , Cv +k)⊤ and Bv = (Bv +1, . . . , Bv +k)⊤. In the same way, we denote by � +C +v and � +B +v the +respective estimators. +Different choices for H: To see the high flexibility of (6), we like to discuss some specific cases. +The most prominent one is the k-sample scenario H0,Cv : {P kCv = 0k×1} = {Cv +1 = . . . = Cv +k}, where +P k = Ik − 1k×k/k and Ik is the k × k-dimensional unity matrix. Turning to a more complex scenario, +we next consider a two-way layout with two factors A and E possessing a and e levels, respectively. By +splitting up the group index i = (iA, iE) we incorporate this scenario in the aforementioned k-sample +framework. In particular, we obtain k = a · e subgroups. Now, we divide the subgroup-specific MCV +Cv +iA,iE = Cv0 + Cvα +iA + Cvϵ +iE + Cvαϵ +iAiE +into a general effect Cv0, the two main effects Cvα +iA , Cvϵ +iE, and an interaction effect Cvαϵ +iAiE. Here, the +usual side conditions � +iA Cvα +iA = � +iB Cvϵ +iE = � +iA Cvαϵ +iAiE = � +iE Cvαϵ +iAiE = 0 ensure the identifiability of +the aforementioned effects. Related null hypotheses are: +• HA +0,Cv : {HACv = 0} = {Cvα +iA = 0 ∀iA}, HA = P a ⊗ (11×e/e) (no main effect A). +• HE +0,Cv : {HECv = 0} = {Cvϵ +iE = 0 ∀iE}, HE = (11×a/a) ⊗ P e (no main effect E). +• HAE +0,Cv : {HAECv = 0} = {Cvαϵ +iAiE = 0 ∀iA, iE}, HAE = P a ⊗ P e (no interaction). +Hereby, ⊗ is the Kronecker product. Clearly, we can replace Cv by Bv to get respective null hypotheses +for the latter. The described strategy can be extended, in a straightforward manner, to higher-way +layouts and hierarchical designs with nested factors, see e.g. Sec. S1.3 of the supplement from Ditzhaus +et al. (2021b) or Sec. 4 of Pauly et al. (2015). +3.2 +Wald-type tests +How to test global null hypothesis of the form (6) is discussed in several papers. In general, quadratic +forms are built on the estimating vectors, here � +C +v or � +B +v. Popular examples are (modified) ANOVA- +type statistics (e.g. Brunner et al., 1997; Friedrich and Pauly, 2018; Sattler et al., 2022) and Wald-type +statistics (e.g. Pauly et al., 2015; Smaga, 2015, 2017; Ditzhaus et al., 2021b). In particular, the latter +is usually an asymptotically pivotal statistic which is beneficial for the resampling procedures proposed +later. For the results, we require the following classical assumption of non-vanishing groups +ni +n → κi ∈ (0, 1) for all i = 1, . . . , k. +(7) +5 + +This limit and all following ones are meant as the total sample size n = �k +i=1 ni tends to ∞. This is in +line with the prior convention from Section 2, where we considered only one group, i.e. n = n1. Now, +we can formulate and motivate the Wald-type statistics for (6): +Sn,Cv(H) = n(H � +C +v)⊤(H �ΣCvH⊤)+H � +C +v, Sn,Bv(H) = n(H � +B +v)⊤(H �ΣBvH⊤)+H � +B +v, +where �ΣCv = diag((n/n1)�σ2 +1,Cv, . . . , (n/nk)�σ2 +k,Cv) and �ΣBv = diag((n/n1)�σ2 +1,Bv, . . .), and A+ denotes +the Moore–Penrose inverse. By Lemma 1, these estimators are consistent for ΣCv = diag(κ−1 +1 σ2 +1,Cv, . . . , κ−1 +k σ2 +k,Cv) +and ΣBv = diag(κ−1 +1 σ2 +1,Bv, . . .). Whenever Assumption 2 holds for all i, the limiting covariance matrices +ΣCv and ΣBv are regular. Finally, we can deduce from Theorem 1, Lemma 1 and Theorem 9.2.2 of +Rao and Mitra (1971) that the limits of Sn,Cv(H) and Sn,Bv(H) under H0,Cv and H0,Bv, respectively, +are chi-squared distributed with rank(H) degrees of freedom. Moreover, the same arguments yield that +n−1Sn,Cv(H) and n−1Sn,Bv(H) always converge in probability to (HCv)⊤(HΣCvH⊤)+HCv and +(HBv)⊤(HΣBvH⊤)+HBv. In the proofs, we show that these limits are positive under alternatives +H1,Cv : HCv ̸= 0 or H1,Bv : HBv ̸= 0. +Theorem 2. Let (7) as well as Assumptions 1 and 2 be fulfilled for all subgroups i. +(i) Under H0,Cv : HCv = 0, Sn,Cv(H) tends in distribution to Z ∼ χ2 +rank(H). +(ii) Under H1,Cv : HCv ̸= 0, Sn,Cv(H) diverges, i.e. Sn,Cv(H) +p→ ∞. +(iii) Under H0,Bv : HBv = 0, Sn,Bv(H) tends in distribution to Z ∼ χ2 +rank(H). +(iv) Under H1,Bv : HBv ̸= 0, Sn,Bv(H) diverges, i.e. Sn,Bv(H) +p→ ∞. +As a result of Theorem 2, we obtain asymptotically valid tests ϕn,Cv = 1{Sn,Cv(H) > χ2 +rank(H),1−α} +for the testing problems H0,Cv vs. H1,Cv, i.e. they have an asymptotic level α and an asymptotic power +of 1, similarly for Bv. Here, χ2 +rank(H),1−α denotes the (1 − α)-quantile of a chi-square distribution with +rank(H) degrees of freedom. +The convergences rate of Wald-type statistics is known to be rather slow. The simulations of +Ditzhaus and Smaga (2022) confirmed this general impression for the specific variant v = VN. The +new simulation study in Section 5 underpins that for the other variants. To tackle this problem, we +follow a studentized permutation strategy. Moreover, we also consider a (pooled) bootstrap test, which +is of particular interest for local null hypotheses testing, see Section 4. +3.3 +Permutation and bootstrapping +Resampling procedures are popular and well-accepted tools to improve the tests’ performance and, in +particular, their control of the type-1 error. Due to our good experience with permutation procedures +for Wald-type statistics (Ditzhaus et al., 2021a; Ditzhaus and Smaga, 2022; Smaga, 2015, 2017), we +propose to follow this successful and powerful strategy also for the underlying problem. A remarkable +advantage of permuting over other resampling strategies is its finite exactness under exchangeability, +i.e. under the more restrictive null hypothesis � +H0 : X11 +d= . . . d= Xk1. For more general (potentially +nonexchangeable) null hypotheses, it is not clear whether the permutation strategy leads indeed to a +valid testing procedure. But in case of studentized statistics, as Wald-type statistics, this desirable +validity was proven in various other settings and we provide a proof in the underlying set-up. The +(pooled) bootstrap, i.e. drawing from the pooled data with replacement, is closely related to the +permutation approach. We consider it here as well because the permutation strategy is not appropriate +for simultaneous testing as discussed in Section 4. +Let us become more specific. We first group all data together and denote the resulting pooled data by +X = (Xij)i=1,...,k;j=1,...,ni. Now, we draw with or without replacement from X to obtain a permutation +(Xπ = (Xπ +ij)i=1,...,k;j=1,...,ni) or a bootstrap (Xb = (Xb +ij)i=1,...,k;j=1,...,ni) sample, respectively. Since no +observation can be drawn twice for Xπ, the permuted observations mutually depend from each. In +contrast to that, we draw for the bootstrap sample with replacement and, hence, some individuals +6 + +appear multiple times and some do not appear at all in Xb. In particular, the bootstrap observations +are independent from each other as the original observations. In Section 4, we explain why this property +is beneficial for the simultaneous testing and why the permutation fails there. +To differentiate between the quantities from the previous Sections for the different samples, we add +the superscript π or b to them when they rely on the permutation or bootstrap sample. For example, +Sπ +n,Cv(H) denotes the permuted test statistic. Since we draw from the pooled data, the assumptions +need to be translated from the specific groups to the pooled distribution. Therefore, we introduce the +expectation µ0 = �k +i=1 κiµi and the covariance matrix Σ0 for the (asymptotic) pooled distribution +P0 = �k +i=1 κiP Xi1, where the matrix is given by its entries [Σ0]ℓm = (�k +i=1 κiE(Xℓ1Xm1))−[µ0]ℓ[µ0]m. +It is easy to check, e.g. via projections, that Assumption 2 is true for the pooled distribution when +this is the case for all (sub-)groups i. Consequently, we just need, in addition to the conditions from +Theorem 1, that Assumption 1 is fulfilled for the pooled quantities: +Theorem 3. In addition to the assumptions of Theorem 1, we suppose that Assumption 1 is fulfilled +for the pooled quantities µ0 and Σ0. Then the following statements are valid under the null hypotheses +H0,Cv, H0,Bv and their respective alternatives: +(a) The permutation statistics Sπ +n,Cv(H) and Sπ +n,Bv(H) always mimic the null distribution limit of +Sn,Cv(H) and Sn,Bv(H) asymptotically. In formulas (exemplarily for Cv): +sup +x∈R +��� Pr +� +Sπ +n,Cv(H) ≤ x | X +� +− χ2 +rank(H)(x) +��� +p→ 0. +(b) The statement in (a) is also true for the bootstrap statistics Sb +n,Cv(H) and Sb +n,Bv(H). +Theorem 3 justifies the validity of the permutation and bootstrap method. To accept this, let +qπ +n,1−α,Cv(X) be the (1 − α)-quantile of the permutation distribution R ∋ x �→ Pr(Sπ +n,Cv(H) ≤ x | X). +Then Theorem 3 ensures that qπ +n,1−α,Cv(X) approximates always (!) the quantile χ2 +rank(H),1−α of the +test ϕn,Cv from Section 3.2. Consequently, the asymptotic properties of ϕn,Cv, namely asymptotic +exactness under the null and consistency for all alternatives, can be transferred to its permutation +counterpart ϕπ +n,Cv = 1{Sn,Cv(H) > qπ +n,1−α,Cv(X)} (cf. Janssen and Pauls, 2003, Lem. 1 and Theo. +7). Clearly, the same is true when we consider B instead of C and/or the bootstrap instead of the +permutation method. +4 +Multiple testing +4.1 +Local null hypotheses and the multiple contrast tests +While testing for main and interaction effects in factorial designs is of high interest, often a more +in-depth analysis is wanted to check which part of the equation systems HCv = 0 or HBv = 0, +respectively, is not true. This leads to the multiple testing problem +H0,ℓ,Cv : h⊤ +ℓ Cv = 0 +(ℓ = 1, . . . , r) +(8) +for contrast vectors hℓ ∈ Rk, i.e. h⊤ +ℓ 1k×1 = 0. The intersection �r +ℓ=1 H0,ℓ,Cv of the local null hypotheses +coincides with the global null hypothesis H0,Cv from (6) with H = (h1, . . . , hr)⊤. In the easiest case, +we are interested in group differences, i.e. the global null hypotheses is H0,Cv : Cv +1 = . . . = Cv +k. +Prominent examples to split the latter into local null hypotheses are Tukey’s all-pairs comparison +7 + +(Tukey, 1953) +H0,Cv : +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Cv +1 = Cv +2 +Cv +1 = Cv +3 +... +Cv +1 = Cv +k +Cv +2 = Cv +3 +... +Cv +k−1 = Cv +k +⇔ H0,Cv : +� +� +� +� +� +� +� +� +� +� +� +� +−1 +1 +0 +. . . +. . . +0 +0 +−1 +0 +1 +0 +. . . +. . . +0 +... +... +... +... +... +... +... +−1 +0 +0 +0 +. . . +. . . +1 +0 +−1 +1 +0 +. . . +. . . +0 +... +... +... +... +... +... +... +0 +. . . +. . . +. . . +0 +−1 +1 +� +� +� +� +� +� +� +� +� +� +� +� +Cv = 0 +or Dunnet’s multiple-to-one comparison (Dunnett, 1955) +H0,Cv : +� +� +� +� +� +� +� +� +� +� +� +Cv +1 = Cv +2 +Cv +1 = Cv +3 +... +Cv +1 = Cv +k +⇔ H0,Cv : +� +� +� +� +� +−1 +1 +0 +. . . +. . . +0 +−1 +0 +1 +0 +. . . +0 +... +... +... +... +... +... +−1 +0 +. . . +. . . +0 +1 +� +� +� +� +� Cv = 0. +Further proposals can be found in Bretz et al. (2001). In principle, we can consider different strategies +from multiple testing, e.g. the Bonferroni or Holm correction, to adjust the type-1 errors. However, +this leads typically to a significant power loss. A more promising approach are multiple contrast tests +(Bretz et al., 2001; Hothorn et al., 2008; Gunawardana and Konietschke, 2019). For them, we test +the single null hypotheses H0,ℓ,Cv by Sn,Cv(h⊤ +ℓ ) from Section 3.2 and then incorporate the explicit +dependence structure of them to obtain a valid testing procedure. In detail, we consider the following +max-type statistic +Sn,max,Cv(H) = max +ℓ=1,...,r(Sn,Cv(h⊤ +ℓ ))1/2 = max +ℓ=1,...,r |T v +ℓ,n|, +T v +ℓ,n = √n +h⊤ +ℓ �C +v +� +h⊤ +ℓ �ΣCvhℓ +. +By Theorem 1, (T v +1,n, . . . , T v +r,n) converges in distribution to a multivariate normal distribution with +standard normal distributed marginals and correlation matrix RCv given by +[RCv]ℓm = +h⊤ +ℓ ΣCvhm +� +h⊤ +ℓ ΣCvhℓ +� +h⊤ +mΣCvhm +. +(9) +The studentization of the T v +ℓ,n’s, i.e. dividision by an estimator for the asymptotic variance, ensures +that each null hypothesis is (asymptotically) treated in the same way. Thus, the equicoordinate +(1 − α)-quantile q1−α,max,Cv(R) of a N(0, RCv)-distribution serves as a “fair” critical value. Such +quantiles can be determined numerically by computer software, e.g. the function qmvnorm() from the +R-package mvtnorm (R Core Team, 2022; Genz et al., 2021; Genz and Bretz, 2009). In practice, RCv +needs to be estimated by �RCv, where we replace ΣCv in (9) by its estimator �ΣCv. In summary, we +obtain an asymptotically exact test ϕn,max,Cv = 1{Sn,max,Cv(H) > q1−α,max,Cv( �RCv)} for the global +null hypothesis H0,Cv. In contrast to the test from Section 3.2, this test also provides additional +information in case of a rejection, namely which local null hypotheses (8) caused this rejection. In +detail, we reject the local null hypothesis H0,ℓ,Cv when T n +ℓ,n > q1−α,max,Cv( �RCv). Moreover, multiple +max-type contrast tests can be inverted to obtain simultaneous confidence intervals for all contrasts +h⊤ +ℓ Cv. All these theoretical properties are summarized in the following theorem: +Theorem 4. Let (7) as well as Assumptions 1 and 2 be fulfilled for all (sub-)groups i. +(a) The test ϕn,max,Cv is asymptotically exact for H0,Cv, i.e. EH0,Cv (ϕn,max,Cv) → α. +8 + +(b) Suppose that the first r′ ≤ r null hypotheses and the remaining r − r′ alternatives, i.e. H1,ℓ,Cv : +h⊤ +ℓ Cv ̸= 0 for ℓ = r′ + 1, . . . , r, are true. Then +lim sup +n→∞ Pr +� r′ +� +ℓ=1 +{|T v +ℓ,n| > q1−α,max,Cv( �RCv)} +� +≤ α +and +lim +n→∞ Pr +� +r� +ℓ=r′+1 +{|T v +ℓ,n| > q1−α,max,Cv( �RCv)} +� += 1. +(c) An asymptotically valid simultaneous confidence interval for h⊤ +ℓ Cv is given by +Pr +� r� +ℓ=1 +� +h⊤ +ℓ �C +v ∈ +� +h⊤ +ℓ Cv ± n−1/2 +� +h⊤ +ℓ �ΣCvhℓ q1−α,max,Cv( �RCv) +��� +→ 1 − α. +(d) All statements are true for Bv instead of Cv when adjusting the estimators properly. +For a performance improvement for small n, we modify the bootstrap from Section 3.3. +4.2 +Bootstrapping +It is not straightforward how or even whether the resampling strategies from Section 3.3 can also be +used for multiple contrast tests. Let us first have a closer look on the bootstrap strategy. In the proofs, +we show that given the data X almost surely +n1/2� +�Cvb +1 − �Cv +0, . . . , �Cvb +k − �Cv +0 +�⊤ +d→ Gb +Cv ∼ N(0k×1, �ΣCv), +where �Cv +0 is the MCV estimator based on the pooled data and �ΣCv = diag(�σ2 +1,Cv, . . . , �σ2 +k,Cv). In +general, the covariance matrices �ΣCv and ΣCv differ. That is why we cannot approximate the limiting +null distribution of (T v +1,n, . . . , T v +r,n) by (T v,b +1,n, . . . , T v,b +r,n) directly. For the Wald-type statistic, we faced +a similar problem and solved it by studentization, i.e. by eliminating the dependence of the limit +distribution on the covariance structure ΣCv or Σb +Cv, respectively. Translated to the present setting, +we would studentize (T v +1,n, . . . , T v +r,n) first, and then taking the maximum of its entries. In formulas, +we would end up with maxr +ℓ=1[�Σ +−1/2 +Cv +(T v +1,n, . . . , T v +r,n)⊤]ℓ. This is indeed a valid testing procedure for +the global H0,Cv. But we cannot match the entries [�Σ +−1/2 +Cv +(T v +1,n, . . . , T v +r,n)⊤]ℓ with the respective local +null hypothesis H0,ℓ,Cv anymore. That is why we need to approximate (T v +1,n, . . . , T v +r,n) directly and +not just a transformation of it. For this purpose, we can find different strategies for other testing +problems in the literature, e.g. wild bootstrapping (c.f. Umlauft et al., 2019; Konietschke et al., 2021) +or group-wise bootstrapping (c.f. Wechsung and Konietschke, 2021). However, we like to exemplify +that the pooled bootstrap with a certain modified studentization can also be applied. In detail, we +approximate n1/2( �C +v − Cv) by +n1/2 �Σ +1/2 +Cv (�Σ +b +Cv)−1/2( �C +vb − �C +v +0), +�C +v +0 = �Cv +0 · 1k×1. +In words, we first studentize ( �C +vb − �C +v +0) and then multiple the result by �Σ +1/2 +Cv leading to the correct +asymptotic covariance structure. The respective multiple contrast statistic becomes +Sb +n,max,Cv(H) = n1/2 max +ℓ=1,...,r |T v,b +ℓ,n|, +T v,b +ℓ,n = h⊤ +ℓ �Σ +1/2 +Cv (�Σ +b +Cv)−1/2( �C +vb − �C +v +0) +� +h⊤ +ℓ �ΣCvhℓ +. +Now, let qb +1−α,max,Cv(X) be the conditional, equicoordinate (1 − α)-quantile of n1/2(T v,b +1,n, . . . , T v,b +r,n)⊤ +given the data X and ϕb +n,max,Cv = 1{Sn,max,Cv(H) > qb +1−α,max,Cv(X)} be the bootstrap multiple +contrast test. Then we can transfer indeed all asymptotic properties from ϕn,max,Cv to ϕb +n,max,Cv, and, +moreover, obtain bootstrap-based simultaneous confidence intervals: +9 + +Theorem 5. In addition to the assumptions of Theorem 1, we suppose that Assumption 1 is fulfilled +for the pooled quantities µ0 and Σ0. Then the statements of Theorem 4a–c remain true when we +replace ϕn,max,Cv and q1−α,max,Cv( �RCv) by their bootstrap counterparts ϕb +n,max,Cv and qb +1−α,max,Cv(X), +respectively. The analogue for B is true. +At a first glance, a similar result might also be reachable for the permutation procedure but it is +not. The asymptotic covariance structure of n1/2( �C +vπ − �C +v +0) is more complicated than of the bootstrap +one due to the strong dependence within the permutation sample. In particular, the permutation +covariance matrix is neither diagonal nor regular. +5 +Simulation study +Complementing the theoretical findings, we conducted an extensive simulation study to investigate the +type-1 error level and power of the 40 tests proposed in the previous sections: +• the sixteen asymptotic tests: ϕCRR, ϕBRR, ϕCVV, ϕBVV, ϕCVN, ϕBVN, ϕCAZ, ϕBAZ, ϕmax,CRR, +ϕmax,BRR, ϕmax,CVV, ϕmax,BVV, ϕmax,CVN, ϕmax,BVN, ϕmax,CAZ, ϕmax,BAZ, +• the eight permutation tests: ϕπ +CRR, ϕπ +BRR, ϕπ +CVV, ϕπ +BVV, ϕπ +CVN, ϕπ +BVN, ϕπ +CAZ, ϕπ +BAZ, +• the sixteen bootstrap tests: ϕb +CRR, ϕb +BRR, ϕb +CVV, ϕb +BVV, ϕb +CVN, ϕb +BVN, ϕb +CAZ, ϕb +BAZ, ϕb +max,CRR, +ϕb +max,BRR, ϕb +max,CVV, ϕb +max,BVV, ϕb +max,CVN, ϕb +max,BVN, ϕb +max,CAZ, ϕb +max,BAZ. +We omit the subscript n here for the sake of clarity. For the multiple contrast tests, we use the Tukey’s +all-pairs comparison (see Section 4). The final simulation results shall serve as a first guideline for +practical use of the R-package GFDmcv consisting of all these methods. +5.1 +Simulation setup +We considered inferring the null hypotheses in (6) in a multivariate one-way layout. The 5-dimensional +data (d = 5) were generated for k = 4 groups. +The mean vector was generated once from the +normal distribution N(0, 1), and µ1, . . . , µk were set to this vector. The covariance matrices Σi +were based on the completely symmetric matrix (1 − ρ)Id + ρ1d1⊤ +d . We considered ρ = 0.1, 0.4, 0.7 +for small, moderate, and large correlation, respectively. Of course, the MCVs usually measure the +variability differently, see Section 1 or Albert and Zhang (2010) for a more detailed discussion on the +differences. Thus, to compare the tests’ properties in a unified way, we set the same values of all Cv +i , +v ∈ {RR, VV, VN, AZ}, i.e. for given i = 1, . . . , k, CRR +i += CVV +i += CVN +i += CAZ +i +. To obtain this, the +matrices Σi were multiplied by appropriate constants av +i , dependent on v. The observations were +generated from three distributions: the normal (N), the Student (t5), and the chi-square (χ2 +10), i.e. +the symmetric, heavily tailed, and skewed distributions, respectively. For simplicity, we consider equal +sample sizes in all groups. Namely, we set n1 = · · · = nk = 30, 50, 70, 100, 150, 200 for the type-1 error +control investigation, and n1 = · · · = nk = 30, 50 for power comparison. +The significance level was set to α = 5%. For generating the data under H0,Cv, i.e. to investigate +the type-1 error control, we set Cv +i = 0.1, 0.5, 1, 1.5 for i and all variants v. For power comparison, we +consider the following three alternative hypotheses: +• H(1) +1,Cv : Cv +1 = Cv +2 = Cv +3 = 0.1 ̸= 0.15 = Cv +4; +• H(2) +1,Cv : Cv +1 = Cv +2 = Cv +3 = 0.5 ̸= 0.7 = Cv +4; +• H(3) +1,Cv : Cv +1 = Cv +2 = Cv +3 = 1 ̸= 1.5 = Cv +4. +Empirical sizes and powers of the tests were computed as the proportion of rejections of the null +hypothesis based on 1000 simulation replications. The p-values of the permutation and bootstrap tests +were estimated by 1000 resampling samples. The simulation experiments and real data example of +Section 6 were performed in the R programming language (R Core Team, 2022). Due to the twenty-four +resampling tests and 270 simulation scenarios, a part of the calculations was made at the Pozna´n +Supercomputing and Networking Center. +10 + +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 30 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 50 +Figure 1: Box-and-whisker plots for the empirical sizes (in %) of the asymptotic tests. The solid, +dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial +proportion confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], respectively. +5.2 +Simulation results +The simulation results are summarized by the box-and-whisker plots in Figures 1-3 and Figures S1-S13 +in the supplementary materials. The complete list of empirical sizes and powers is presented in +Tables S3-S8 in the supplementary materials. First, we focus on the type-1 error level of the tests, and +then we consider their power. +Type-1 error level For proper maintaining the type-1 error, the empirical sizes should belong to +the binomial proportion 95% and 99% confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], respectively +(Duchesne and Francq, 2015). In the figures, these limits are presented by the horizontal lines. It +is apparent that the type-1 error control of the asymptotic tests is unstable and primarily lead to +liberal decisions, i.e. their empirical sizes are (much) greater than 6.8% in most cases (see Figure 1 +and Figure S1 in the supplement). The record-breaking empirical sizes about 35% are obtained for the +ϕBRR. For larger values of MCVs, the asymptotic tests tend to be quite conservative. Overall, the +performance of the asymptotic tests improves with increasing sample sizes, but the converge speed is +rather slow (see Figure S1 in the supplement). The fastest convergence is observed for ϕmax,CVV. +Contrary, the permutation and bootstrap Wald-type procedures from Section 3.3 for the global null +hypotheses perform very well (Figure 2) and control the type-1 error level accurately in the majority +of settings. In particular, their empirical sizes are always smaller than the upper limits of the binomial +proportion confidence intervals, even for small sample sizes. Only the bootstrap strategy based on +the MCV Cv lead partly to slight conservative decisions for ni = 50, 70. Moreover, the boxs’ for the +permutation tests are slightly narrower than ones for bootstrapping in case of smaller sizes. This slight +visible advantage of permuting can be explained by its finite exactness under exchangeability. +Turning now to the bootstrap multiple tests, we can observe that the type-1 error control is still +satisfactory for the standardized mean vectors B. However, the decisions based on the MCVs become +very conservative reaching down to values below 1%. The latter type-1 error rates improve very slowly +and for the largest sample size of ni = 200 the conservativness is still clearly present. In the supplement, +all results are shown in tables and, in particular, a detailed comparison between the bootstrap and +asymptotic multiple contrast test can be made. In summary, the bootstrap procedures converge much +slower to the desired 5%-benchmark than the asymptotic test. But, under the small sample sizes, the +bootstrap procedure is conservative while the asymptotic strategy is rather liberal, and, thus, only +the first controls the type-1 error rate. Nevertheless, the overall performance of the bootstrap is not +satisfactory, which opens the door for future research. Here, we like to point out that we already studied +11 + +a group-wise bootstrap procedure, i.e. drawing just from the respective groups, a wild bootstrap +procedure, a Bayesian bootstrap procedure, and a parametric bootstrap procedure without much more +success. Despite the unsatisfactory results for the MCV, we like to highlight the good performance of +this (pooled) bootstrap strategy for the standardized means Bv. To the best of our knowledge, this +was the first time that such a pooled bootstrap procedure was applied for multiple contrast tests. +Power Let us now discuss the results of the power comparison. The resulting empirical powers +are presented in Figure 3 and Figures S8-S13 in the supplement. Since the too liberal character of +the asymptotic tests is unacceptable, we do not consider them in power investigation to avoid unfair +comparisons and the potential of results’ misinterpretation. +First, let us consider the comparison for Wald-type and multiple contrast testing procedures for +a given definition of MCV, since the conclusions presented below are the same for each Cv and Bv, +v ∈ {RR, VV, VN, AZ}. From Figure 3, we can observe that the Wald-type permutation and bootstrap +tests have very similar power, but the bootstrap ones seem to be slightly less powerful. The ϕb +max,Bv +tests characterize similar power to their Wald-type counterparts. Unfortunately, the ϕb +max,Cv tests show +their conservative character and have considerably smaller power than the other tests. In general, each +test based on a given MCV is at least slightly less powerful than its analog based on the reciprocal +of MCV. For each test, the power is similar under the normal and χ2 +10 distributions, while for the +t5-distribution, it is much smaller (Figures S8-S10). Thus, for the heavy-tailed distribution, the +convergence may be slower. This is especially evident for the ϕb +max,Cv tests due to their extremely +conservative character in such a case. Let us also notice that the power of all tests decreases with +the increase of the value of the MCV, which was also observed for earlier methods for variability +comparison proposed in Aerts and Haesbroeck (2017), Ditzhaus and Smaga (2022), and Pauly and +Smaga (2020). +Now, we want to consider how the behavior of the tests depends on the MCV definition. We can +observe that the power of all tests for different coefficients is not the same, which was expected due to +various conceptions of measuring multivariate variability. The general observation is as follows: The +tests based on the Van Valen coefficient are usually the most powerful and outperform the procedures +for Rayment’s MCV, which are followed by the tests for MCVs proposed by Albert and Zhang, and +Voinov and Nikulin. We can also observe that, for different amounts of correlation, the power of most +tests is stable, but sometimes greater differences appear for the Van Valen coefficient. A possible reason +for the latter is that the VV variant does not take the dependence of variables into account. +Recommendation To sum up, the Wald-type permutation and bootstrap tests as well as the multiple +bootstrap tests based on reciprocals of MCVs control the type-1 error level even for small sample sizes. +Moreover, these testing procedures have sensible power, which however depends on the coefficient used. +For these reasons, these tests can be recommended for practical use, and thus, it is now available to +infer about each of the four multivariate coefficients of variation. Note however that the advantage of +the ϕb +max,Bv tests over the Wald-type tests is that they simultaneously verify the contrasts’ significance, +in particular, can perform post hoc testing. Unfortunately, the ϕb +max,Cv tests are very conservative and +hence less powerful, which gives us a direction for future research. +6 +Real data application +In this section, we consider, as an illustrative data example, the external quality assessment for clinical +laboratories (Libeer, 1993; Sciacovelli et al., 2018; WHO, 2022). Moreover, we present further simulation +results, which mimic the set-up from this data example. +6.1 +Analysis of EQA data set +In clinical laboratories, controlling analytical performance and maintaining inter-laboratory variability +within acceptable limits are important issues that are even a concern in External Quality Assessment +(EQA) schemes. They are organized nationally or internationally by government health agencies or +private companies. In the case when the available data are multivariate, Zhang et al. (2010) proposed +12 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure 2: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained from all cases considered. The solid, dashed, and dotted lines represent the +significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] +and [3.2%, 6.8%], respectively. +to use the multivariate coefficient of variation for comparing the inter-laboratory reproducibility of +13 + +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure 3: Box-and-whisker plots for the empirical powers (in %) of the permutation and bootstrap +tests obtained from all cases considered. +assay techniques used by clinical laboratories. Naturally, the lower the MCV, the better the analytical +performance. However, the simple values of particular MCV do not mean significant differences in +the techniques considered. That is why we illustrate the use of tests proposed for comparing four +techniques based on electrophoretic data sets from the French and Belgian national EQA programs, +which was also considered by Zhang et al. (2010) and Aerts and Haesbroeck (2017). +The serum protein electrophoresis (SPE) is a laboratory test profile consisting of five fractions +summing up to 100% of total proteins. The fractions are albumin, α1, α2, β, and γ globulins. The SPE +can be assayed in different ways depending on the media or the analytical principle. In the experiment, +the following four techniques were compared: HT cellulose acetate (CH), HT agarose gel (acid blue) +(EH), HT agarose gel (amido black) (JH), and BCP capillary zone (GB). These four SPE techniques +use distinct support mediums, staining colors, or analytical principles. Thus, we want to compare the +techniques by testing for equal MCVs. This was first done by Aerts and Haesbroeck (2017), but they +considered the MCV of Voinov and Nikulin only. We extend this study using the new Wald-type and +multiple contrast tests for all four variants. But first, the data need to be transformed and cleaned. +Due to the compositional nature of the electrophoretic data, a one-to-one transformation from the +5-dimensional to the 4-dimensional space is required. For this purpose, the well-known isometric +log-ratio transformation is used. Second, similarly to Aerts and Haesbroeck (2017), we remove the +outliers from the data set, which we detected by computing the robust Mahalanobis distances of the +observations. After that, we have four samples of 4-dimensional observations with sizes n1 = 133, +n2 = 112, n3 = 74, n4 = 62. +First of all, we calculate the estimators (3) of MCVs for each technique and each MCV variant, see +Table 1. It is apparent that the techniques do not perform equally well. In particular, the GB technique +leads to the smallest MCV value for each variant and, thus, seems to be the most stable technique +in terms of measurement variability. To check whether this first impression can be underpinned with +statistical certainty, we first performed the different tests for the global hypotheses (6) of technique-wise +equality of MCVs. Hereby, we considered Tukey’s contrast matrix for the different multiple contrast +tests. For each MCV, all tests clearly rejected the global null hypotheses with p-values very close to +zero (results are not shown). However, these rejections are just partially informative and we are rather +interested in a more in-depth analysis with pairwise comparisons of the different techniques. Therefore, +we can invert, as explained in Section 4, the multiple contrast tests into simultaneous confidence +intervals, see Table 2. These intervals indeed confirm our first impression and the GB technique leads +to a significant lower MCV and grater value of its reciprocal compared to all three other techniques. +14 + +Table 1: Values of MCVs’ estimators for four techniques. +MCV +CH +EH +JH +GB +�CRR +i +0.0559 +0.0525 +0.0489 +0.0249 +�CVV +i +0.1421 +0.1235 +0.1244 +0.0627 +�CVN +i +0.0688 +0.0558 +0.0619 +0.0238 +�CAZ +i +0.0985 +0.0719 +0.0811 +0.0343 +Furthermore, it can be seen that the confidence intervals for the bootstrap approach are slightly wider +than for the asymptotic one. This can be explained by the liberality of the asymptotic strategy which +we observed in the simulation study. Another interesting observation is that a significant difference +between the CH and EH techniques can only be detected by the AZ variant. Here, we like to point out +again that the four variants are, in fact, different measures and, thus, opposite inference results may +appear. Aerts and Haesbroeck (2017) also considered pairwise tests for the underlying data set (see +Table 6 in their paper) but for the specific variant v =VN only. In their analysis, they could detect a +significant difference of CH and EH as well. However, they did not adjusted for multiplicity. Thus, +their result is not contradicting ours for v =VN and need to be taken with a pinch of salt. +6.2 +Simulation study based on EQA data set +To check the appropriateness of the results for the EQA data set, we conduct an additional simulation +study based on this set. To mimic the observation given in the data example, we generated the +simulation data with four samples using: +• the samples sizes (n1 = 133, n2 = 112, n3 = 74, n4 = 62) from the data example, +• for checking the type-1 control, in each group, the mean and covariance matrix were set to the +sample mean and sample covariance matrix of the pooled data, +• for power investigation, the mean and covariance matrix in the i-th group was equal to the +quantities of the i-th sample from the data set. +The 4-dimensional data were generated from the same three distributions as in Section 5, i.e. the +normal, t5, and χ2 +10 distributions. We have calculated the empirical sizes and powers for the global +hypotheses (6) and the same for multiple contrast tests. +The results regarding the type-1 error control of the (global) tests from Sections 3 and 4 (see +Table S1 in the supplement) are comparable to the ones from the prior simulation study in Section 5 +and, thus, not discussed in detail again. Moreover, all these tests have a power close to 100% (results +not shown). This confirms the appropriately of rejection of the global null hypotheses for the EQA data +set. Let us turn to the post hoc testing problem and have a closer look on the asymptotic and bootstrap +tests from Section 4. The power results are summarized in Figure 4 and are shown more detailed +in Table S2 from the supplement. Although the bootstrap ϕb +max,Cv tests may still be conservative +(Table S1 in the supplement), the empirical powers of ϕb +max,Cv and ϕb +max,Bv tests are very similar for +given v ∈ {RR, VV, VN, AZ}. For CH-EH, CH-JH, and EH-JH comparisons, they are summarised in +Figure 4. In the first case, the power of the tests is varied. The tests based on the MCV of Albert and +Zhang are the most powerful, while the other tests have much smaller power. This clearly explains the +rejections and nonrejections for the CH-EH comparison. On the other hand, for EH vs. JH, all tests +have almost trivial power, which follows from the smallest differences in estimators of MCV’s among +all comparisons (see Table 1). In the case of CH-JH, the empirical power of all testing procedures +increases (even up to about 60% for the ϕb +max,CRR and ϕb +max,BRR tests). This increase of power in +comparison to EH-JH can be explained by the increase in the differences in estimators (Table 1). For +CH-GB, EH-GB, and JH-GB, the empirical powers of all tests were very close to 100%, so we do not +15 + +Table 2: Estimates of the contrasts h⊤ +ℓ �C +v and h⊤ +ℓ �B +v with respective simultaneous 95%-confidence +intervals [95%-L,95%-U] based on the asymptotic and bootstrap procedures of Section 4. The bold +values represent the cases, where significant differences are obtained. +Comparison +Variant v +Method +h⊤ +ℓ �C +v +95%-L +95%-U +h⊤ +ℓ �B +v +95%-L +95%-U +CH-EH +RR +asym +-0.003 +-0.010 +0.003 +1.181 +-1.102 +3.464 +boot +-0.011 +0.004 +-1.290 +3.652 +VV +asym +-0.019 +-0.040 +0.002 +1.059 +-0.114 +2.231 +boot +-0.041 +0.004 +-0.195 +2.312 +VN +asym +-0.013 +-0.028 +0.002 +3.392 +-0.418 +7.202 +boot +-0.031 +0.005 +-0.797 +7.581 +AZ +asym +-0.027 +-0.046 +-0.007 +3.751 +0.969 +6.534 +boot +-0.047 +-0.006 +0.862 +6.640 +CH-JH +RR +asym +-0.007 +-0.016 +0.002 +2.587 +-0.729 +5.903 +boot +-0.017 +0.003 +-1.002 +6.176 +VV +asym +-0.018 +-0.042 +0.007 +0.999 +-0.392 +2.390 +boot +-0.043 +0.008 +-0.488 +2.486 +VN +asym +-0.007 +-0.027 +0.013 +1.625 +-3.248 +6.499 +boot +-0.031 +0.017 +-3.733 +6.984 +AZ +asym +-0.017 +-0.041 +0.006 +2.173 +-0.920 +5.266 +boot +-0.042 +0.007 +-1.039 +5.385 +CH-GB +RR +asym +-0.031 +-0.037 +-0.025 +22.217 +17.301 +27.132 +boot +-0.038 +-0.024 +16.896 +27.537 +VV +asym +-0.079 +-0.098 +-0.061 +8.922 +6.549 +11.294 +boot +-0.100 +-0.059 +6.386 +11.458 +VN +asym +-0.045 +-0.057 +-0.033 +27.480 +17.938 +37.021 +boot +-0.060 +-0.030 +16.989 +37.971 +AZ +asym +-0.064 +-0.081 +-0.047 +19.034 +12.492 +25.577 +boot +-0.082 +-0.046 +12.241 +25.827 +EH-JH +RR +asym +-0.004 +-0.012 +0.005 +1.406 +-1.913 +4.724 +boot +-0.013 +0.006 +-2.186 +4.997 +VV +asym +0.001 +-0.021 +0.023 +-0.059 +-1.463 +1.345 +boot +-0.022 +0.024 +-1.560 +1.442 +VN +asym +0.006 +-0.013 +0.025 +-1.767 +-7.032 +3.499 +boot +-0.017 +0.029 +-7.556 +4.023 +AZ +asym +0.009 +-0.012 +0.031 +-1.578 +-5.104 +1.947 +boot +-0.013 +0.032 +-5.239 +2.082 +EH-GB +RR +asym +-0.028 +-0.033 +-0.022 +21.035 +16.118 +25.953 +boot +-0.033 +-0.022 +15.713 +26.358 +VV +asym +-0.061 +-0.076 +-0.045 +7.863 +5.483 +10.243 +boot +-0.078 +-0.044 +5.319 +10.408 +VN +asym +-0.032 +-0.043 +-0.021 +24.088 +14.340 +33.835 +boot +-0.045 +-0.019 +13.370 +34.805 +AZ +asym +-0.038 +-0.052 +-0.023 +15.283 +8.525 +22.041 +boot +-0.053 +-0.023 +8.266 +22.300 +JH-GB +RR +asym +-0.024 +-0.031 +-0.016 +19.630 +14.156 +25.104 +boot +-0.032 +-0.015 +13.705 +25.554 +VV +asym +-0.062 +-0.081 +-0.042 +7.922 +5.427 +10.417 +boot +-0.083 +-0.041 +5.255 +10.589 +VN +asym +-0.038 +-0.056 +-0.021 +25.854 +15.644 +36.065 +boot +-0.059 +-0.017 +14.628 +37.081 +AZ +asym +-0.047 +-0.066 +-0.028 +16.861 +9.970 +23.753 +boot +-0.067 +-0.027 +9.706 +24.017 +16 + +0 +20 +40 +60 +80 +100 +Empirical powers (%) +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +Asymptotic MCT tests +Bootstrap MCT tests +CH−EH +0 +20 +40 +60 +80 +100 +Empirical powers (%) +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +Asymptotic MCT tests +Bootstrap MCT tests +CH−JH +0 +20 +40 +60 +80 +100 +Empirical powers (%) +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +C +RR +B +RR +C +VV +B +VV +C +VN +B +VN +C +AZ +B +AZ +Asymptotic MCT tests +Bootstrap MCT tests +EH−JH +Figure 4: Box-and-whisker plots for the empirical powers (in %) of the multiple contrast tests obtained +for three “nontrivial” contrasts for simulation based on the EQA data. +draw them. However, of course, they justify the recognition of significant differences for these contrasts +using all multiple contrast tests proposed. +7 +Summary and discussion +In this paper, we discussed testing procedures for the multivariate coefficients of variation. To the +best of our knowledge, such testing procedures were just developed for the special MCV version +of Voinov and Nikulin (1996). To be more concrete, Aerts and Haesbroeck (2017) proposed (semi- +)parametric procedures for the k-sample setting and, recently, Ditzhaus and Smaga (2022) suggested a +nonparametric approach for general factorial designs. However, also the other MCV variants (Reyment, +1960; Van Valen, 1974; Albert and Zhang, 2010) are relevant for applications. That is why we followed +the successful path of Ditzhaus and Smaga (2022) and adopted their results to the other MCV variants +as well as their reciprocals, the so-called standardized means. In particular, we developed respective +permutation procedure with a more satisfactory type-1 error control in small sample sizes. +Furthermore, we provided a post hoc strategy for a more in-depth analysis. Therefore, we combined +the well-known tool of multiple contrast tests (Mukerjee et al., 1987; Bretz et al., 2001; Konietschke +et al., 2018; Umlauft et al., 2019; Gunawardana and Konietschke, 2019; Wechsung and Konietschke, +2021) with our theoretical results. This resulted into asymptotically valid tests, which are rather +liberal for small sample sizes, and a novel pooled bootstrap strategy. To the best of our knowledge, +this pooled bootstrap idea was not used in the context of multiple testing procedures before. The +simulation results confirm its usage for the reciprocals of the MCVs, by a more convincing type-1 error +control under small sample sizes. However, for the MCVs itself the bootstrap procedures become quite +conservative and we cannot completely recommend its usage. Here, further research is required to +obtain a better resampling strategy for MCV post hoc testing. +All procedures are shown to be asymptotically valid and consistent by empirical process theory +(van der Vaart and Wellner, 1996) and are extensively studied in simulations. To motivate their use, +17 + +we provide the R package GFDmcv. Currently, the package can be requested via email and will be +soon publicly available on CRAN. +A further interesting aspect is inferring paired data, as in the recent investigation regarding the +repeatability and reliability of GABA (Gamma-aminobutyric acid) measurements taken from the +same patient group (Duda et al., 2021). Under the assumption of normality, Shoukri et al. (2008) +already discussed this issue for comparing two univariate CV. Conceptionally, this situation is not +different to our set-up and, in more detail, corresponds to the one-sample scenario with k = 1 and +d = 2. Consequently, the nonparametric methodology of comparing two (or more) CVs or even MCVs +in paired data settings can be directly deduced from the theory presented in Section 2. However, a +detailed investigation would overload this paper and is, thus, postponed to future research. Moreover, +as already raised by Ditzhaus and Smaga (2022), the problem of outliers, as in the EQA data set, shall +be tackled nonparametrically by considering robust estimators for µ and Σ. For first (semi-)parametric +solutions in this context, we refer to Aerts and Haesbroeck (2017). +Acknowledgement +The authors would like to thank Professor Adelin Albert (Facult´e de M´edecine, University of Li`ege) +for sharing the electrophoretic data sets from the French and Belgian national EQA programs used +in Zhang et al. (2010). A part of calculations for the simulation study was made at the Pozna´n +Supercomputing and Networking Center (grant no. 382). +Supplementary materials +In the supplementary material, all results of simulation studies of Sections 5 and 6 are presented. +A +Proofs +A.1 +Proof of Theorem 1 +From the proof of Theorem 1 in Ditzhaus and Smaga (2022), we obtain +n1/2 +1 +� +�µ − µ +vec(�Σ) − vec(Σ) +� +d +−→ Dψ(µ)G, +(10) +where +G ∼N +� +0, +� +Σ +Ψ⊤ +3 +Ψ3 +Ψ4 +�� +, Dψ(x) = +� Id +0d×d2 +� +D(x) +Id2 +� +, +Ψ3, Ψ4, � +D are defined in (4), Id is the d × d-dimensional unity matrix and 0d×d2 is the d × d2- +dimensional zero matrix. Here and subsequently, we consider a matrix as an element of Rℓ, ℓ ∈ N, by +vectorization vec(A) = (A11, . . . , Ad1, A21, . . . , Add)⊤ for A = (Aij)i,j=1,...,d. In this way, we can equip +GLsym(Rd) with the Euclidean norm. +To verify the first statement for the different choices of C, we apply the δ-method and, later, its +permutation/bootstrap analogue. For the latter, we require differentiability in a stronger sense (see e.g. +van der Vaart and Wellner, 1996, Theorem 3.9.5), so-called uniform differentiability. In our specific +multivariate set-up, we call a map Φ : (Rd \ {0}) × GLsym(Rd) → R uniformly differentiable when +t−1 +n +� +Φ[an + tnbn, An + tnBn] − Φ[an, An] +� +→ DΦ(a, A) +� +b +vec(B) +� +as n = n1 → ∞ +(11) +for a (Jacobi) matrix DΦ(a, A) ∈ Rd+d2 as well as for all An → A ∈ GLsym(Rd), Bn → B ∈ Rd×d, +an → a ∈ Rd \ {0}, bn → b ∈ Rd and tn → 0. Setting an = a and An = A, we get (standard) +differentiability. +18 + +Lemma 3. The maps ΦRR, ΦVV, ΦAZ : (Rd \ {0}) × GLsym(Rd) → R defined as +ΦRR(a, A) = +� +det(A)1/d +(a⊤a) +, +ΦVV(a, A) = +� +tr(A) +a⊤a +and +ΦAZ(a, A) = +� +a⊤Aa +(a⊤a)2 +are uniformly differentiable in the sense of (11) with (Jacobi) matrices +DΦRR(a, A) = 1 +2d +� det(A) +(A⊤a)d +�1/(2d)−1 � +− 2d det(A) +a⊤ +(a⊤a)d+1 , det(A) +� +vec(A−1) +�⊤ +(a⊤a)d +� +, +DΦVV(a, A) = 1 +2 +�tr(A) +a⊤a +�−1/2 � +−2 tr(A) +a⊤ +(a⊤a)2 , +1 +a⊤a (vec(Id))⊤ +� +, +DΦAZ(a, A) = 1 +2 +� a⊤Aa +(a⊤a)2 +�−1/2 � +−4a⊤Aa +a⊤ +(a⊤a)3 + 2 a⊤A +(a⊤a)2 , a⊤ ⊗ a⊤ +(a⊤a)2 +� +. +Proof of Lemma 3: We first consider the maps +�ΦRR(a, A) = det(A) +(a⊤a)d , +�ΦVV(a, A) = tr(A) +a⊤a +and +�ΦAZ(a, A) = a⊤Aa +(a⊤a)2 . +Let An → A ∈ GLsym(Rd), Bn → B ∈ Rd×d, an → a ∈ Rd \ {0}, bn → b ∈ Rd and tn → 0. Then we +first observe that +t−1 +n +� +det(An + tnBn) − det(An) +� +→ det(A)tr(A−1B). +For the fixed choice An = A, this follows from the differentiability of the determinant, e.g. shown in +Theorem 8.1 from Magnus and Neudecker (2019). For arbitrary An, we recall that the determinant +can be expressed as a polynomial of the matrix’s entries and polynomials are uniformly differentiable +in the sense of (11). +Secondly, we obtain from the linearity of the trace +t−1 +n +� +tr(An + tnBn) − tr(An) +� += tr(Bn) → tr(B). +Thirdly, we recall that A is symmetric and, thus, +t−1 +n +� +(an + tnbn)⊤(An + tnBn)(an + tnbn) − a⊤ +n Anan +� += 2b⊤ +n Anan + anBnan + O(tn) +→ 2b⊤Aa + aBa. +In particular, setting An = Id and Bn = 0d×d we obtain +t−1 +n +� +(an + tnbn)⊤(an + tnbn) − a⊤ +n an +� +→ 2b⊤a. +Consequently, applying the chain rule yields the results. In detail, we get +t−1 +n +� +�ΦRR[an + tnbn, An + tnBn] − �ΦRR[an, An] +� +→ det(A)tr(A−1B)(a⊤a)d − d(a⊤a)d−1(2b⊤a) det(A) +(a⊤a)2d += det(A)tr(A−1B)(a⊤a)d − d(a⊤a)d−1(2b⊤a) det(A) +(a⊤a)2d += −2d det(A) +a⊤b +(a⊤a)d+1 + det(A)tr(A−1B) +(a⊤a)d += +� +−2d det(A) +a⊤ +(a⊤a)d+1 , det(A) +� +vec(A−1) +�⊤ +(a⊤a)d +� � +b +vec(B) +� +, +19 + +where we used the (easy-to-see) equation tr(E⊤F ) = (vec(E))⊤ vec(F ) for respective matrices E, F +in the last equation. Moreover, +t−1 +n +� +�ΦVV[an + tnbn, An + tnBn] − �ΦVV[an, An] +� +→ tr(B)(a⊤a) − 2(b⊤a)tr(A) +(a⊤a)2 += +� +−2tr(A) +a⊤ +(a⊤a)2 , +1 +a⊤a (vec(Id))⊤ +� � +b +vec(B) +� +. +Finally, +t−1 +n +� +�ΦAZ[an + tnbn, An + tnBn] − �ΦAZ[an, An] +� +→ (2b⊤Aa + aBa)(a⊤a)2 − 2(a⊤a)(2b⊤a)a⊤Aa +(a⊤a)4 += +� +−4a⊤Aa +a⊤ +(a⊤a)3 + 2 a⊤A +(a⊤a)2 , a⊤ ⊗ a⊤ +(a⊤a)2 +� � +b +vec(B) +� +, +where we used the following equality for general matrices E, F and G with appropriate dimensions +such that the respective multiplications are well-defined: +vec(EF G) = (G⊤ ⊗ E)vec(F ). +The statements regarding the maps ΦRR, ΦVV and ΦAZ follow easily from the chain rule, when the +(clearly uniformly differentiable) maps ϕ1, ϕ2 : (0, ∞) → R defined by ϕ1(x) = x1/(2d) and ϕ2(x) = x1/2 +are applied to them. +□ +Combining Lemma 3, (10), the δ-method and the definition of Av(µ, Σ) yields +n1/2 +1 +� +�Cv − Cv� +d +−→ DΦv(µ, Σ)Dψ(µ)G ∼ N(0, σ2 +Cv), +The results for Bv = 1/Cv follows by the chain rule with the map ϕ3(x) = x−1, x ̸= 0. +A.2 +Proof of Lemma 2 +We first like to note that the proof for the case v = VN can be found in the appendix of Ditzhaus +and Smaga (2022). In fact, we follow their arguments and adopt them to the different variants +v = RR, VV, AZ. +First, we define, for abbreviation, +� +X = (X11, . . . , X1d, X11X11, . . . , X11X1d, X12X11, . . . , X1dX1d)⊤. +(12) +It is easy to see that the covariance matrix of � +Xi equals +Σ� +X = +� +Σ +Ψ⊤ +3 +Ψ3 +Ψ4 +� +. +We observe that σ2 +Cv = 0 implies degeneracy of Av(µ, Σ)� +X. For v = RR, the latter translates to: +There exists a constant �c ∈ R such that +� +−2d det(Σ) +µ⊤ +i +(µ⊤µ)d+1 + +det(Σ)(vec(Σ−1)) +⊤ +(µ⊤µ)d +� +D(µ), +det(Σ)(vec(Σ−1)) +⊤ +(µ⊤µ)d +� +� +X = �c +(13) +with probability one. Define for m, s ∈ {1, . . . , d} +as = +� +− 2d det(Σ) +µ⊤ +(µ⊤µ)d+1 + det(Σ) +� +vec(Σ−1) +�⊤ +(µ⊤µ)d +� +D(µ) +� +s ∈ R, +bsm = +�det(Σ) +� +vec(Σ−1) +�⊤ +(µ⊤µ)d +� +sd−d+m = det(Σ)[Σ−1]m,s +(µ⊤µ)d +∈ R. +(14) +20 + +Now, we can simplify (13) to +d +� +m=1 +(amX1m + bmmX2 +1m) + +d +� +s,m=1;s̸=m +bsmX1sX1m = �c +(15) +with probability one. Since Σ−1 is nonsingular covariance matrix itself, the diagonal entries need +to be positive and, thus, bmm ̸= 0 holds for all m ∈ {1, . . . , d}. Now, fix some m. Given the other +components (X1s)s̸=m, the left hand side of (15) is a polynomial in X1m of degree two and, thus, X1m +can take at most two different values to solve (15). This contradicts Assumption 2. +This proof strategy can be used, in the same way, for the other two variants v ∈ {VV, AZ}. For +this purpose, we just need to consider the respective counterparts of bsm from (14) and show bmm ̸= 0 +for some m. Let us start with v = VV: +bVV +mm = +� +1 +µ⊤µ (vec(Id))⊤ � +md−d+m = +1 +µ⊤µ ̸= 0 for all m = 1, . . . , d. +Finally, we consider v = AZ: +bAZ +mm = +�µ⊤ ⊗ µ⊤ +(µ⊤µ)2 +� +md−d+m = +[µ]2 +m +(µ⊤µ)2 ̸= 0 +for all m = 1, . . . , d with [µ]m ̸= 0. By Assumption 1, we have µ ̸= 0 and, thus, bAZ +mm ̸= 0 for some m. +A.3 +Proof of Theorem 2 +As already mentioned in the paper, it remains to show that (HCv)⊤(HΣCvH⊤)+HCv as well as +(HBv)⊤(HΣBvH⊤)+HBv are positive under any alternative H1,Cv : HCv ̸= 0 or H1,Bv : HBv ̸= 0, +respectively. We just give the proof for C and note that the statement for B follows by simply +interchanging the letters C and B. +For the proof, we need some (well-known) properties of the Moore–Penrose inverse for a matrix A, +which can be found e.g. in Rao and Mitra (1971): (I) (A⊤)+ = (A+)⊤, (II) (A⊤A)+ = A+(A⊤)+, +and (III) AA+A = A. Now, suppose that the alternative H1,Cv : HCv ̸= 0 is true. By Lemma 2, the +root Σ1/2 +Cv = diag(κ−1/2 +1 +σ1,Cv, . . . , κ−1/2 +k +σk,Cv) of the covariance matrix is regular. Thus, there is some +v ∈ Rk \ {0k×1} such that Cv = Σ1/2 +Cv v. From this and (I)–(III) we obtain +0k×1 ̸= HCv = HΣ1/2 +Cv v = HΣ1/2 +Cv (HΣ1/2 +Cv )+HΣ1/2 +C v = HΣ1/2 +Cv +� +(HΣ1/2 +Cv )+HCv� +. +This implies (HΣ1/2 +Cv )+HCv ̸= 0. Consequently, +(HCv)⊤(HΣCvH⊤)+HCv = (HCv)⊤(Σ1/2 +Cv H⊤)+(HΣ1/2 +Cv )+HCv += +� +(HΣ1/2 +Cv )+HCv�⊤� +(HΣ1/2 +Cv )+HCv� +> 0. +A.4 +Proof of Theorem 3(a) +One difficulty and significant difference to the asymptotic approach is that the permutation sample is +dependent and, thus, the groups need to be considered simultaneously. The reason for the latter is +that we pool the data and then draw without (!) replacement. The action of pooling the data is also +important for deriving the theory. Thus, we introduce, on the one hand, the pooled estimators �µ0, �Σ0, +�Cv +0 and �Bv +0 depending on all observations and not just observations from a specific group. On the other +hand, let Y ∼ P0 be a random, d-dimensional vector following the asymptotic pooled distribution +P0 = �k +i=1 κiP Xi1. Moreover, we denote by µY , ΣY , ΨY 3, ΨY 4 the respective theoretical quantities +from the main paper but for Y instead of Xi1. In particular, ΣY is the covariance matrix of Y . +21 + +Again, we benefit from the preparatory work of Ditzhaus and Smaga (2022), who already verified +(see their (12)) that given the data almost surely +n1/2 +� +� +� +� +� +� +� +�µπ +1 − �µ0 +vec(�Σ +π +1) − vec(�Σ0) +... +�µπ +k − �µ0 +vec(�Σ +π +k) − vec(�Σ0) +� +� +� +� +� +� +� +d +−→ Gπ = +� +� +� +Gπ +1... +Gπ +k +� +� +� . +(16) +Furthermore, they showed that Gπ is centered, kd(d+1)-dimensional normal distributed with covariance +structure +� +� +� +γ(1, 1)Σπ +. . . +γ(1, k)Σπ +... +... +... +γ(k, 1)Σπ +. . . +γ(k, k)Σπ +� +� +� = +� +� +� +� +κ−1 +1 Σπ +0d′×(k−2)d′ +0d′×d′ +0(k−2)d′×d′ +... +0(k−2)d′×d′ +0d′×d′ +. . . +κ−1 +k Σπ +� +� +� +� − +� +� +� +Σπ +. . . +Σπ +... +... +... +Σπ +. . . +Σπ +� +� +� , +(17) +where d′ = d(d + 1), +Σπ = +� +ΣY +Ψ⊤ +Y 3 +ΨY 3 +ΨY 4 +� +and +γ(i, i′) = κ−1 +i 1{i = i′} − 1 +(i, i′ = 1, . . . , k). +To obtain the asymptotic normality of the permuted MCVs, we apply the δ-method, similar to the +proof of Theorem 1. However, we need to respect that we center the permutation quantities �Cvπ +i +and +�Bvπ +i +by �Cv +0 and �Bv +0, respectively, which both change with growing sample sizes. That is why we need +a stronger form of the δ-method (see e.g. van der Vaart and Wellner, 1996, Theorem 3.9.5) which +requires uniform differentiability in the sense of (11). By Lemma 3 the maps ΦRR, ΦVV, ΦAZ fulfil this +requirement and, hence, we get that given the observations almost surely +n1/2� +�Cvπ +1 +− �Cv +0, . . . , �Cvπ +k +− �Cv +0 +�⊤ +d +−→ +� +� +� +� +DΦv(µY , ΣY ) +01×(k−2)d′ +01×d′ +0(k−2)×d′ +... +0(k−2)×d′ +01×d′ +01×(k−2)d′ +DΦv(µY , ΣY ) +� +� +� +� Gπ = Gπ +Cv. +To get the analogue results for �Bvπ +i +(v ∈ {RR, VV, AZ}) we apply the (uniform) δ-method to ϕ3 ◦ Φv +instead. We leave the additional writing effort to the interested readers and proceed just with the C’s. +Clearly, Gπ +Cv follows a centered, multidimensional normal distribution and, in regard to (17), we can +simplify its covariance matrix to +�ΣCv − (DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤)1k×k, +where +�ΣCv = diag(κ−1 +1 DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤, . . . , κ−1 +k DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤) += diag(σ2 +1,Cv,Y , . . . , σ2 +k,Cv,Y ) +and σ2 +1,Cv,Y is the pooled counterpart of σ2 +1,Cv. Now, it becomes clear why we require the matrix H to +be a contrast matrix. Namely, this implies H( �Cv +0 · 1k×1) = 0 as well as H1k×k = 0, and, consequently, +it follows +n1/2H �C +π = n1/2H +� +�Cvπ +1 +− �Cv +0, . . . , �Cvπ +k +− �Cv +0 +�⊤ +d +−→ HGπ +Cv ∼ N(0, H �ΣCvH⊤) +given the observations almost surely. As in Section 3.2, this is the first important ingredient for the +asymptotic convergence of the Wald-type statistic. The second is the convergence of (H �Σ +π +CvH⊤)+ +to (H �ΣCvH⊤)+. Therefore, it remains to argue (1) σ2 +i,Cv,Y > 0 for all i = 1, . . . , k which implies the +22 + +regularity of �ΣCv, and (2) �Σ +π +Cv converges (conditionally) in probability to �ΣCv given the observations +almost surely. +Hereby, (1) follows directly from Lemma 2 and the simple observation that Assumption 2 can +directly be transferred to the pooled distribution. For (2), we recall from Ditzhaus and Smaga (2022) +that +� +�µπ +i +vec(�Σ +π +i ) +� +p +−→ +� +µY +vec(ΣY ) +� +, +i ∈ {1, . . . , k}, +given the observations almost surely. Then (conditional) convergence in probability of �Σ +π +Cv to �ΣCv +follows from the continuous mapping theorem. +A.5 +Proof of Theorem 3(b) +In principle, the proof follows the same argumentation as for Theorem 3(a). However, the bootstrap +procedure is slightly easier to handle because we draw with (!) replacement and, thus, the groups are +still independent. The proof of Ditzhaus and Smaga (2022) for our (16) can be adopted to get an +analogous result for the bootstrap. Here, one just need to replace Theorems 3.7.1 and 3.7.2 of van der +Vaart and Wellner (1996) for the permutation procedure by the bootstrap counterparts Theorem 3.7.6 +and 3.7.7 in their argumentation via empirical processes. All these results originally cover only the +two-sample case (k = 2) but can be directly extended to k ≥ 3, as argued e.g. by Ditzhaus et al. +(2021a) in their Lemma 9 and Remark 1. Finally, we can obtain that given the observations almost +surely +n1/2 +� +� +� +� +� +� +� +� +�µb +1 − �µ0 +vec(�Σ +b +1) − vec(�Σ0) +... +�µb +k − �µ0 +vec(�Σ +b +k) − vec(�Σ0) +� +� +� +� +� +� +� +� +d +−→ Gb = +� +� +� +Gb +1... +Gb +k +� +� +� , +(18) +where Gb is centered, kd(d + 1)-dimensional normal distributed with covariance structure +� +� +� +� +κ−1 +1 Σπ +0d′×(k−2)d′ +0d′×d′ +0(k−2)d′×d′ +... +0(k−2)d′×d′ +0d′×d′ +. . . +κ−1 +k Σπ +� +� +� +� . +(19) +Applying the (uniform) δ-method we get, given the observations almost surely, +n1/2� +�Cvb +1 − �Cv +0, . . . , �Cvb +k − �Cv +0 +�⊤ +d +−→ +� +� +� +� +DΦv(µY , ΣY ) +01×(k−2)d′ +01×d′ +0(k−2)×d′ +... +0(k−2)×d′ +01×d′ +01×(k−2)d′ +DΦv(µY , ΣY ) +� +� +� +� Gb = Gb +Cv ∼ N(0k×1, �ΣCv), +(20) +where �ΣCv = diag(σ2 +1,Cv,Y , . . . , σ2 +k,Cv,Y ). The rest of the proof follows the arguments from Section 3.2 +and from the proof’s end of Theorem 3(a). To avoid unnecessary repetition, we leave the details to +the interested reader. As an intermediate result, we just like to mentioned that given the data almost +surely +�Σ +b +Cv +p→ �ΣCv +(21) +while �ΣCv is regular as already argued in the proof of Theorem 3(a). +23 + +A.6 +Proof of Theorem 4 +The statements in (a) and (c) follow immediately from Theorem 1 as discussed briefly before Theorem 4. +The key step is to follow from Theorem 1 that +�√nh⊤ +1 �C +v − h⊤ +1 Cv +� +h⊤ +1 �ΣCvh1 +, . . . , √nh⊤ +r �C +v − h⊤ +r Cv +� +h⊤ +r �ΣCvhr +�⊤ += √ndiag((h⊤ +1 �ΣCvh1)−1/2, . . . , (h⊤ +r �ΣCvhr)−1/2)H( �C +v − Cv) d→ Z ∼ N(0r×1, RCv). +(22) +Then (c) follows by a simple inversion of this convergence statement and for (a) we just need to remind +ourselves that h⊤ +ℓ Cv = 0 for all ℓ = 1, . . . , r under H0,Cv. +Now, suppose H1,ℓ,Cv : h⊤ +ℓ Cv ̸= 0 is true. Then we can deduce from Slutzky’s Lemma and (22) +|T v +ℓ,n| ≥ √n +|h⊤ +ℓ Cv| +� +h⊤ +ℓ �ΣCvhℓ +− +��� +√nh⊤ +ℓ �C +v − h⊤ +ℓ Cv +� +h⊤ +ℓ �ΣCvhℓ +��� +p→ ∞. +From this, we obtain the second statement of (b). +Now, let H0,1,Cv, . . . , H0,r′,Cv be true for r′ ≤ r and define �R +′ = ([ �R]ℓm)ℓ,m=1,...,r′. By (a) +Pr +� +max +ℓ=1,...,r′ |T v +ℓ,n| > q1−α( �R +′) +� +→ α. +Moreover, it is easy to see that q1−α,max,Cv( �R) ≥ q1−α( �R +′), e.g. by noting Sn,max,Cv(H) ≥ maxℓ=1,...,r′ |T v +ℓ,n|. +In summary, we obtain +Pr +� r′ +� +ℓ=1 +{|T v +ℓ,n| > q1−α,max,Cv( �R)} +� += Pr +� +max +ℓ=1,...,r′ |T v +ℓ,n| > q1−α,max,Cv( �R) +� +≤ Pr +� +max +ℓ=1,...,r′ |T v +ℓ,n| > q1−α( �R +′) +� +→α +proving the first statement of (b). +Clearly, an analogue of (22) is true for B instead of C, see Theorem 1. Consequently, (d) follows +from the same arguments as used for (a)–(c). +A.7 +Proof of Theorem 5 +To prove the statement, we need a bootstrap analogue of (22). 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Accreditation and Quality Assurance 15:351–357 +27 + +Supplementary materials to +Inference for all variants of the multivariate coefficient of variation +in factorial designs +Marc Ditzhaus1 and �Lukasz Smaga2,∗ +1Faculty of Mathematics, Otto-von-Guericke University Magdeburg, Germany +2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland +January 31, 2023 +This supplement contains all results of simulation studies of Sections 5 and 6 of the main paper. +They are summarized in Figures S1-S13 and presented in Tables S1-S8. +List of Figures +S1 +Box-and-whisker plots for the empirical sizes of the asymptotic tests obtained from all +cases considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +S2 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained under normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +S3 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained under t5-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +S4 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained under χ2 +10-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +S5 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained for ρ = 0.1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +S6 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained for ρ = 0.4 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +S7 +Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests +obtained for ρ = 0.7 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +S8 +Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained under normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +S9 +Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained under t5-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +S10 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained under χ2 +10-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +S11 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained for ρ = 0.1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +S12 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained for ρ = 0.4 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +S13 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests +obtained for ρ = 0.7 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +List of Tables +S1 +Empirical sizes of all tests obtained in simulation study based on real data example . . +13 +S2 +Empirical powers of the multiple contrast tests obtained in simulation based on real +data example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1 +arXiv:2301.12009v1 [stat.ME] 27 Jan 2023 + +S3 +Empirical sizes of all tests obtained under normal distribution . . . . . . . . . . . . . . +15 +S4 +Empirical sizes of all tests obtained under t5-distribution . . . . . . . . . . . . . . . . . +23 +S5 +Empirical sizes of all tests obtained under χ2 +10-distribution . . . . . . . . . . . . . . . . +31 +S6 +Empirical powers of the permutation and bootstrap tests under normal distribution +. +39 +S7 +Empirical powers of the permutation and bootstrap tests under t5-distribution +. . . . +41 +S8 +Empirical powers of the permutation and bootstrap tests under χ2 +10-distribution . . . . +43 +2 + +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 30 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 50 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 70 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 100 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 150 +0 +5 +10 +15 +20 +25 +30 +35 +Empirical sizes (%) +CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ +Asymptotic Wald−type tests +Asymptotic MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S1: Box-and-whisker plots for the empirical sizes (as percentages) of the asymptotic tests +obtained from all cases considered. The solid, dashed, and dotted lines represent the significance level +α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], +respectively. +3 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S2: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained under normal distribution. The solid, dashed, and dotted lines represent the +significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] +and [3.2%, 6.8%], respectively. +4 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S3: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained under t5-distribution. The solid, dashed, and dotted lines represent the +significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] +and [3.2%, 6.8%], respectively. +5 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S4: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained under χ2 +10-distribution. The solid, dashed, and dotted lines represent the +significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] +and [3.2%, 6.8%], respectively. +6 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S5: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.1. The solid, dashed, and dotted lines represent the significance level +α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], +respectively. +7 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S6: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.4. The solid, dashed, and dotted lines represent the significance level +α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], +respectively. +8 + +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 70 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 100 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 150 +0 +1 +2 +3 +4 +5 +6 +7 +Empirical sizes (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 200 +Figure S7: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.7. The solid, dashed, and dotted lines represent the significance level +α = 5% and the 95% and 99% binomial proportion confidence intervals [3.6%, 6.4%] and [3.2%, 6.8%], +respectively. +9 + +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S8: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained under normal distribution. +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S9: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained under t5-distribution. +10 + +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S10: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained under χ2 +10-distribution. +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S11: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.1. +11 + +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S12: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.4. +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 30 +0 +20 +40 +60 +80 +100 +Empirical powers (%) +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +CRR +BRR +CVV +BVV +CVN +BVN +CAZ +BAZ +Permutation Wald−type tests +Bootstrap Wald−type tests +Bootstrap MCT tests +n1 = n2 = n3 = n4 = 50 +Figure S13: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and +bootstrap tests obtained for ρ = 0.7. +12 + +Table S1: Empirical sizes (as percentages) of all tests obtained in +simulation study based on real data example of Section 6 in the main +paper (Method: a - asymptotic Wald-type test; π - permutation +Wald-type test; b - bootstrap Wald-type test; max, a - asymptotic +multiple contrast test; max, b - bootstrap multiple contrast test) +Distr. +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +N +8.7 +8.0 +7.2 +6.9 +8.4 +7.6 +7.1 +6.9 +a +4.7 +4.9 +4.9 +4.7 +4.8 +4.8 +4.2 +4.8 +π +4.8 +5.1 +4.8 +4.9 +4.5 +4.4 +4.5 +4.7 +b +7.8 +7.7 +6.3 +6.5 +7.4 +7.0 +7.0 +6.2 +max, a +4.1 +5.4 +3.5 +5.0 +3.4 +5.3 +3.5 +4.4 +max, b +t5 +14.4 +16.7 +9.8 +12.3 +16.4 +17.2 +9.4 +12.9 +a +5.0 +5.0 +4.3 +4.5 +5.6 +5.3 +4.9 +4.6 +π +4.5 +4.9 +4.1 +4.8 +5.5 +4.8 +4.2 +4.2 +b +12.5 +15.1 +6.4 +11.5 +13.1 +16.1 +6.5 +12.1 +max, a +1.2 +4.2 +1.1 +4.1 +1.3 +5.0 +1.1 +3.8 +max, b +χ2 +10 +10.8 +10.8 +7.8 +7.9 +10.5 +9.9 +8.9 +8.7 +a +5.5 +5.5 +4.4 +5.2 +5.1 +5.5 +3.9 +4.7 +π +5.7 +5.8 +4.2 +5.0 +4.9 +4.9 +4.1 +4.7 +b +9.9 +9.7 +7.3 +7.4 +10.0 +8.8 +7.5 +7.8 +max, a +3.6 +4.9 +3.2 +5.2 +3.2 +5.6 +2.4 +4.5 +max, b +13 + +Table S2: Empirical powers (as percentages) of the multiple contrast +tests obtained in simulation study based on real data example of +Section 6 in the main paper +Distr. +Test +CH-EH +CH-JH +CH-GB +EH-JH +EH-GB +JH-GB +N +ϕmax,CRR +14.5 +62.4 +100 +17.7 +100 +100.0 +ϕmax,BRR +14.2 +59.8 +100 +16.7 +100 +100.0 +ϕmax,CV V +37.3 +27.3 +100 +1.5 +100 +100.0 +ϕmax,BV V +39.2 +23.4 +100 +1.7 +100 +100.0 +ϕmax,CV N +42.9 +10.3 +100 +5.2 +100 +100.0 +ϕmax,BV N +42.1 +8.5 +100 +6.9 +100 +100.0 +ϕmax,CAZ +81.7 +28.9 +100 +7.3 +100 +100.0 +ϕmax,BAZ +81.5 +24.5 +100 +10.2 +100 +100.0 +ϕb +max,CRR +8.3 +51.1 +100 +11.4 +100 +100.0 +ϕb +max,BRR +9.1 +49.5 +100 +11.7 +100 +100.0 +ϕb +max,CV V +32.6 +23.8 +100 +1.3 +100 +100.0 +ϕb +max,BV V +36.3 +21.7 +100 +1.5 +100 +100.0 +ϕb +max,CV N +27.5 +5.3 +100 +1.7 +100 +100.0 +ϕb +max,BV N +33.5 +5.6 +100 +4.7 +100 +100.0 +ϕb +max,CAZ +79.5 +26.5 +100 +6.4 +100 +100.0 +ϕb +max,BAZ +79.8 +23.3 +100 +8.8 +100 +100.0 +t5 +ϕmax,CRR +7.4 +29.0 +100.0 +10.0 +100.0 +99.8 +ϕmax,BRR +8.9 +28.7 +100.0 +10.6 +99.9 +99.6 +ϕmax,CV V +13.2 +15.2 +99.3 +1.2 +98.4 +96.4 +ϕmax,BV V +17.4 +16.0 +98.1 +2.1 +95.0 +95.3 +ϕmax,CV N +25.6 +9.4 +99.9 +4.7 +99.2 +99.3 +ϕmax,BV N +27.5 +10.1 +99.0 +7.4 +97.9 +98.8 +ϕmax,CAZ +44.8 +19.7 +99.3 +3.2 +95.0 +96.8 +ϕmax,BAZ +46.8 +20.4 +97.9 +6.6 +90.1 +94.7 +ϕb +max,CRR +2.0 +9.5 +99.9 +1.7 +99.6 +98.4 +ϕb +max,BRR +2.8 +12.5 +99.7 +2.6 +99.5 +98.5 +ϕb +max,CV V +10.3 +12.6 +98.6 +0.8 +97.4 +94.7 +ϕb +max,BV V +15.4 +14.7 +98.0 +1.7 +94.6 +95.0 +ϕb +max,CV N +6.1 +2.5 +97.5 +0.5 +93.4 +91.3 +ϕb +max,BV N +13.7 +4.2 +97.8 +2.2 +93.7 +94.8 +ϕb +max,CAZ +39.8 +17.4 +98.1 +2.3 +93.1 +95.4 +ϕb +max,BAZ +45.6 +19.2 +97.7 +5.9 +89.6 +93.8 +χ2 +10 +ϕmax,CRR +9.1 +48.5 +100.0 +15.5 +100.0 +100.0 +ϕmax,BRR +9.3 +46.1 +100.0 +14.8 +100.0 +100.0 +ϕmax,CV V +29.0 +23.1 +100.0 +1.7 +100.0 +100.0 +ϕmax,BV V +29.5 +20.9 +100.0 +2.3 +100.0 +100.0 +ϕmax,CV N +35.7 +11.7 +100.0 +3.8 +100.0 +100.0 +ϕmax,BV N +35.5 +10.5 +100.0 +5.5 +100.0 +100.0 +ϕmax,CAZ +66.2 +26.1 +100.0 +5.2 +99.7 +99.8 +ϕmax,BAZ +64.2 +23.2 +99.9 +7.4 +98.9 +99.5 +ϕb +max,CRR +4.2 +32.7 +100.0 +7.0 +100.0 +100.0 +ϕb +max,BRR +5.0 +32.0 +100.0 +6.9 +100.0 +100.0 +ϕb +max,CV V +25.7 +20.6 +100.0 +1.1 +100.0 +100.0 +ϕb +max,BV V +28.2 +19.5 +100.0 +2.1 +100.0 +100.0 +ϕb +max,CV N +17.6 +4.7 +100.0 +0.9 +100.0 +99.9 +ϕb +max,BV N +25.3 +5.2 +100.0 +2.9 +99.9 +100.0 +ϕb +max,CAZ +61.1 +23.2 +100.0 +3.5 +99.7 +99.8 +ϕb +max,BAZ +62.8 +22.5 +99.9 +6.9 +98.7 +99.5 +14 + +Table S3: Empirical sizes (as percentages) of all tests obtained under +normal distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.1) +30 +17.5 +17.7 +9.0 +8.9 +15.1 +12.6 +11.2 +10.7 +a +6.1 +5.7 +6.0 +5.7 +4.5 +4.3 +5.3 +5.2 +π +5.7 +6.0 +6.1 +6.0 +3.9 +4.3 +5.0 +5.0 +b +15.9 +16.1 +8.4 +8.7 +12.2 +11.0 +9.1 +9.3 +max, a +1.7 +6.2 +4.1 +6.2 +0.2 +5.4 +1.9 +4.3 +max, b +50 +9.7 +9.4 +7.5 +7.9 +10.3 +9.3 +7.6 +7.1 +a +4.6 +4.4 +5.1 +5.6 +5.3 +4.7 +4.2 +4.3 +π +4.6 +4.5 +4.9 +5.3 +4.9 +5.2 +4.5 +4.3 +b +8.7 +9.2 +6.8 +7.0 +8.8 +9.0 +6.3 +6.2 +max, a +2.6 +5.3 +5.0 +6.0 +1.3 +6.0 +2.4 +4.1 +max, b +70 +10.8 +10.2 +5.6 +5.9 +9.6 +8.9 +6.6 +6.1 +a +6.2 +5.7 +5.1 +4.8 +5.1 +6.1 +4.5 +4.2 +π +5.6 +5.4 +5.0 +4.5 +5.4 +5.9 +4.4 +4.3 +b +9.8 +9.5 +5.1 +5.0 +8.4 +8.2 +5.8 +5.4 +max, a +3.4 +6.2 +4.0 +4.5 +2.6 +6.2 +2.6 +4.2 +max, b +100 +7.2 +7.6 +6.9 +7.2 +6.9 +6.3 +6.9 +6.9 +a +5.1 +5.2 +6.1 +5.7 +4.0 +3.9 +4.9 +5.0 +π +5.0 +5.1 +6.2 +6.0 +4.1 +4.3 +5.3 +5.5 +b +6.2 +6.2 +6.4 +6.6 +5.9 +5.6 +6.7 +6.2 +max, a +4.2 +5.4 +5.5 +6.1 +2.3 +4.3 +4.2 +4.6 +max, b +150 +7.3 +7.3 +5.6 +5.6 +5.2 +5.2 +6.0 +6.1 +a +5.4 +5.4 +4.8 +5.0 +3.6 +4.4 +4.7 +4.8 +π +5.3 +5.7 +4.7 +5.3 +3.8 +4.4 +4.5 +4.8 +b +6.7 +7.0 +5.4 +5.4 +4.1 +4.2 +6.2 +6.3 +max, a +4.8 +5.8 +4.8 +5.0 +2.3 +3.8 +3.9 +5.2 +max, b +200 +7.0 +6.9 +5.5 +5.1 +5.5 +4.9 +6.4 +6.7 +a +5.0 +5.1 +5.0 +4.7 +4.0 +4.0 +5.8 +5.4 +π +5.8 +5.7 +5.1 +4.9 +4.1 +3.8 +5.2 +5.4 +b +6.3 +6.2 +5.8 +5.9 +4.9 +4.9 +6.3 +5.9 +max, a +4.7 +5.5 +5.1 +5.6 +3.0 +4.4 +4.7 +5.3 +max, b +0.1 +(0.5, 0.5, 0.5, 0.5) +30 +8.1 +10.2 +6.9 +7.2 +12.6 +13.3 +9.9 +10.2 +a +5.6 +5.5 +5.1 +4.8 +6.1 +4.5 +5.4 +4.6 +π +5.4 +5.8 +4.7 +4.6 +4.8 +4.2 +5.8 +5.1 +b +5.8 +9.5 +6.1 +6.7 +8.9 +12.4 +6.4 +9.6 +max, a +0.2 +5.7 +2.3 +4.1 +0.0 +5.4 +0.5 +5.3 +max, b +50 +5.4 +6.4 +5.1 +5.2 +8.1 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percentages) of all tests obtained +under t5-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.1) +30 +32.1 +34.9 +11.8 +17.4 +25.8 +28.4 +15.1 +20.2 +a +5.7 +5.8 +4.7 +4.7 +5.3 +4.8 +4.9 +4.7 +π +5.3 +6.0 +4.2 +4.1 +4.5 +4.7 +4.3 +4.2 +b +27.1 +32.5 +8.3 +15.9 +21.3 +25.9 +10.3 +19.0 +max, a +0.3 +5.8 +0.7 +3.4 +0.1 +5.9 +0.8 +3.5 +max, b +50 +19.9 +23.0 +10.3 +14.9 +18.5 +21.9 +10.4 +14.0 +a +5.1 +4.4 +5.2 +4.9 +5.1 +5.8 +4.7 +3.9 +π +4.4 +4.6 +5.1 +4.6 +5.3 +5.4 +4.7 +3.2 +b +16.1 +20.9 +7.1 +13.4 +15.1 +19.8 +7.7 +12.5 +max, a +0.3 +3.8 +0.7 +3.7 +0.3 +6.1 +0.4 +3.8 +max, b +70 +16.2 +19.4 +9.5 +13.4 +15.2 +18.1 +8.8 +13.4 +a +4.7 +5.7 +6.0 +5.8 +4.3 +4.7 +4.6 +4.4 +π +4.5 +5.2 +5.7 +5.2 +3.6 +4.5 +4.4 +4.0 +b +13.6 +18.7 +7.3 +12.3 +12.0 +16.2 +7.1 +12.3 +max, a +0.7 +5.1 +0.8 +4.6 +0.7 +4.9 +0.7 +4.0 +max, b +100 +16.0 +17.8 +9.0 +10.3 +15.2 +17.9 +10.8 +13.9 +a +5.5 +5.4 +5.9 +5.8 +5.5 +5.8 +5.1 +5.1 +π +4.9 +5.0 +5.6 +5.4 +5.7 +5.9 +5.4 +5.2 +b +13.2 +15.8 +6.4 +9.4 +11.6 +17.0 +6.9 +11.9 +max, a +1.2 +4.7 +1.9 +4.0 +1.0 +5.9 +0.9 +4.5 +max, b +150 +12.0 +13.2 +6.1 +7.2 +12.5 +14.5 +7.8 +10.8 +a +5.0 +5.7 +4.3 +4.2 +4.3 +4.5 +4.8 +4.9 +π +4.9 +5.5 +4.1 +4.1 +4.6 +5.0 +5.2 +4.9 +b +9.9 +12.9 +4.4 +6.1 +9.3 +13.2 +5.2 +8.8 +max, a +1.5 +4.4 +1.2 +3.0 +1.0 +5.4 +1.3 +4.5 +max, b +200 +10.8 +11.1 +5.8 +7.4 +11.3 +13.1 +8.0 +10.0 +a +5.2 +6.1 +4.6 +4.8 +6.0 +5.3 +4.7 +5.4 +π +5.0 +5.5 +4.4 +4.7 +5.2 +5.6 +4.5 +4.9 +b +8.3 +10.7 +4.3 +6.8 +8.7 +11.6 +5.1 +7.9 +max, a +1.3 +3.9 +1.1 +2.6 +0.9 +4.5 +1.3 +4.3 +max, b +0.1 +(0.5, 0.5, 0.5, 0.5) +30 +16.7 +22.1 +12.8 +18.4 +21.4 +25.3 +12.1 +17.9 +a +6.0 +5.6 +6.0 +6.2 +4.8 +5.3 +5.1 +6.0 +π +6.1 +5.9 +6.0 +5.8 +4.2 +5.0 +5.3 +5.2 +b +12.0 +20.2 +8.6 +16.6 +14.9 +22.6 +8.0 +16.3 +max, a +0.1 +5.6 +0.3 +4.0 +0.0 +5.9 +0.1 +4.7 +max, b +50 +13.4 +16.2 +10.9 +15.4 +18.7 +22.2 +9.5 +14.7 +a +4.8 +4.9 +5.7 +5.7 +6.1 +5.8 +5.3 +5.4 +π +4.6 +5.3 +5.3 +4.7 +5.8 +6.1 +5.5 +5.7 +b +9.0 +15.6 +7.4 +13.9 +13.9 +21.1 +6.5 +13.1 +max, a +0.0 +5.1 +0.4 +4.4 +0.1 +5.9 +0.2 +4.3 +max, b +70 +12.9 +15.4 +7.5 +11.3 +11.7 +15.5 +9.7 +13.7 +a +6.1 +5.8 +4.6 +4.0 +4.1 +4.4 +5.7 +5.3 +π +5.9 +5.9 +4.1 +3.6 +4.3 +4.7 +5.3 +5.3 +b +9.4 +14.0 +4.8 +10.6 +8.2 +15.1 +5.9 +12.2 +max, a +0.5 +6.3 +1.0 +3.0 +0.2 +4.3 +0.6 +4.8 +max, b +100 +10.9 +13.8 +7.5 +8.8 +10.8 +13.4 +8.1 +11.6 +a +5.3 +5.3 +5.4 +5.1 +4.7 +5.4 +4.6 +5.2 +π +5.4 +5.0 +5.0 +5.3 +4.8 +5.0 +4.9 +5.2 +b +7.8 +13.6 +5.6 +9.0 +8.3 +11.5 +5.4 +10.4 +max, a +23 + +Table S4: Empirical sizes (as percentages) of all tests obtained +under t5-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.5 +4.2 +1.1 +3.8 +0.7 +4.8 +0.5 +5.0 +max, b +150 +8.7 +10.9 +6.5 +8.2 +9.7 +12.4 +7.2 +9.2 +a +4.2 +4.9 +5.0 +4.9 +5.4 +5.4 +4.2 +4.6 +π +4.0 +4.4 +4.5 +4.0 +5.4 +5.0 +4.3 +4.4 +b +7.2 +10.0 +4.1 +7.1 +7.1 +11.7 +4.5 +7.6 +max, a +0.7 +4.7 +1.1 +2.7 +0.8 +4.9 +1.1 +3.9 +max, b +200 +8.7 +10.5 +8.1 +9.2 +9.2 +10.9 +5.8 +8.2 +a +5.8 +5.7 +6.0 +6.0 +5.4 +4.6 +4.4 +4.2 +π +5.7 +5.7 +6.2 +6.3 +4.7 +4.9 +4.0 +4.3 +b +7.0 +9.9 +6.0 +8.0 +6.3 +9.4 +3.9 +7.6 +max, a +1.4 +4.7 +2.7 +5.5 +0.4 +4.5 +0.8 +3.8 +max, b +0.1 +(1, 1, 1, 1) +30 +3.2 +9.9 +10.9 +15.5 +9.8 +18.4 +5.6 +13.1 +a +4.1 +4.3 +4.2 +5.2 +4.5 +5.8 +4.9 +6.0 +π +3.5 +4.7 +3.7 +4.7 +3.6 +5.9 +4.2 +6.1 +b +0.5 +9.9 +6.1 +16.0 +4.0 +16.7 +2.2 +11.5 +max, a +0.0 +4.7 +0.0 +4.2 +0.0 +5.9 +0.0 +5.0 +max, b +50 +4.9 +9.1 +7.9 +12.5 +7.2 +13.5 +5.9 +10.4 +a +4.6 +5.6 +4.8 +5.0 +3.7 +4.6 +5.6 +4.6 +π +5.0 +5.1 +4.6 +5.0 +3.7 +4.8 +5.0 +4.1 +b +2.0 +8.3 +4.5 +10.8 +2.2 +11.6 +1.8 +9.2 +max, a +0.0 +5.4 +0.5 +3.8 +0.0 +4.4 +0.0 +4.4 +max, b +70 +5.5 +9.6 +7.2 +10.8 +7.5 +12.1 +6.4 +12.4 +a +5.1 +5.5 +5.1 +4.9 +4.0 +4.5 +5.9 +6.3 +π +5.2 +5.5 +4.6 +4.9 +3.8 +4.2 +5.8 +6.1 +b +2.5 +8.6 +4.6 +9.5 +2.8 +11.3 +2.7 +11.4 +max, a +0.0 +4.9 +0.8 +3.9 +0.0 +4.3 +0.2 +6.1 +max, b +100 +5.0 +8.1 +7.9 +10.6 +7.6 +12.4 +3.9 +8.9 +a +4.5 +4.9 +6.0 +5.6 +4.9 +4.0 +3.4 +4.2 +π +4.8 +5.3 +5.3 +5.3 +4.4 +3.9 +3.5 +4.2 +b +2.6 +7.7 +5.1 +9.4 +2.9 +11.0 +1.6 +8.2 +max, a +0.2 +4.4 +1.1 +4.2 +0.0 +4.2 +0.1 +3.6 +max, b +150 +5.2 +8.4 +6.4 +7.9 +7.5 +9.9 +6.1 +8.0 +a +5.0 +5.8 +4.5 +5.1 +5.2 +5.3 +5.4 +5.0 +π +4.4 +6.2 +4.9 +5.2 +4.9 +5.1 +5.3 +5.0 +b +3.2 +8.5 +4.5 +7.3 +4.0 +9.6 +2.4 +7.7 +max, a +0.2 +5.9 +1.0 +4.1 +0.1 +4.9 +0.5 +4.1 +max, b +200 +5.6 +6.6 +5.7 +6.9 +6.4 +9.6 +5.7 +7.5 +a +4.9 +5.0 +4.7 +4.8 +5.0 +4.9 +4.9 +4.5 +π +5.1 +5.1 +4.8 +4.4 +5.1 +5.3 +4.8 +4.4 +b +3.4 +6.8 +3.8 +5.9 +4.5 +8.3 +2.8 +7.4 +max, a +0.5 +4.3 +2.0 +3.8 +0.4 +4.8 +0.5 +4.4 +max, b +0.1 +(1.5, 1.5, 1.5, 1.5) +30 +1.3 +7.8 +7.1 +12.9 +4.5 +14.2 +2.3 +9.3 +a +6.0 +5.4 +4.7 +4.7 +5.2 +5.6 +4.6 +4.8 +π +5.1 +5.4 +3.9 +4.2 +4.3 +5.3 +3.9 +5.0 +b +0.1 +7.1 +3.6 +13.7 +0.8 +12.9 +1.1 +8.7 +max, a +0.0 +4.1 +0.1 +3.8 +0.0 +5.8 +0.0 +5.4 +max, b +50 +1.2 +7.5 +7.2 +11.2 +3.8 +10.9 +1.9 +8.7 +a +5.3 +4.6 +4.8 +4.5 +4.6 +4.2 +5.0 +5.5 +π +24 + +Table S4: Empirical sizes (as percentages) of all tests obtained +under t5-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +4.8 +5.3 +4.5 +4.5 +3.9 +4.1 +4.6 +5.3 +b +0.3 +6.5 +3.1 +9.6 +0.1 +10.5 +0.3 +8.9 +max, a +0.0 +4.8 +0.1 +3.7 +0.0 +4.8 +0.0 +6.1 +max, b +70 +1.3 +5.9 +6.8 +11.3 +4.0 +10.6 +4.4 +11.2 +a +4.8 +4.5 +5.2 +5.4 +5.0 +5.1 +5.9 +6.1 +π +4.2 +4.5 +4.6 +4.7 +4.6 +5.4 +6.2 +5.8 +b +0.1 +5.4 +3.4 +10.2 +0.4 +9.5 +0.9 +9.6 +max, a +0.0 +3.9 +0.2 +3.9 +0.0 +5.4 +0.0 +6.1 +max, b +100 +2.0 +5.8 +5.6 +8.3 +3.9 +9.3 +4.6 +8.5 +a +4.3 +3.8 +4.9 +4.8 +5.1 +4.9 +6.1 +6.0 +π +4.4 +4.1 +4.6 +4.1 +5.1 +4.9 +6.2 +5.5 +b +0.5 +4.8 +3.3 +7.2 +0.8 +8.6 +0.7 +8.5 +max, a +0.0 +3.5 +0.4 +4.2 +0.0 +4.9 +0.0 +5.5 +max, b +150 +3.3 +5.7 +5.7 +7.6 +4.6 +9.8 +4.8 +8.5 +a +4.5 +3.7 +5.2 +5.6 +5.4 +6.0 +5.9 +4.6 +π +4.8 +4.0 +5.0 +4.9 +4.9 +5.6 +6.2 +4.1 +b +0.5 +4.7 +3.9 +6.9 +1.5 +9.3 +0.8 +8.2 +max, a +0.0 +3.4 +1.3 +4.2 +0.0 +5.7 +0.0 +4.6 +max, b +200 +3.9 +6.6 +5.0 +6.8 +4.6 +8.5 +4.5 +7.2 +a +5.0 +5.3 +4.1 +4.1 +5.2 +5.3 +5.4 +5.2 +π +5.2 +5.9 +4.3 +4.0 +4.9 +5.0 +5.2 +4.8 +b +1.5 +6.5 +3.6 +5.7 +1.8 +7.8 +1.5 +6.5 +max, a +0.1 +5.7 +1.0 +3.5 +0.0 +5.5 +0.0 +5.1 +max, b +0.4 +(0.1, 0.1, 0.1, 0.1) +30 +31.9 +35.5 +13.3 +18.4 +26.3 +29.6 +16.6 +21.9 +a +5.2 +5.3 +5.3 +5.0 +5.2 +5.5 +5.8 +6.3 +π +5.0 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t5-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +1.3 +1.4 +7.1 +12.6 +0.7 +11.6 +5.0 +5.1 +max, a +0.0 +3.6 +0.7 +3.5 +0.0 +6.2 +1.0 +1.6 +max, b +70 +2.7 +2.1 +8.4 +10.4 +5.2 +10.9 +8.0 +5.9 +a +5.6 +5.4 +4.9 +4.8 +6.0 +5.5 +6.2 +4.5 +π +4.3 +5.1 +3.9 +4.5 +5.4 +6.0 +4.5 +4.6 +b +1.2 +2.2 +5.4 +10.2 +0.4 +9.9 +7.9 +5.6 +max, a +0.0 +4.9 +0.7 +4.4 +0.0 +6.0 +1.0 +1.6 +max, b +100 +2.7 +2.9 +9.3 +12.4 +5.7 +11.3 +9.4 +8.1 +a +5.0 +5.4 +6.0 +6.3 +6.1 +5.9 +4.8 +4.8 +π +3.7 +4.8 +6.2 +6.3 +6.1 +5.6 +4.4 +5.4 +b +1.6 +1.9 +6.7 +11.3 +0.5 +10.3 +9.1 +8.4 +max, a +0.4 +5.1 +1.1 +6.0 +0.0 +5.8 +1.5 +1.4 +max, b +150 +3.5 +3.1 +8.5 +10.1 +4.4 +7.6 +12.2 +10.7 +a +5.1 +5.1 +5.6 +5.7 +4.9 +5.0 +5.5 +5.1 +π +5.0 +5.3 +5.5 +5.1 +4.3 +4.9 +4.4 +5.5 +b +2.2 +2.7 +5.2 +9.4 +1.4 +6.5 +11.7 +11.0 +max, a +0.7 +4.7 +1.2 +4.5 +0.0 +5.0 +0.8 +2.1 +max, b +200 +3.1 +2.3 +5.5 +6.3 +3.1 +6.4 +11.3 +10.9 +a +4.1 +3.9 +4.2 +3.9 +3.7 +3.4 +4.3 +4.8 +π +4.0 +3.7 +4.1 +4.0 +3.6 +3.3 +4.5 +5.4 +b +2.7 +2.5 +4.1 +6.0 +0.8 +6.2 +10.0 +10.8 +max, a +1.2 +3.7 +1.0 +3.9 +0.0 +3.6 +0.9 +1.4 +max, b +30 + +Table S5: Empirical sizes (as percentages) of all tests obtained +under χ2 +10-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.1) +30 +23.6 +24.2 +12.0 +13.0 +18.6 +19.8 +12.7 +14.3 +a +5.8 +6.0 +4.5 +4.6 +4.2 +4.1 +5.3 +4.8 +π +5.8 +6.0 +4.4 +5.0 +3.8 +3.9 +5.0 +4.7 +b +21.2 +22.5 +9.3 +12.5 +14.7 +17.2 +10.4 +14.0 +max, a +1.2 +6.3 +1.9 +3.9 +0.4 +5.4 +1.1 +4.4 +max, b +50 +13.7 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+6.3 +4.8 +5.1 +π +5.8 +5.8 +5.5 +5.6 +5.6 +6.0 +4.5 +5.4 +b +6.0 +6.4 +7.6 +8.6 +8.8 +9.9 +11.1 +14.3 +max, a +3.4 +6.2 +3.1 +5.4 +1.7 +5.9 +0.5 +3.2 +max, b +100 +4.1 +4.2 +6.7 +7.4 +6.5 +7.0 +11.5 +14.3 +a +3.3 +3.8 +4.6 +4.1 +3.9 +4.3 +4.8 +4.3 +π +3.8 +4.0 +4.3 +4.0 +4.3 +4.7 +4.7 +5.1 +b +4.0 +3.9 +5.5 +6.9 +5.7 +6.4 +8.9 +12.2 +max, a +2.5 +4.5 +2.4 +4.7 +2.3 +4.3 +0.7 +3.2 +max, b +36 + +Table S5: Empirical sizes (as percentages) of all tests obtained +under χ2 +10-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +150 +5.9 +5.7 +6.5 +7.0 +8.6 +8.6 +11.1 +12.8 +a +5.6 +5.6 +4.4 +4.4 +6.1 +5.8 +3.8 +3.9 +π +5.7 +5.2 +4.3 +4.7 +5.9 +6.1 +4.2 +4.0 +b +5.0 +4.9 +5.5 +6.5 +7.4 +8.1 +8.5 +10.8 +max, a +4.1 +5.6 +2.9 +4.6 +4.0 +5.9 +0.6 +2.1 +max, b +200 +5.3 +5.4 +6.5 +6.9 +7.3 +6.6 +10.0 +11.5 +a +5.5 +5.2 +5.2 +5.2 +5.4 +5.5 +4.7 +4.4 +π +5.0 +4.8 +5.6 +5.3 +5.4 +5.2 +5.1 +4.4 +b +4.4 +4.6 +5.8 +6.1 +5.6 +5.8 +8.4 +9.9 +max, a +4.2 +4.8 +4.2 +5.2 +3.1 +5.0 +1.2 +2.9 +max, b +0.7 +(1, 1, 1, 1) +30 +2.6 +2.3 +9.0 +9.1 +8.7 +11.1 +7.6 +4.6 +a +5.4 +5.8 +4.8 +4.6 +5.8 +5.6 +3.8 +3.6 +π +4.1 +5.2 +4.7 +4.7 +4.6 +4.9 +3.9 +4.0 +b +1.5 +1.7 +6.4 +9.0 +3.6 +11.0 +7.1 +6.1 +max, a +0.0 +6.2 +1.2 +4.6 +0.0 +5.9 +1.0 +1.0 +max, b +50 +3.7 +2.7 +7.6 +7.4 +7.5 +9.6 +11.3 +9.9 +a +5.6 +5.8 +4.5 +4.7 +5.7 +5.8 +4.6 +5.5 +π +4.3 +4.6 +4.3 +4.8 +5.3 +5.1 +4.2 +5.9 +b +2.9 +1.9 +5.5 +6.9 +3.8 +8.5 +11.7 +9.7 +max, a +0.7 +5.6 +1.2 +4.4 +0.0 +5.9 +1.1 +1.6 +max, b +70 +2.6 +2.1 +6.6 +6.6 +5.9 +7.4 +13.1 +12.3 +a +4.5 +4.7 +5.1 +5.0 +4.3 +5.0 +5.1 +5.0 +π +3.9 +4.4 +5.0 +5.0 +4.1 +4.8 +4.7 +6.0 +b +1.8 +1.6 +6.0 +7.2 +3.5 +6.2 +12.9 +12.0 +max, a +0.8 +4.6 +3.0 +5.3 +0.2 +5.4 +1.6 +1.6 +max, b +100 +3.0 +2.2 +5.9 +6.4 +5.0 +5.0 +14.6 +15.0 +a +5.1 +4.8 +4.4 +4.2 +4.5 +3.9 +4.9 +4.9 +π +4.8 +4.6 +4.3 +4.1 +4.6 +3.9 +4.6 +4.9 +b +3.3 +2.4 +4.3 +5.1 +4.0 +5.2 +14.0 +14.3 +max, a +2.5 +5.8 +1.9 +3.1 +0.5 +5.2 +1.0 +1.8 +max, b +150 +3.5 +2.3 +7.2 +7.0 +6.9 +6.8 +13.6 +15.6 +a +5.8 +5.2 +6.1 +6.0 +6.1 +5.7 +4.7 +4.2 +π +5.6 +4.7 +6.0 +6.0 +6.0 +5.9 +4.4 +5.4 +b +3.2 +1.9 +6.3 +6.8 +5.3 +7.0 +12.6 +15.1 +max, a +2.7 +5.4 +4.1 +5.8 +1.7 +5.9 +1.0 +2.1 +max, b +200 +4.1 +3.7 +5.8 +6.2 +5.1 +5.1 +14.2 +16.6 +a +5.5 +5.4 +4.6 +5.3 +4.4 +4.6 +4.8 +4.5 +π +5.1 +5.4 +5.0 +5.5 +4.7 +4.3 +4.6 +4.8 +b +3.0 +2.7 +4.3 +4.6 +4.3 +5.6 +13.0 +15.3 +max, a +3.3 +4.4 +3.3 +4.6 +2.3 +5.4 +0.6 +2.5 +max, b +0.7 +(1.5, 1.5, 1.5, 1.5) +30 +1.1 +1.0 +6.8 +7.6 +3.3 +8.5 +3.7 +3.3 +a +4.7 +5.1 +5.1 +4.7 +4.9 +5.1 +5.9 +5.7 +π +3.4 +3.9 +4.6 +5.1 +4.1 +4.4 +6.2 +5.7 +b +0.3 +1.0 +5.0 +6.8 +0.3 +8.0 +2.4 +4.1 +max, a +0.0 +3.7 +1.2 +4.8 +0.0 +5.5 +0.9 +2.3 +max, b +50 +1.5 +1.6 +6.3 +7.3 +3.6 +7.0 +5.9 +4.5 +a +4.3 +4.3 +5.3 +5.4 +5.1 +5.2 +4.1 +4.4 +π +2.8 +3.9 +5.2 +5.2 +4.3 +5.1 +3.9 +4.5 +b +37 + +Table S5: Empirical sizes (as percentages) of all tests obtained +under χ2 +10-distribution (Method: a - asymptotic Wald-type test; π - +permutation Wald-type test; b - bootstrap Wald-type test; max, a +- asymptotic multiple contrast test; max, b - bootstrap multiple +contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.6 +0.9 +5.4 +6.2 +0.4 +6.5 +6.9 +4.7 +max, a +0.0 +3.9 +2.2 +5.2 +0.0 +6.0 +1.2 +0.9 +max, b +70 +1.8 +2.1 +6.3 +6.6 +2.2 +5.2 +8.4 +7.5 +a +5.1 +4.8 +5.4 +4.9 +4.7 +4.4 +4.9 +5.6 +π +3.0 +4.6 +5.5 +5.1 +4.4 +3.8 +4.7 +6.2 +b +1.3 +1.6 +4.5 +6.1 +0.4 +4.4 +9.1 +7.7 +max, a +0.1 +4.3 +2.9 +5.3 +0.0 +4.6 +1.2 +2.0 +max, b +100 +2.2 +1.5 +5.5 +5.7 +2.0 +4.8 +13.2 +10.1 +a +4.6 +4.9 +4.8 +4.9 +3.9 +4.3 +5.5 +4.9 +π +3.4 +3.9 +4.5 +5.1 +3.6 +3.8 +5.5 +5.5 +b +1.8 +1.1 +4.8 +5.2 +0.7 +4.5 +14.1 +10.4 +max, a +0.5 +4.8 +2.9 +4.7 +0.0 +5.2 +1.6 +2.0 +max, b +150 +3.0 +2.4 +4.8 +5.1 +3.0 +4.8 +13.8 +12.7 +a +5.5 +5.6 +4.6 +4.7 +4.2 +4.3 +5.4 +6.2 +π +5.1 +5.6 +4.7 +4.6 +4.3 +4.5 +5.0 +5.8 +b +2.9 +2.2 +3.8 +4.4 +1.3 +4.3 +14.0 +14.0 +max, a +2.5 +5.3 +2.8 +3.8 +0.2 +5.5 +1.8 +2.2 +max, b +200 +2.9 +2.7 +5.5 +5.4 +4.3 +5.7 +15.3 +16.0 +a +5.4 +5.7 +5.2 +5.1 +5.6 +5.6 +5.6 +6.1 +π +4.6 +4.8 +4.8 +4.9 +5.7 +5.6 +5.2 +5.8 +b +2.4 +2.3 +5.3 +5.5 +2.5 +4.9 +14.6 +15.0 +max, a +2.3 +4.8 +4.7 +5.5 +0.7 +5.2 +1.6 +2.5 +max, b +38 + +Table S6: Empirical powers (as percentages) of the permutation and +bootstrap tests under normal distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.15) +30 +97.8 +98.9 +99.8 +99.8 +29.9 +45.1 +39.1 +52.4 +π +97.8 +98.8 +99.7 +99.8 +27.8 +44.9 +37.7 +51.4 +b +91.4 +99.0 +98.8 +99.9 +5.9 +42.6 +21.2 +42.4 +max, b +50 +100.0 +100.0 +100.0 +100.0 +64.1 +77.9 +67.6 +80.6 +π +100.0 +100.0 +100.0 +100.0 +62.9 +77.6 +67.6 +80.4 +b +100.0 +100.0 +100.0 +100.0 +39.5 +75.0 +55.2 +74.3 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +29.7 +44.4 +77.3 +86.1 +11.0 +19.3 +13.6 +24.5 +π +29.5 +45.4 +77.0 +85.5 +9.6 +18.9 +13.3 +24.6 +b +0.8 +44.3 +67.7 +84.1 +0.0 +19.5 +1.3 +21.5 +max, b +50 +56.1 +72.2 +98.0 +99.0 +22.2 +35.9 +25.9 +40.3 +π +56.0 +72.6 +98.0 +99.1 +20.5 +35.0 +25.8 +39.9 +b +25.5 +71.9 +97.0 +98.5 +2.4 +36.3 +9.1 +36.1 +max, b +(1, 1, 1, 1.5) +30 +10.9 +16.3 +48.1 +66.3 +8.5 +13.1 +8.1 +12.4 +π +10.0 +16.9 +47.2 +67.5 +7.5 +12.5 +7.3 +12.9 +b +0.0 +15.5 +15.5 +64.1 +0.0 +13.9 +0.0 +14.5 +max, b +50 +16.4 +30.2 +84.3 +91.6 +11.6 +22.7 +12.3 +24.2 +π +15.9 +30.3 +84.8 +92.1 +10.5 +22.6 +12.0 +24.4 +b +0.0 +28.8 +74.2 +91.1 +0.0 +22.9 +0.0 +23.8 +max, b +0.4 +(0.1, 0.1, 0.1, 0.15) +30 +98.9 +99.6 +95.5 +97.8 +30.3 +47.9 +38.1 +53.9 +π +98.9 +99.7 +95.5 +97.7 +28.1 +46.8 +37.8 +53.4 +b +93.1 +99.5 +89.5 +97.1 +6.5 +43.6 +19.8 +47.0 +max, b +50 +100.0 +100.0 +99.8 +99.8 +64.5 +78.0 +67.9 +79.1 +π +100.0 +100.0 +99.8 +99.9 +64.2 +78.0 +67.0 +79.0 +b +99.9 +100.0 +99.6 +99.7 +42.1 +75.1 +55.2 +73.5 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +27.8 +42.3 +68.9 +77.8 +10.9 +21.5 +12.2 +21.1 +π +26.4 +42.9 +68.7 +77.7 +9.5 +20.6 +12.2 +21.3 +b +2.0 +42.4 +50.8 +74.1 +0.2 +21.3 +1.8 +16.7 +max, b +50 +55.6 +67.3 +94.2 +96.7 +21.5 +36.2 +24.6 +34.5 +π +54.1 +67.3 +93.7 +96.0 +20.5 +35.3 +24.0 +34.0 +b +28.8 +67.1 +89.6 +95.5 +3.2 +34.6 +10.4 +29.5 +max, b +(1, 1, 1, 1.5) +30 +8.8 +16.6 +48.1 +65.0 +9.7 +12.9 +9.9 +16.4 +π +7.6 +16.7 +48.8 +64.9 +8.5 +12.9 +8.6 +16.8 +b +0.0 +15.6 +20.1 +62.3 +0.0 +13.2 +0.0 +13.5 +max, b +50 +14.4 +29.8 +80.5 +89.6 +11.7 +20.7 +12.8 +22.3 +π +13.4 +29.9 +80.2 +89.3 +10.8 +20.6 +12.5 +22.9 +b +0.0 +29.3 +67.6 +87.5 +0.0 +20.2 +0.2 +18.3 +max, b +0.7 +(0.1, 0.1, 0.1, 0.15) +30 +97.6 +99.2 +72.4 +80.7 +31.7 +47.9 +23.9 +39.4 +π +97.5 +99.3 +71.3 +80.3 +28.6 +47.4 +22.7 +38.9 +b +90.6 +99.0 +48.4 +73.5 +7.3 +44.8 +10.9 +29.3 +max, b +50 +100.0 +100.0 +94.8 +97.0 +60.7 +76.0 +47.8 +65.8 +π +100.0 +100.0 +95.2 +97.2 +59.9 +75.4 +48.3 +65.4 +b +100.0 +100.0 +90.1 +95.1 +38.7 +73.4 +34.0 +58.5 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +21.2 +37.6 +46.2 +55.7 +10.4 +21.2 +7.8 +9.0 +π +20.6 +37.5 +44.5 +55.0 +9.3 +20.4 +6.8 +9.4 +b +3.3 +36.8 +24.2 +50.2 +0.0 +20.7 +1.9 +3.4 +max, b +50 +43.6 +59.2 +76.4 +83.0 +19.5 +35.4 +9.1 +12.3 +π +43.3 +59.5 +76.9 +83.3 +18.6 +35.4 +8.6 +13.0 +b +39 + +Table S6: Empirical powers (as percentages) of the permutation and +bootstrap tests under normal distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +29.0 +59.9 +65.8 +79.6 +2.5 +35.9 +2.9 +5.8 +max, b +(1, 1, 1, 1.5) +30 +7.3 +14.5 +37.8 +54.4 +8.6 +12.9 +6.7 +11.2 +π +5.0 +13.6 +36.7 +54.0 +7.6 +12.8 +5.7 +10.7 +b +0.0 +13.2 +14.7 +49.4 +0.0 +14.5 +0.8 +3.8 +max, b +50 +16.0 +29.9 +70.0 +82.2 +11.6 +21.0 +9.3 +12.2 +π +13.1 +27.7 +69.8 +82.2 +10.5 +20.7 +8.6 +12.8 +b +0.6 +28.4 +55.4 +80.0 +0.0 +22.2 +2.1 +3.8 +max, b +40 + +Table S7: Empirical powers (as percentages) of the permutation +and bootstrap tests under t5-distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.15) +30 +68.4 +76.9 +63.8 +72.6 +22.8 +35.3 +21.2 +32.3 +π +67.4 +77.9 +60.9 +70.0 +20.2 +34.5 +20.3 +30.1 +b +21.1 +71.1 +28.8 +63.0 +0.5 +32.5 +3.2 +25.4 +max, b +50 +90.2 +94.5 +86.7 +92.1 +37.8 +49.8 +40.9 +51.3 +π +89.9 +94.3 +85.5 +90.3 +35.6 +49.5 +39.2 +48.8 +b +59.9 +90.1 +61.7 +85.3 +6.7 +47.4 +15.5 +42.8 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +20.3 +31.6 +38.5 +45.1 +9.7 +15.9 +13.1 +16.3 +π +18.8 +33.3 +36.3 +42.8 +8.6 +15.3 +11.6 +16.2 +b +0.1 +28.4 +12.6 +38.0 +0.1 +15.6 +0.3 +13.0 +max, b +50 +35.0 +51.8 +65.2 +72.8 +17.8 +26.6 +21.0 +29.3 +π +35.6 +53.0 +64.2 +71.6 +16.3 +26.3 +20.0 +28.3 +b +3.6 +46.5 +37.8 +67.5 +1.1 +26.5 +2.9 +24.5 +max, b +(1, 1, 1, 1.5) +30 +11.2 +17.6 +31.1 +43.7 +9.0 +11.9 +10.2 +14.3 +π +11.0 +18.2 +29.0 +41.9 +7.6 +11.7 +8.4 +13.9 +b +0.0 +16.6 +3.9 +37.6 +0.0 +14.1 +0.0 +13.4 +max, b +50 +17.4 +31.0 +57.4 +68.9 +11.7 +22.0 +12.7 +21.7 +π +17.1 +31.6 +55.0 +66.6 +11.3 +22.6 +11.8 +22.4 +b +0.0 +28.2 +19.9 +62.8 +0.0 +22.8 +0.0 +21.9 +max, b +0.4 +(0.1, 0.1, 0.1, 0.15) +30 +69.7 +79.4 +55.6 +64.5 +21.4 +32.2 +22.4 +33.9 +π +67.2 +79.5 +53.6 +62.8 +19.7 +31.8 +21.5 +32.4 +b +20.9 +74.2 +20.7 +56.2 +1.0 +31.2 +4.1 +25.7 +max, b +50 +91.5 +94.4 +78.7 +85.2 +38.1 +50.8 +40.4 +53.9 +π +90.4 +93.8 +78.0 +84.0 +35.0 +50.3 +38.9 +50.9 +b +62.9 +90.8 +52.2 +78.7 +9.5 +47.8 +16.7 +44.2 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +18.9 +28.1 +36.0 +43.3 +12.5 +19.1 +11.4 +14.9 +π +18.7 +29.0 +35.0 +40.9 +10.0 +18.7 +10.4 +14.1 +b +0.0 +27.8 +9.4 +36.8 +0.0 +21.1 +0.3 +11.4 +max, b +50 +37.3 +50.4 +56.3 +64.6 +18.1 +28.2 +16.0 +22.4 +π +36.9 +51.0 +54.9 +62.8 +16.9 +28.4 +15.4 +21.9 +b +4.4 +48.5 +27.9 +56.8 +0.4 +27.9 +1.7 +18.5 +max, b +(1, 1, 1, 1.5) +30 +11.2 +17.7 +31.4 +44.4 +7.9 +13.6 +8.0 +13.2 +π +10.5 +18.0 +28.2 +42.1 +6.4 +13.5 +6.2 +13.2 +b +0.0 +15.7 +5.7 +37.5 +0.0 +14.0 +0.2 +9.5 +max, b +50 +15.3 +27.6 +54.4 +68.7 +10.4 +20.4 +12.6 +20.3 +π +14.4 +28.2 +52.0 +67.2 +9.7 +20.0 +10.5 +19.2 +b +0.0 +26.6 +20.2 +61.1 +0.0 +20.5 +0.3 +15.3 +max, b +0.7 +(0.1, 0.1, 0.1, 0.15) +30 +66.8 +76.0 +40.0 +49.0 +23.0 +32.3 +17.6 +28.3 +π +65.5 +76.6 +38.5 +46.7 +20.5 +32.2 +16.8 +27.4 +b +19.5 +71.8 +10.3 +41.9 +0.9 +29.9 +1.8 +21.6 +max, b +50 +89.8 +92.7 +62.2 +68.9 +38.8 +50.5 +31.0 +44.0 +π +88.4 +92.8 +60.4 +67.2 +36.1 +49.5 +29.0 +42.3 +b +59.5 +88.7 +30.4 +62.5 +6.9 +46.6 +9.5 +37.5 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +18.2 +27.0 +25.1 +31.1 +9.1 +13.9 +9.0 +9.5 +π +18.1 +28.3 +24.0 +28.7 +8.7 +13.4 +7.3 +9.1 +b +0.6 +26.2 +5.1 +25.7 +0.1 +14.0 +0.8 +2.9 +max, b +50 +35.1 +48.8 +42.2 +50.0 +18.0 +25.8 +9.9 +11.7 +π +35.3 +49.8 +41.3 +49.3 +16.0 +25.5 +8.5 +12.0 +b +41 + +Table S7: Empirical powers (as percentages) of the permutation +and bootstrap tests under t5-distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +8.0 +47.7 +14.6 +43.9 +0.4 +25.2 +1.8 +6.3 +max, b +(1, 1, 1, 1.5) +30 +7.9 +13.8 +25.7 +36.1 +9.8 +13.0 +7.9 +9.7 +π +5.4 +12.7 +23.9 +33.3 +7.5 +13.3 +5.9 +9.4 +b +0.0 +10.0 +4.2 +29.1 +0.0 +13.8 +0.8 +2.4 +max, b +50 +13.2 +24.3 +46.0 +58.8 +12.2 +19.8 +9.2 +10.3 +π +11.9 +24.6 +43.5 +55.8 +10.5 +19.8 +7.8 +9.4 +b +0.3 +23.2 +16.2 +51.0 +0.0 +20.1 +1.0 +2.7 +max, b +42 + +Table S8: Empirical powers (as percentages) of the permutation and +bootstrap tests under χ2 +10-distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +0.1 +(0.1, 0.1, 0.1, 0.15) +30 +96.1 +97.8 +96.9 +98.8 +27.2 +42.6 +35.2 +48.2 +π +95.8 +97.9 +96.5 +98.8 +25.2 +40.4 +33.3 +45.7 +b +82.2 +97.2 +88.8 +97.4 +4.1 +39.6 +16.8 +40.2 +max, b +50 +99.9 +100.0 +100.0 +100.0 +59.1 +72.0 +63.9 +76.0 +π +100.0 +100.0 +100.0 +100.0 +57.5 +71.8 +64.1 +75.4 +b +99.6 +100.0 +99.7 +100.0 +27.3 +67.5 +43.5 +70.1 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +29.7 +47.5 +77.7 +85.4 +13.6 +25.4 +19.4 +31.3 +π +29.0 +48.3 +76.3 +85.4 +12.1 +24.4 +18.9 +31.4 +b +2.4 +47.9 +62.5 +81.5 +0.4 +25.3 +4.5 +25.8 +max, b +50 +57.3 +71.4 +96.8 +98.3 +30.4 +45.7 +35.3 +50.2 +π +57.3 +71.5 +97.0 +98.2 +29.9 +45.4 +35.6 +50.6 +b +29.7 +72.4 +94.5 +97.4 +9.6 +46.5 +22.5 +48.1 +max, b +(1, 1, 1, 1.5) +30 +9.6 +19.3 +50.6 +69.4 +6.2 +12.5 +7.7 +16.9 +π +8.7 +18.8 +48.9 +69.4 +5.7 +12.2 +6.8 +17.1 +b +0.0 +18.8 +22.4 +67.1 +0.0 +14.7 +0.0 +17.0 +max, b +50 +17.3 +32.1 +83.6 +90.7 +12.1 +27.9 +13.7 +31.9 +π +17.0 +32.2 +84.0 +90.8 +11.9 +27.5 +13.8 +33.2 +b +0.0 +32.9 +71.9 +89.7 +0.0 +28.5 +0.0 +31.8 +max, b +0.4 +(0.1, 0.1, 0.1, 0.15) +30 +97.6 +99.0 +89.2 +95.1 +28.0 +41.1 +32.1 +45.7 +π +97.6 +99.2 +88.8 +94.8 +25.5 +40.6 +30.8 +44.4 +b +83.4 +99.1 +72.9 +90.4 +2.6 +37.5 +11.9 +38.2 +max, b +50 +100.0 +100.0 +99.3 +99.7 +58.0 +70.9 +58.2 +71.4 +π +100.0 +100.0 +98.9 +99.7 +57.1 +69.8 +57.2 +69.6 +b +99.9 +100.0 +96.1 +99.2 +29.9 +68.6 +37.8 +65.2 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +28.5 +45.8 +66.3 +73.9 +13.0 +22.9 +14.4 +21.1 +π +28.0 +45.8 +65.7 +73.5 +11.9 +22.5 +13.6 +21.2 +b +4.0 +49.4 +49.7 +70.9 +0.2 +24.3 +3.4 +15.8 +max, b +50 +59.7 +72.9 +92.6 +95.2 +30.1 +43.9 +24.8 +36.1 +π +59.0 +73.5 +92.3 +95.1 +28.8 +43.6 +24.3 +36.4 +b +36.1 +73.4 +86.4 +93.6 +9.0 +43.7 +13.4 +30.3 +max, b +(1, 1, 1, 1.5) +30 +9.4 +16.6 +52.7 +69.3 +7.5 +14.8 +9.3 +16.3 +π +8.1 +16.9 +49.0 +68.0 +6.3 +13.6 +8.6 +16.0 +b +0.0 +17.6 +25.1 +65.2 +0.0 +14.9 +0.3 +11.3 +max, b +50 +17.0 +34.2 +88.0 +92.9 +12.8 +30.0 +13.5 +26.5 +π +15.4 +34.1 +87.0 +93.4 +11.8 +29.3 +13.2 +26.9 +b +0.0 +35.3 +78.2 +92.9 +0.0 +31.3 +0.8 +21.1 +max, b +0.7 +(0.1, 0.1, 0.1, 0.15) +30 +97.9 +98.4 +63.5 +74.8 +28.1 +44.5 +19.4 +34.7 +π +97.8 +98.4 +62.7 +72.8 +25.6 +43.3 +19.1 +33.9 +b +86.5 +98.0 +38.2 +66.7 +2.6 +41.2 +5.5 +26.0 +max, b +50 +100.0 +100.0 +93.1 +95.9 +53.9 +67.9 +33.8 +50.0 +π +100.0 +100.0 +92.2 +96.1 +52.5 +67.0 +34.0 +49.3 +b +99.6 +99.9 +83.5 +94.3 +26.6 +64.4 +17.3 +43.4 +max, b +(0.5, 0.5, 0.5, 0.7) +30 +26.2 +39.8 +47.1 +57.6 +14.4 +22.3 +9.1 +9.9 +π +24.6 +40.0 +47.3 +57.9 +13.3 +22.8 +8.5 +10.6 +b +4.5 +41.2 +28.0 +50.0 +0.2 +24.5 +1.1 +3.5 +max, b +50 +55.3 +70.2 +77.1 +83.7 +28.5 +45.2 +10.9 +12.5 +π +53.9 +70.0 +76.0 +82.9 +28.2 +45.7 +10.5 +13.1 +b +43 + +Table S8: Empirical powers (as percentages) of the permutation and +bootstrap tests under χ2 +10-distribution (Method: π - permutation +Wald-type test; b - bootstrap Wald-type test; max, b - bootstrap +multiple contrast test) (continued) +ρ +(C1, C2, C3, C4) +ni +ϕCRR +ϕBRR +ϕCV V +ϕBV V +ϕCV N +ϕBV N +ϕCAZ +ϕBAZ +Method +35.0 +70.5 +64.9 +79.3 +6.3 +44.4 +1.9 +5.2 +max, b +(1, 1, 1, 1.5) +30 +8.7 +16.7 +45.3 +63.7 +7.1 +14.4 +7.6 +9.8 +π +4.6 +14.6 +44.1 +62.0 +5.7 +13.4 +6.9 +10.1 +b +0.0 +15.0 +19.4 +59.4 +0.0 +15.5 +1.5 +4.0 +max, b +50 +16.7 +30.0 +74.5 +86.5 +12.6 +26.8 +10.7 +13.9 +π +13.7 +28.5 +75.4 +86.4 +11.5 +26.9 +10.1 +14.5 +b +0.5 +29.5 +62.8 +82.4 +0.0 +27.9 +2.3 +5.3 +max, b +44 + diff --git a/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/load_file.txt b/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3bd0865e5e42a052ce76452963696a9c4f91ee6a --- /dev/null +++ b/KNFLT4oBgHgl3EQfKy8R/content/tmp_files/load_file.txt @@ -0,0 +1,15043 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf,len=15042 +page_content='Inference for all variants of the multivariate coefficient of variation in factorial designs Marc Ditzhaus1 and �Lukasz Smaga2,∗ 1Faculty of Mathematics, Otto von Guericke University Magdeburg, Germany 2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland January 31, 2023 Abstract The multivariate coefficient of variation (MCV) is an attractive and easy-to-interpret effect size for the dispersion in multivariate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Recently, the first inference methods for the MCV were proposed by Ditzhaus and Smaga (2022) for general factorial designs covering k-sample settings but also complex higher-way layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, two questions are still pending: (1) The theory on inference methods for MCV is primarily derived for one special MCV variant while there are several reasonable proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2) When rejecting a global null hypothesis in factorial designs, a more in-depth analysis is typically of high interest to find the specific contrasts of MCV leading to the aforementioned rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In this paper, we tackle both by, first, extending the aforementioned nonparametric permutation procedure to the other MCV variants and, second, by proposing a max-type test for post hoc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To improve the small sample performance of the latter, we suggest a novel studentized bootstrap strategy and prove its asymptotic validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The actual performance of all proposed tests and post hoc procedures are compared in an extensive simulation study and illustrated by a real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Keywords: Coefficient of variation, factorial designs, multiple testing, bootstrap and permutation procedures, multivariate analysis, standardized mean 1 Introduction The coefficient of variation (CV) is defined as the standard deviation divided by the population mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By this, it becomes a powerful, unit-free measure of dispersion and is used in diverse areas, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' in medicine for reliability and reproducibility of measurements (Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021), for risk evaluation in finance (Ferri and Jones, 1979) or in psychology (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Furthermore, it is used in control charts for monitoring (Jalilibal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, as pointed out by Yeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2016) in the context of control charts: “There are many situations where multiple characteristics need to be monitored simultaneously.” This is certainly true apart from control charts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' when various medical measurements are taken from the same patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For such scenarios, the CV can be extended to the multivariate setting in various ways (Reyment, 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Van Valen, 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Voinov and Nikulin, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Albert and Zhang, 2010): CRR = � (det Σ)1/d µ⊤µ , CVV = � trΣ µ⊤µ, CVN = � 1 µ⊤Σ−1µ, CAZ = � µ⊤Σµ (µ⊤µ)2 , (1) which all reduces to CV in the univariate case (d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here µ ̸= 0 denotes the mean vector of a d-dimensional random variable and Σ is the corresponding covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The standardized means as the reciprocal of Cv are of their own interest: Bv = 1 Cv (v = RR, VV, VN, AZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2) ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Email address: ls@amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='pl 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='12009v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='ME] 27 Jan 2023 The differences between the four variants are discussed in great detail by Albert and Zhang (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' One remarkable difference is that CRR and CVN require a regular matrix Σ while the other two variants do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This regularity assumptions becomes rather restrictive for high-dimensional scenarios, such as microarray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, the variant CVV is the only one which does not take the covariance of the different measurements into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' When we turn to the inference problem, the literature regarding the MCV becomes scare while for the univariate CV various two- and k-sample testing proposal can be found in the literature, see Aerts and Haesbroeck (2017) and Pauly and Smaga (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Recently, Ditzhaus and Smaga (2022) addressed the remaining question for more general CV testing procedures, namely in complex factorial designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The latter are highly relevant for various fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' in biomedicine or psychology (GISSI-2, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Baigent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Cassidy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Kurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2015), where the k-sample set-up is often too narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Methods for factorial designs allow to discuss main effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' of the gender, measurement, site) and even interaction effects: ’it is desirable for reports of factorial trials to include estimates of the interaction between the treatments’ (Lubsen and Pocock, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In addition to the extension towards factorial designs, Ditzhaus and Smaga (2022) proposed their nonparametric methods directly for CVN in the multivariate set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This complemented the prior proposal of Aerts and Haesbroeck (2017), which, in contrast, relied on (semi-)parametric model assumptions and whose convergence rate was rather slow, see the simulation results of Ditzhaus and Smaga (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To the best of our knowledge, these two proposals are the only ones discussing the inference problem for MCV and, moreover, both restricted their study to CVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, the natural question arises whether the inference strategies can be transferred to the other MCV variants from (1), especially to be able to study also settings with non-regular covariance matrices Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For this purpose, we will adopt the results of Ditzhaus and Smaga (2022) and, in particular, derive permutation versions with a better performance under small sample sizes, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In this way, we add a further chapter to the success story of studentized permutation tests in complex factorial designs (Pauly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Friedrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Harrar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Ditzhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' While classical permutation tests for exchangeable data settings are well-known, it is less known that the studentized permutation versions are also valid beyond exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Beside global null hypothesis testing in k-sample settings or inferring main/interaction effects in factorial designs, often a more in-depth analysis is wanted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' multiple pairwise comparisons, to get a better picture of the underlying effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Since Bonferroni correction leads partially to a significant power loss, strategies incorporating the concrete dependence structure are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefor, multiple contrast tests and corresponding simultaneous confidence intervals are well established for means (Mukerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2001), the relative treatment effect (Umlauft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Gunawardana and Konietschke, 2019), and the area under the receiving operating curve (Konietschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Wechsung and Konietschke, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A respective multiple strategy for the MCV is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In Section 2, we derive a central limit theorem for all MCV variants and their reciprocals in the one-sample setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we discuss assumptions such that the limit is not degenerated and how the limiting variances can be estimated consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' All these results lie the foundation for the Wald-type statistics to infer main and interaction effects in terms of MCVs and standardized means in general factorial designs, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we develop respective permutation and bootstrap counterparts of these Wald-type statistics and prove their asymptotic validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In Section 4, we discuss the issue of simultaneous inference and present max-type multiple contrast tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Again, we complement this by an asymptotically valid resampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' An exhaustive simulation study is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The tests’ applicability are illustrated by analyzing data of external quality assessment in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Finally, Section 7 summarizes the major results of the paper and discusses further research possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' All proofs and additional simulation results are in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 2 2 The nonparametric framework Consider n1 independent, identically distributed d-dimensional random variables Xj = (Xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Xjd)⊤ (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Hereby, we suppose no specific conditions on the distributions of Xj except the following assumptions on the moments to ensure the well-definedness of Cj and Bj: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let µ ̸= 0 and E(X4 jℓ) < ∞ for all j and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we suppose: (a) For CRR, CVN, BRR, and BVN, we consider only regular matrices Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (b) For CVV and BVV, we assume Σ ̸= 0d×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (c) For CAZ and BAZ, we suppose µ⊤Σµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here and below, we denote by 0d×d a d × d-dimensional matrix consisting of zeros only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Assumption 1 ensures that Cv, Bv are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Clearly, a ⇒ c ⇒ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, CVV seems to be the most general variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' But, as mentioned in the introduction, CVV does not take the covariance structure into account and only combines the marginal variability into one standardized effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For statistical inference, we estimate the MCVs or their reciprocals, respectively, by plugging-in the sample mean �µ and covariance matrix �Σ: �CRR = � (det �Σ)1/d �µ⊤ �µ , �CVV = � tr�Σ �µ⊤ �µ , �CVN = � 1 �µ⊤ �Σ −1 �µ , �CAZ = � �µ⊤ �Σ�µ (�µ⊤ �µ)2 , (3) and �Bv = 1/ �Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By extending the results of Ditzhaus and Smaga (2022) for CVN and BVN, we are able to derive central limit theorems for all these estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The respective asymptotic variances have a rather complex structure and depend on several quantities: ARR(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σ) = � −2d det(Σ) µ⊤ (µ⊤µ)d+1 + det(Σ) � vec(Σ−1) �⊤ (µ⊤µ)d � D(µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' det(Σ) � vec(Σ−1) �⊤ (µ⊤µ)d � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' AVV(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σ) = � −2 tr(Σ) µ⊤ (µ⊤µ)2 + 1 µ⊤µ (vec(Id))⊤ � D(µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 1 µ⊤µ (vec(Id))⊤ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' AVN(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σ) = � 2 µ⊤Σ−1 − [(µ⊤Σ−1) ⊗ (µ⊤Σ−1)] � D(µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' −(µ⊤Σ−1) ⊗ (µ⊤Σ−1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' AAZ(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σ) = � −4 µ⊤Σµ µ⊤ (µ⊤µ)3 + 2 µ⊤Σ (µ⊤µ)2 + µ⊤ ⊗ µ⊤ (µ⊤µ)2 � D(µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' µ⊤ ⊗ µ⊤ (µ⊤µ)2 � where ⊗ is the Kronecker product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' and the matrices � D(x) ∈ Rd2×d for x = (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , xd)⊤ ∈ Rd, Ψi3 ∈ Rd2×d as well as Ψi4 ∈ Rd2×d2 are given by their entries [ � D(x)]ad−d+r,s = −xrI{s = a ̸= r} − 2xsI{s = r = a} − xaI{r = s ̸= a} [Ψ3]ad−d+r,s = E(X1aX1rX1s) − E(X1aX1r)E(X1s) (4) [Ψ4]ad−d+r,bd−d+s = E(X1aX1rX1bX1s) − E(X1aX1r)E(X1bX1s) for a, b, r, s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, we are able to formulate the central limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here and subsequent, all limits are meant as n1 → ∞ unless stated explicitly otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Theorem 1 (Central limit theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let v ∈ {RR, VV, VN, AZ} and Assumption 1 be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The estimators �Cv and �Bv i are asymptotically normal: n1/2 1 � �Cv − Cv� d −→ ZCv ∼ N(0, σ2 Cv), and n1/2 1 � �Bv − Bv� d −→ ZBv ∼ N(0, σ2 Bv) 3 with asymptotic variances σ2 Bv = (Cv)−4σ2 Cv and σ2 Cv = Sv 4 Av(µ, Σ) � Σ Ψ⊤ 3 Ψ3 Ψ4 � Av(µ, Σ)⊤, where SRR = d−2(CRR)2−4d, SVV = (CVV)−2, SVN = (CVN)6, SAZ = (CAZ)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The typically unknown variances, σ2 Bv and σ2 Cv, can be naturally estimated by replacing the expectations and covariances by their empirical counterparts, for instance: [ �Ψ3]ad−d+r,s = � n−1 1 n1 � j=1 XjaXjrXjs � − � n−1 1 n1 � j=1 XjaXjr �� n−1 1 n1 � j=1 Xjs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In this way, we obtain �σ2 Cv = �Sv 4 Av(�µ, �Σ) � �Σ �Ψ ⊤ 3 �Ψ3 �Ψ4 � Av(�µ, �Σ)⊤, �σ2 Bv = ( �Cv)−4�σ2 Cv, (5) where �SRR = d−2( �CRR)2−4d, �SVV = ( �CVV)−2, �SVN = ( �CVN)6, �SAZ = ( �CAZ)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A direct consequence of the continuous mapping theorem and the strong law of large numbers is Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Under Assumption 1, �σ2 Cv p→ σ2 Cv and �σ2 Bv p→ σ2 Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In general, there is no guarantee that the limits from Theorem 1 are not degenerated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' σ2 Cv = 0 and, equivalently, σ2 Bv = 0 might be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' As in the univariate case (Pauly and Smaga, 2020) and for v = VN in the multivariate setting (Ditzhaus and Smaga, 2022), degeneracy can just appear in rather unusual scenarios of the following kind: Definition 1 (Conditional two-point distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let Y = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Yd)⊤ ∈ Rd be a multivariate random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We call the rth coordinate Yr conditionally two-point distributed if it is (conditionally) degenerated or it just takes (conditionally) two different values with positive probability, both given the remaining components (Ys)s=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='s̸=r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For example, the coordinates of (Y1, Y2, Y1 + Y 2 2 ) for arbitrarily distributed Y1, Y2 are conditionally two-point distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The same is true for the coordinates of (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Yd) with binomial distributed Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' These extreme cases need to be excluded: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' No coordinate of X1 is conditionally two-point distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In fact, a weaker assumption is also enough as illustrated in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In detail, it is sufficient to suppose that the ℓth coordinate of X1 is not conditionally two-point distributed for some ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, we then additionally require [µ]ℓ ̸= 0 in case of v = AZ and [µ⊤Σ−1]ℓ ̸= 0 for v = VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' From our point of view, Assumption 2 is easier to check, in particular later for the resampling procedures, and we do not loose much of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Under Assumptions 1 and 2 we have σ2 Cv > 0 and, thus, σ2 Bv > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Theorem 1 and Lemma 1 [ �Cv ± n−1/2 1 �σCvz1−α/2] and [ �Bv ± n−1/2 1 �σBvz1−α/2] are asymptotically valid confidence intervals for Cv and Bv, respectively, where z1−α/2 is the (1 − α/2)-quantile of N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, these results serve as the foundation for inference methods in more complex models as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 4 3 Global testing for factorial designs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 Factorial designs with respective null hypotheses From the easy one-sample scenario from the previous section, we immediately turn to general factorial designs covering two- and k-sample settings as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Notationally, factorial designs can be incorporated in a k-sample framework by interpreting the groups as subgroups for different factor combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We explain this concept below in more detail but first start with introducing the concrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For this purpose, we add another index i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k, k ∈ N, to all quantities from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, let Xij = (Xij1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Xijd)⊤ (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , ni), where Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Xini are identically distributed for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k and all observations X11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Xknk are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Depending on the research question, we choose a contrast matrix H ∈ Rr×k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' H1k×1 = 0r×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We now like to infer the following null hypothesis in terms of the chosen H and by this cover a huge variety of testing problems: H0,Cv : HCv = 0r×1, H0,Bv : HBv = 0r×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (6) Here, Cv = (Cv 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Cv k)⊤ and Bv = (Bv 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Bv k)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the same way, we denote by � C v and � B v the respective estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Different choices for H: To see the high flexibility of (6), we like to discuss some specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The most prominent one is the k-sample scenario H0,Cv : {P kCv = 0k×1} = {Cv 1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' = Cv k}, where P k = Ik − 1k×k/k and Ik is the k × k-dimensional unity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Turning to a more complex scenario, we next consider a two-way layout with two factors A and E possessing a and e levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By splitting up the group index i = (iA, iE) we incorporate this scenario in the aforementioned k-sample framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, we obtain k = a · e subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, we divide the subgroup-specific MCV Cv iA,iE = Cv0 + Cvα iA + Cvϵ iE + Cvαϵ iAiE into a general effect Cv0, the two main effects Cvα iA , Cvϵ iE, and an interaction effect Cvαϵ iAiE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, the usual side conditions � iA Cvα iA = � iB Cvϵ iE = � iA Cvαϵ iAiE = � iE Cvαϵ iAiE = 0 ensure the identifiability of the aforementioned effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Related null hypotheses are: HA 0,Cv : {HACv = 0} = {Cvα iA = 0 ∀iA}, HA = P a ⊗ (11×e/e) (no main effect A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' HE 0,Cv : {HECv = 0} = {Cvϵ iE = 0 ∀iE}, HE = (11×a/a) ⊗ P e (no main effect E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' HAE 0,Cv : {HAECv = 0} = {Cvαϵ iAiE = 0 ∀iA, iE}, HAE = P a ⊗ P e (no interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Hereby, ⊗ is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Clearly, we can replace Cv by Bv to get respective null hypotheses for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The described strategy can be extended, in a straightforward manner, to higher-way layouts and hierarchical designs with nested factors, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 of the supplement from Ditzhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2021b) or Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 4 of Pauly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 Wald-type tests How to test global null hypothesis of the form (6) is discussed in several papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In general, quadratic forms are built on the estimating vectors, here � C v or � B v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Popular examples are (modified) ANOVA- type statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Brunner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Friedrich and Pauly, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Sattler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2022) and Wald-type statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Pauly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Smaga, 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Ditzhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, the latter is usually an asymptotically pivotal statistic which is beneficial for the resampling procedures proposed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the results, we require the following classical assumption of non-vanishing groups ni n → κi ∈ (0, 1) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (7) 5 This limit and all following ones are meant as the total sample size n = �k i=1 ni tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This is in line with the prior convention from Section 2, where we considered only one group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' n = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, we can formulate and motivate the Wald-type statistics for (6): Sn,Cv(H) = n(H � C v)⊤(H �ΣCvH⊤)+H � C v, Sn,Bv(H) = n(H � B v)⊤(H �ΣBvH⊤)+H � B v, where �ΣCv = diag((n/n1)�σ2 1,Cv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , (n/nk)�σ2 k,Cv) and �ΣBv = diag((n/n1)�σ2 1,Bv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ), and A+ denotes the Moore–Penrose inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Lemma 1, these estimators are consistent for ΣCv = diag(κ−1 1 σ2 1,Cv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , κ−1 k σ2 k,Cv) and ΣBv = diag(κ−1 1 σ2 1,Bv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Whenever Assumption 2 holds for all i, the limiting covariance matrices ΣCv and ΣBv are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Finally, we can deduce from Theorem 1, Lemma 1 and Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 of Rao and Mitra (1971) that the limits of Sn,Cv(H) and Sn,Bv(H) under H0,Cv and H0,Bv, respectively, are chi-squared distributed with rank(H) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, the same arguments yield that n−1Sn,Cv(H) and n−1Sn,Bv(H) always converge in probability to (HCv)⊤(HΣCvH⊤)+HCv and (HBv)⊤(HΣBvH⊤)+HBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the proofs, we show that these limits are positive under alternatives H1,Cv : HCv ̸= 0 or H1,Bv : HBv ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let (7) as well as Assumptions 1 and 2 be fulfilled for all subgroups i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (i) Under H0,Cv : HCv = 0, Sn,Cv(H) tends in distribution to Z ∼ χ2 rank(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (ii) Under H1,Cv : HCv ̸= 0, Sn,Cv(H) diverges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Sn,Cv(H) p→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (iii) Under H0,Bv : HBv = 0, Sn,Bv(H) tends in distribution to Z ∼ χ2 rank(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (iv) Under H1,Bv : HBv ̸= 0, Sn,Bv(H) diverges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Sn,Bv(H) p→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' As a result of Theorem 2, we obtain asymptotically valid tests ϕn,Cv = 1{Sn,Cv(H) > χ2 rank(H),1−α} for the testing problems H0,Cv vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' H1,Cv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' they have an asymptotic level α and an asymptotic power of 1, similarly for Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, χ2 rank(H),1−α denotes the (1 − α)-quantile of a chi-square distribution with rank(H) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The convergences rate of Wald-type statistics is known to be rather slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The simulations of Ditzhaus and Smaga (2022) confirmed this general impression for the specific variant v = VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The new simulation study in Section 5 underpins that for the other variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To tackle this problem, we follow a studentized permutation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we also consider a (pooled) bootstrap test, which is of particular interest for local null hypotheses testing, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 Permutation and bootstrapping Resampling procedures are popular and well-accepted tools to improve the tests’ performance and, in particular, their control of the type-1 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Due to our good experience with permutation procedures for Wald-type statistics (Ditzhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Ditzhaus and Smaga, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Smaga, 2015, 2017), we propose to follow this successful and powerful strategy also for the underlying problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A remarkable advantage of permuting over other resampling strategies is its finite exactness under exchangeability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' under the more restrictive null hypothesis � H0 : X11 d= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' d= Xk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For more general (potentially nonexchangeable) null hypotheses, it is not clear whether the permutation strategy leads indeed to a valid testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' But in case of studentized statistics, as Wald-type statistics, this desirable validity was proven in various other settings and we provide a proof in the underlying set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The (pooled) bootstrap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' drawing from the pooled data with replacement, is closely related to the permutation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We consider it here as well because the permutation strategy is not appropriate for simultaneous testing as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let us become more specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We first group all data together and denote the resulting pooled data by X = (Xij)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, we draw with or without replacement from X to obtain a permutation (Xπ = (Xπ ij)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',ni) or a bootstrap (Xb = (Xb ij)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',ni) sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Since no observation can be drawn twice for Xπ, the permuted observations mutually depend from each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In contrast to that, we draw for the bootstrap sample with replacement and, hence, some individuals 6 appear multiple times and some do not appear at all in Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, the bootstrap observations are independent from each other as the original observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In Section 4, we explain why this property is beneficial for the simultaneous testing and why the permutation fails there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To differentiate between the quantities from the previous Sections for the different samples, we add the superscript π or b to them when they rely on the permutation or bootstrap sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For example, Sπ n,Cv(H) denotes the permuted test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Since we draw from the pooled data, the assumptions need to be translated from the specific groups to the pooled distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefore, we introduce the expectation µ0 = �k i=1 κiµi and the covariance matrix Σ0 for the (asymptotic) pooled distribution P0 = �k i=1 κiP Xi1, where the matrix is given by its entries [Σ0]ℓm = (�k i=1 κiE(Xℓ1Xm1))−[µ0]ℓ[µ0]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' It is easy to check, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' via projections, that Assumption 2 is true for the pooled distribution when this is the case for all (sub-)groups i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, we just need, in addition to the conditions from Theorem 1, that Assumption 1 is fulfilled for the pooled quantities: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In addition to the assumptions of Theorem 1, we suppose that Assumption 1 is fulfilled for the pooled quantities µ0 and Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then the following statements are valid under the null hypotheses H0,Cv, H0,Bv and their respective alternatives: (a) The permutation statistics Sπ n,Cv(H) and Sπ n,Bv(H) always mimic the null distribution limit of Sn,Cv(H) and Sn,Bv(H) asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In formulas (exemplarily for Cv): sup x∈R ��� Pr � Sπ n,Cv(H) ≤ x | X � − χ2 rank(H)(x) ��� p→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (b) The statement in (a) is also true for the bootstrap statistics Sb n,Cv(H) and Sb n,Bv(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Theorem 3 justifies the validity of the permutation and bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To accept this, let qπ n,1−α,Cv(X) be the (1 − α)-quantile of the permutation distribution R ∋ x �→ Pr(Sπ n,Cv(H) ≤ x | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then Theorem 3 ensures that qπ n,1−α,Cv(X) approximates always (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=') the quantile χ2 rank(H),1−α of the test ϕn,Cv from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, the asymptotic properties of ϕn,Cv, namely asymptotic exactness under the null and consistency for all alternatives, can be transferred to its permutation counterpart ϕπ n,Cv = 1{Sn,Cv(H) > qπ n,1−α,Cv(X)} (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Janssen and Pauls, 2003, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 1 and Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Clearly, the same is true when we consider B instead of C and/or the bootstrap instead of the permutation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 4 Multiple testing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 Local null hypotheses and the multiple contrast tests While testing for main and interaction effects in factorial designs is of high interest, often a more in-depth analysis is wanted to check which part of the equation systems HCv = 0 or HBv = 0, respectively, is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This leads to the multiple testing problem H0,ℓ,Cv : h⊤ ℓ Cv = 0 (ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , r) (8) for contrast vectors hℓ ∈ Rk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' h⊤ ℓ 1k×1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The intersection �r ℓ=1 H0,ℓ,Cv of the local null hypotheses coincides with the global null hypothesis H0,Cv from (6) with H = (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , hr)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the easiest case, we are interested in group differences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the global null hypotheses is H0,Cv : Cv 1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' = Cv k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Prominent examples to split the latter into local null hypotheses are Tukey’s all-pairs comparison 7 (Tukey, 1953) H0,Cv : � � � � � � � � � � � � � � � � � � � � � � � � � � � Cv 1 = Cv 2 Cv 1 = Cv 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Cv 1 = Cv k Cv 2 = Cv 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Cv k−1 = Cv k ⇔ H0,Cv : � � � � � � � � � � � � −1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 0 −1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 1 0 −1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 −1 1 � � � � � � � � � � � � Cv = 0 or Dunnet’s multiple-to-one comparison (Dunnett, 1955) H0,Cv : � � � � � � � � � � � Cv 1 = Cv 2 Cv 1 = Cv 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Cv 1 = Cv k ⇔ H0,Cv : � � � � � −1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 −1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' −1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0 1 � � � � � Cv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Further proposals can be found in Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In principle, we can consider different strategies from multiple testing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the Bonferroni or Holm correction, to adjust the type-1 errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, this leads typically to a significant power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A more promising approach are multiple contrast tests (Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Hothorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Gunawardana and Konietschke, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For them, we test the single null hypotheses H0,ℓ,Cv by Sn,Cv(h⊤ ℓ ) from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 and then incorporate the explicit dependence structure of them to obtain a valid testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In detail, we consider the following max-type statistic Sn,max,Cv(H) = max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r(Sn,Cv(h⊤ ℓ ))1/2 = max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r |T v ℓ,n|, T v ℓ,n = √n h⊤ ℓ �C v � h⊤ ℓ �ΣCvhℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Theorem 1, (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n) converges in distribution to a multivariate normal distribution with standard normal distributed marginals and correlation matrix RCv given by [RCv]ℓm = h⊤ ℓ ΣCvhm � h⊤ ℓ ΣCvhℓ � h⊤ mΣCvhm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (9) The studentization of the T v ℓ,n’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' dividision by an estimator for the asymptotic variance, ensures that each null hypothesis is (asymptotically) treated in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, the equicoordinate (1 − α)-quantile q1−α,max,Cv(R) of a N(0, RCv)-distribution serves as a “fair” critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Such quantiles can be determined numerically by computer software, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the function qmvnorm() from the R-package mvtnorm (R Core Team, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Genz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Genz and Bretz, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In practice, RCv needs to be estimated by �RCv, where we replace ΣCv in (9) by its estimator �ΣCv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In summary, we obtain an asymptotically exact test ϕn,max,Cv = 1{Sn,max,Cv(H) > q1−α,max,Cv( �RCv)} for the global null hypothesis H0,Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In contrast to the test from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2, this test also provides additional information in case of a rejection, namely which local null hypotheses (8) caused this rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In detail, we reject the local null hypothesis H0,ℓ,Cv when T n ℓ,n > q1−α,max,Cv( �RCv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, multiple max-type contrast tests can be inverted to obtain simultaneous confidence intervals for all contrasts h⊤ ℓ Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' All these theoretical properties are summarized in the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let (7) as well as Assumptions 1 and 2 be fulfilled for all (sub-)groups i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (a) The test ϕn,max,Cv is asymptotically exact for H0,Cv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' EH0,Cv (ϕn,max,Cv) → α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 8 (b) Suppose that the first r′ ≤ r null hypotheses and the remaining r − r′ alternatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' H1,ℓ,Cv : h⊤ ℓ Cv ̸= 0 for ℓ = r′ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , r, are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then lim sup n→∞ Pr � r′ � ℓ=1 {|T v ℓ,n| > q1−α,max,Cv( �RCv)} � ≤ α and lim n→∞ Pr � r� ℓ=r′+1 {|T v ℓ,n| > q1−α,max,Cv( �RCv)} � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (c) An asymptotically valid simultaneous confidence interval for h⊤ ℓ Cv is given by Pr � r� ℓ=1 � h⊤ ℓ �C v ∈ � h⊤ ℓ Cv ± n−1/2 � h⊤ ℓ �ΣCvhℓ q1−α,max,Cv( �RCv) ��� → 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (d) All statements are true for Bv instead of Cv when adjusting the estimators properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For a performance improvement for small n, we modify the bootstrap from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 Bootstrapping It is not straightforward how or even whether the resampling strategies from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 can also be used for multiple contrast tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let us first have a closer look on the bootstrap strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the proofs, we show that given the data X almost surely n1/2� �Cvb 1 − �Cv 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , �Cvb k − �Cv 0 �⊤ d→ Gb Cv ∼ N(0k×1, �ΣCv), where �Cv 0 is the MCV estimator based on the pooled data and �ΣCv = diag(�σ2 1,Cv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , �σ2 k,Cv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In general, the covariance matrices �ΣCv and ΣCv differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' That is why we cannot approximate the limiting null distribution of (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n) by (T v,b 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v,b r,n) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the Wald-type statistic, we faced a similar problem and solved it by studentization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' by eliminating the dependence of the limit distribution on the covariance structure ΣCv or Σb Cv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Translated to the present setting, we would studentize (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n) first, and then taking the maximum of its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In formulas, we would end up with maxr ℓ=1[�Σ −1/2 Cv (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n)⊤]ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This is indeed a valid testing procedure for the global H0,Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' But we cannot match the entries [�Σ −1/2 Cv (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n)⊤]ℓ with the respective local null hypothesis H0,ℓ,Cv anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' That is why we need to approximate (T v 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v r,n) directly and not just a transformation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For this purpose, we can find different strategies for other testing problems in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' wild bootstrapping (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Umlauft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Konietschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021) or group-wise bootstrapping (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Wechsung and Konietschke, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, we like to exemplify that the pooled bootstrap with a certain modified studentization can also be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In detail, we approximate n1/2( �C v − Cv) by n1/2 �Σ 1/2 Cv (�Σ b Cv)−1/2( �C vb − �C v 0), �C v 0 = �Cv 0 · 1k×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In words, we first studentize ( �C vb − �C v 0) and then multiple the result by �Σ 1/2 Cv leading to the correct asymptotic covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The respective multiple contrast statistic becomes Sb n,max,Cv(H) = n1/2 max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r |T v,b ℓ,n|, T v,b ℓ,n = h⊤ ℓ �Σ 1/2 Cv (�Σ b Cv)−1/2( �C vb − �C v 0) � h⊤ ℓ �ΣCvhℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, let qb 1−α,max,Cv(X) be the conditional, equicoordinate (1 − α)-quantile of n1/2(T v,b 1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , T v,b r,n)⊤ given the data X and ϕb n,max,Cv = 1{Sn,max,Cv(H) > qb 1−α,max,Cv(X)} be the bootstrap multiple contrast test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then we can transfer indeed all asymptotic properties from ϕn,max,Cv to ϕb n,max,Cv, and, moreover, obtain bootstrap-based simultaneous confidence intervals: 9 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In addition to the assumptions of Theorem 1, we suppose that Assumption 1 is fulfilled for the pooled quantities µ0 and Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then the statements of Theorem 4a–c remain true when we replace ϕn,max,Cv and q1−α,max,Cv( �RCv) by their bootstrap counterparts ϕb n,max,Cv and qb 1−α,max,Cv(X), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The analogue for B is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' At a first glance, a similar result might also be reachable for the permutation procedure but it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The asymptotic covariance structure of n1/2( �C vπ − �C v 0) is more complicated than of the bootstrap one due to the strong dependence within the permutation sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, the permutation covariance matrix is neither diagonal nor regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 5 Simulation study Complementing the theoretical findings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' we conducted an extensive simulation study to investigate the type-1 error level and power of the 40 tests proposed in the previous sections: the sixteen asymptotic tests: ϕCRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕBRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕCVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕBVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕCVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕBVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕCAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕBAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the eight permutation tests: ϕπ CRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ BRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ CVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ BVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ CVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ BVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ CAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕπ BAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the sixteen bootstrap tests: ϕb CRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb BRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb CVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb BVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb CVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb BVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb CAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb BAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕb max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We omit the subscript n here for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the multiple contrast tests, we use the Tukey’s all-pairs comparison (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The final simulation results shall serve as a first guideline for practical use of the R-package GFDmcv consisting of all these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 Simulation setup We considered inferring the null hypotheses in (6) in a multivariate one-way layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The 5-dimensional data (d = 5) were generated for k = 4 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The mean vector was generated once from the normal distribution N(0, 1), and µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , µk were set to this vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The covariance matrices Σi were based on the completely symmetric matrix (1 − ρ)Id + ρ1d1⊤ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We considered ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 for small, moderate, and large correlation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Of course, the MCVs usually measure the variability differently, see Section 1 or Albert and Zhang (2010) for a more detailed discussion on the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, to compare the tests’ properties in a unified way, we set the same values of all Cv i , v ∈ {RR, VV, VN, AZ}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' for given i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k, CRR i = CVV i = CVN i = CAZ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To obtain this, the matrices Σi were multiplied by appropriate constants av i , dependent on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The observations were generated from three distributions: the normal (N), the Student (t5), and the chi-square (χ2 10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the symmetric, heavily tailed, and skewed distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For simplicity, we consider equal sample sizes in all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Namely, we set n1 = · · · = nk = 30, 50, 70, 100, 150, 200 for the type-1 error control investigation, and n1 = · · · = nk = 30, 50 for power comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The significance level was set to α = 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For generating the data under H0,Cv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' to investigate the type-1 error control, we set Cv i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 for i and all variants v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For power comparison, we consider the following three alternative hypotheses: H(1) 1,Cv : Cv 1 = Cv 2 = Cv 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 = Cv 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' H(2) 1,Cv : Cv 1 = Cv 2 = Cv 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 = Cv 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' H(3) 1,Cv : Cv 1 = Cv 2 = Cv 3 = 1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 = Cv 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Empirical sizes and powers of the tests were computed as the proportion of rejections of the null hypothesis based on 1000 simulation replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The p-values of the permutation and bootstrap tests were estimated by 1000 resampling samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The simulation experiments and real data example of Section 6 were performed in the R programming language (R Core Team, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Due to the twenty-four resampling tests and 270 simulation scenarios, a part of the calculations was made at the Pozna´n Supercomputing and Networking Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 10 0 5 10 15 20 25 30 35 Empirical sizes (%) CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ Asymptotic Wald−type tests Asymptotic MCT tests n1 = n2 = n3 = n4 = 30 0 5 10 15 20 25 30 35 Empirical sizes (%) CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ Asymptotic Wald−type tests Asymptotic MCT tests n1 = n2 = n3 = n4 = 50 Figure 1: Box-and-whisker plots for the empirical sizes (in %) of the asymptotic tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 Simulation results The simulation results are summarized by the box-and-whisker plots in Figures 1-3 and Figures S1-S13 in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The complete list of empirical sizes and powers is presented in Tables S3-S8 in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' First, we focus on the type-1 error level of the tests, and then we consider their power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Type-1 error level For proper maintaining the type-1 error, the empirical sizes should belong to the binomial proportion 95% and 99% confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively (Duchesne and Francq, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the figures, these limits are presented by the horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' It is apparent that the type-1 error control of the asymptotic tests is unstable and primarily lead to liberal decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' their empirical sizes are (much) greater than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8% in most cases (see Figure 1 and Figure S1 in the supplement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The record-breaking empirical sizes about 35% are obtained for the ϕBRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For larger values of MCVs, the asymptotic tests tend to be quite conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Overall, the performance of the asymptotic tests improves with increasing sample sizes, but the converge speed is rather slow (see Figure S1 in the supplement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The fastest convergence is observed for ϕmax,CVV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Contrary, the permutation and bootstrap Wald-type procedures from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 for the global null hypotheses perform very well (Figure 2) and control the type-1 error level accurately in the majority of settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, their empirical sizes are always smaller than the upper limits of the binomial proportion confidence intervals, even for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Only the bootstrap strategy based on the MCV Cv lead partly to slight conservative decisions for ni = 50, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, the boxs’ for the permutation tests are slightly narrower than ones for bootstrapping in case of smaller sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This slight visible advantage of permuting can be explained by its finite exactness under exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Turning now to the bootstrap multiple tests, we can observe that the type-1 error control is still satisfactory for the standardized mean vectors B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, the decisions based on the MCVs become very conservative reaching down to values below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The latter type-1 error rates improve very slowly and for the largest sample size of ni = 200 the conservativness is still clearly present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the supplement, all results are shown in tables and, in particular, a detailed comparison between the bootstrap and asymptotic multiple contrast test can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In summary, the bootstrap procedures converge much slower to the desired 5%-benchmark than the asymptotic test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' But, under the small sample sizes, the bootstrap procedure is conservative while the asymptotic strategy is rather liberal, and, thus, only the first controls the type-1 error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Nevertheless, the overall performance of the bootstrap is not satisfactory, which opens the door for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, we like to point out that we already studied 11 a group-wise bootstrap procedure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' drawing just from the respective groups, a wild bootstrap procedure, a Bayesian bootstrap procedure, and a parametric bootstrap procedure without much more success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Despite the unsatisfactory results for the MCV, we like to highlight the good performance of this (pooled) bootstrap strategy for the standardized means Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To the best of our knowledge, this was the first time that such a pooled bootstrap procedure was applied for multiple contrast tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Power Let us now discuss the results of the power comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The resulting empirical powers are presented in Figure 3 and Figures S8-S13 in the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Since the too liberal character of the asymptotic tests is unacceptable, we do not consider them in power investigation to avoid unfair comparisons and the potential of results’ misinterpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' First, let us consider the comparison for Wald-type and multiple contrast testing procedures for a given definition of MCV, since the conclusions presented below are the same for each Cv and Bv, v ∈ {RR, VV, VN, AZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' From Figure 3, we can observe that the Wald-type permutation and bootstrap tests have very similar power, but the bootstrap ones seem to be slightly less powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The ϕb max,Bv tests characterize similar power to their Wald-type counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Unfortunately, the ϕb max,Cv tests show their conservative character and have considerably smaller power than the other tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In general, each test based on a given MCV is at least slightly less powerful than its analog based on the reciprocal of MCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For each test, the power is similar under the normal and χ2 10 distributions, while for the t5-distribution, it is much smaller (Figures S8-S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, for the heavy-tailed distribution, the convergence may be slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This is especially evident for the ϕb max,Cv tests due to their extremely conservative character in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let us also notice that the power of all tests decreases with the increase of the value of the MCV, which was also observed for earlier methods for variability comparison proposed in Aerts and Haesbroeck (2017), Ditzhaus and Smaga (2022), and Pauly and Smaga (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, we want to consider how the behavior of the tests depends on the MCV definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We can observe that the power of all tests for different coefficients is not the same, which was expected due to various conceptions of measuring multivariate variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The general observation is as follows: The tests based on the Van Valen coefficient are usually the most powerful and outperform the procedures for Rayment’s MCV, which are followed by the tests for MCVs proposed by Albert and Zhang, and Voinov and Nikulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We can also observe that, for different amounts of correlation, the power of most tests is stable, but sometimes greater differences appear for the Van Valen coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A possible reason for the latter is that the VV variant does not take the dependence of variables into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Recommendation To sum up, the Wald-type permutation and bootstrap tests as well as the multiple bootstrap tests based on reciprocals of MCVs control the type-1 error level even for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, these testing procedures have sensible power, which however depends on the coefficient used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For these reasons, these tests can be recommended for practical use, and thus, it is now available to infer about each of the four multivariate coefficients of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Note however that the advantage of the ϕb max,Bv tests over the Wald-type tests is that they simultaneously verify the contrasts’ significance, in particular, can perform post hoc testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Unfortunately, the ϕb max,Cv tests are very conservative and hence less powerful, which gives us a direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 6 Real data application In this section, we consider, as an illustrative data example, the external quality assessment for clinical laboratories (Libeer, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Sciacovelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' WHO, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we present further simulation results, which mimic the set-up from this data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 Analysis of EQA data set In clinical laboratories, controlling analytical performance and maintaining inter-laboratory variability within acceptable limits are important issues that are even a concern in External Quality Assessment (EQA) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' They are organized nationally or internationally by government health agencies or private companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the case when the available data are multivariate, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2010) proposed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure 2: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained from all cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='to use the multivariate coefficient of variation for comparing the inter-laboratory reproducibility of ' metadata={'source': 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure 3: Box-and-whisker plots for the empirical powers (in %) of the permutation and bootstrap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='tests obtained from all cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' assay techniques used by clinical laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Naturally, the lower the MCV, the better the analytical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, the simple values of particular MCV do not mean significant differences in the techniques considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' That is why we illustrate the use of tests proposed for comparing four techniques based on electrophoretic data sets from the French and Belgian national EQA programs, which was also considered by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2010) and Aerts and Haesbroeck (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The serum protein electrophoresis (SPE) is a laboratory test profile consisting of five fractions summing up to 100% of total proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The fractions are albumin, α1, α2, β, and γ globulins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The SPE can be assayed in different ways depending on the media or the analytical principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the experiment, the following four techniques were compared: HT cellulose acetate (CH), HT agarose gel (acid blue) (EH), HT agarose gel (amido black) (JH), and BCP capillary zone (GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' These four SPE techniques use distinct support mediums, staining colors, or analytical principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, we want to compare the techniques by testing for equal MCVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This was first done by Aerts and Haesbroeck (2017), but they considered the MCV of Voinov and Nikulin only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We extend this study using the new Wald-type and multiple contrast tests for all four variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' But first, the data need to be transformed and cleaned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Due to the compositional nature of the electrophoretic data, a one-to-one transformation from the 5-dimensional to the 4-dimensional space is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For this purpose, the well-known isometric log-ratio transformation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Second, similarly to Aerts and Haesbroeck (2017), we remove the outliers from the data set, which we detected by computing the robust Mahalanobis distances of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' After that, we have four samples of 4-dimensional observations with sizes n1 = 133, n2 = 112, n3 = 74, n4 = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' First of all, we calculate the estimators (3) of MCVs for each technique and each MCV variant, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' It is apparent that the techniques do not perform equally well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, the GB technique leads to the smallest MCV value for each variant and, thus, seems to be the most stable technique in terms of measurement variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To check whether this first impression can be underpinned with statistical certainty, we first performed the different tests for the global hypotheses (6) of technique-wise equality of MCVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Hereby, we considered Tukey’s contrast matrix for the different multiple contrast tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For each MCV, all tests clearly rejected the global null hypotheses with p-values very close to zero (results are not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, these rejections are just partially informative and we are rather interested in a more in-depth analysis with pairwise comparisons of the different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefore, we can invert, as explained in Section 4, the multiple contrast tests into simultaneous confidence intervals, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' These intervals indeed confirm our first impression and the GB technique leads to a significant lower MCV and grater value of its reciprocal compared to all three other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 14 Table 1: Values of MCVs’ estimators for four techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' MCV CH EH JH GB �CRR i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0249 �CVV i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0627 �CVN i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0238 �CAZ i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0343 Furthermore, it can be seen that the confidence intervals for the bootstrap approach are slightly wider than for the asymptotic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This can be explained by the liberality of the asymptotic strategy which we observed in the simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Another interesting observation is that a significant difference between the CH and EH techniques can only be detected by the AZ variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, we like to point out again that the four variants are, in fact, different measures and, thus, opposite inference results may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Aerts and Haesbroeck (2017) also considered pairwise tests for the underlying data set (see Table 6 in their paper) but for the specific variant v =VN only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In their analysis, they could detect a significant difference of CH and EH as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, they did not adjusted for multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, their result is not contradicting ours for v =VN and need to be taken with a pinch of salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 Simulation study based on EQA data set To check the appropriateness of the results for the EQA data set, we conduct an additional simulation study based on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To mimic the observation given in the data example, we generated the simulation data with four samples using: the samples sizes (n1 = 133, n2 = 112, n3 = 74, n4 = 62) from the data example, for checking the type-1 control, in each group, the mean and covariance matrix were set to the sample mean and sample covariance matrix of the pooled data, for power investigation, the mean and covariance matrix in the i-th group was equal to the quantities of the i-th sample from the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The 4-dimensional data were generated from the same three distributions as in Section 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' the normal, t5, and χ2 10 distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We have calculated the empirical sizes and powers for the global hypotheses (6) and the same for multiple contrast tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The results regarding the type-1 error control of the (global) tests from Sections 3 and 4 (see Table S1 in the supplement) are comparable to the ones from the prior simulation study in Section 5 and, thus, not discussed in detail again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, all these tests have a power close to 100% (results not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This confirms the appropriately of rejection of the global null hypotheses for the EQA data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let us turn to the post hoc testing problem and have a closer look on the asymptotic and bootstrap tests from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The power results are summarized in Figure 4 and are shown more detailed in Table S2 from the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Although the bootstrap ϕb max,Cv tests may still be conservative (Table S1 in the supplement), the empirical powers of ϕb max,Cv and ϕb max,Bv tests are very similar for given v ∈ {RR, VV, VN, AZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For CH-EH, CH-JH, and EH-JH comparisons, they are summarised in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the first case, the power of the tests is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The tests based on the MCV of Albert and Zhang are the most powerful, while the other tests have much smaller power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This clearly explains the rejections and nonrejections for the CH-EH comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' On the other hand, for EH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' JH, all tests have almost trivial power, which follows from the smallest differences in estimators of MCV’s among all comparisons (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In the case of CH-JH, the empirical power of all testing procedures increases (even up to about 60% for the ϕb max,CRR and ϕb max,BRR tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This increase of power in comparison to EH-JH can be explained by the increase in the differences in estimators (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For CH-GB, EH-GB, and JH-GB, the empirical powers of all tests were very close to 100%, so we do not 15 Table 2: Estimates of the contrasts h⊤ ℓ �C v and h⊤ ℓ �B v with respective simultaneous 95%-confidence intervals [95%-L,95%-U] based on the asymptotic and bootstrap procedures of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The bold values represent the cases, where significant differences are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Comparison Variant v Method h⊤ ℓ �C v 95%-L 95%-U h⊤ ℓ �B v 95%-L 95%-U CH-EH RR asym 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='181 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='AZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='AZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='EH−JH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure 4: Box-and-whisker plots for the empirical powers (in %) of the multiple contrast tests obtained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='for three “nontrivial” contrasts for simulation based on the EQA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' draw them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, of course, they justify the recognition of significant differences for these contrasts using all multiple contrast tests proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 7 Summary and discussion In this paper, we discussed testing procedures for the multivariate coefficients of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To the best of our knowledge, such testing procedures were just developed for the special MCV version of Voinov and Nikulin (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To be more concrete, Aerts and Haesbroeck (2017) proposed (semi- )parametric procedures for the k-sample setting and, recently, Ditzhaus and Smaga (2022) suggested a nonparametric approach for general factorial designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, also the other MCV variants (Reyment, 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Van Valen, 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Albert and Zhang, 2010) are relevant for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' That is why we followed the successful path of Ditzhaus and Smaga (2022) and adopted their results to the other MCV variants as well as their reciprocals, the so-called standardized means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, we developed respective permutation procedure with a more satisfactory type-1 error control in small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Furthermore, we provided a post hoc strategy for a more in-depth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefore, we combined the well-known tool of multiple contrast tests (Mukerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Konietschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Umlauft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Gunawardana and Konietschke, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Wechsung and Konietschke, 2021) with our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This resulted into asymptotically valid tests, which are rather liberal for small sample sizes, and a novel pooled bootstrap strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To the best of our knowledge, this pooled bootstrap idea was not used in the context of multiple testing procedures before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The simulation results confirm its usage for the reciprocals of the MCVs, by a more convincing type-1 error control under small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, for the MCVs itself the bootstrap procedures become quite conservative and we cannot completely recommend its usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, further research is required to obtain a better resampling strategy for MCV post hoc testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' All procedures are shown to be asymptotically valid and consistent by empirical process theory (van der Vaart and Wellner, 1996) and are extensively studied in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To motivate their use, 17 we provide the R package GFDmcv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Currently, the package can be requested via email and will be soon publicly available on CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A further interesting aspect is inferring paired data, as in the recent investigation regarding the repeatability and reliability of GABA (Gamma-aminobutyric acid) measurements taken from the same patient group (Duda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Under the assumption of normality, Shoukri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2008) already discussed this issue for comparing two univariate CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Conceptionally, this situation is not different to our set-up and, in more detail, corresponds to the one-sample scenario with k = 1 and d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, the nonparametric methodology of comparing two (or more) CVs or even MCVs in paired data settings can be directly deduced from the theory presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, a detailed investigation would overload this paper and is, thus, postponed to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, as already raised by Ditzhaus and Smaga (2022), the problem of outliers, as in the EQA data set, shall be tackled nonparametrically by considering robust estimators for µ and Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For first (semi-)parametric solutions in this context, we refer to Aerts and Haesbroeck (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Acknowledgement The authors would like to thank Professor Adelin Albert (Facult´e de M´edecine, University of Li`ege) for sharing the electrophoretic data sets from the French and Belgian national EQA programs used in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A part of calculations for the simulation study was made at the Pozna´n Supercomputing and Networking Center (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 382).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Supplementary materials In the supplementary material, all results of simulation studies of Sections 5 and 6 are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 Proof of Theorem 1 From the proof of Theorem 1 in Ditzhaus and Smaga (2022), we obtain n1/2 1 � �µ − µ vec(�Σ) − vec(Σ) � d −→ Dψ(µ)G, (10) where G ∼N � 0, � Σ Ψ⊤ 3 Ψ3 Ψ4 �� , Dψ(x) = � Id 0d×d2 � D(x) Id2 � , Ψ3, Ψ4, � D are defined in (4), Id is the d × d-dimensional unity matrix and 0d×d2 is the d × d2- dimensional zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here and subsequently, we consider a matrix as an element of Rℓ, ℓ ∈ N, by vectorization vec(A) = (A11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Ad1, A21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , Add)⊤ for A = (Aij)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In this way, we can equip GLsym(Rd) with the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To verify the first statement for the different choices of C, we apply the δ-method and, later, its permutation/bootstrap analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the latter, we require differentiability in a stronger sense (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' van der Vaart and Wellner, 1996, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5), so-called uniform differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In our specific multivariate set-up, we call a map Φ : (Rd \\ {0}) × GLsym(Rd) → R uniformly differentiable when t−1 n � Φ[an + tnbn, An + tnBn] − Φ[an, An] � → DΦ(a, A) � b vec(B) � as n = n1 → ∞ (11) for a (Jacobi) matrix DΦ(a, A) ∈ Rd+d2 as well as for all An → A ∈ GLsym(Rd), Bn → B ∈ Rd×d, an → a ∈ Rd \\ {0}, bn → b ∈ Rd and tn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Setting an = a and An = A, we get (standard) differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 18 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The maps ΦRR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ΦVV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ΦAZ : (Rd \\ {0}) × GLsym(Rd) → R defined as ΦRR(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = � det(A)1/d (a⊤a) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ΦVV(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = � tr(A) a⊤a and ΦAZ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = � a⊤Aa (a⊤a)2 are uniformly differentiable in the sense of (11) with (Jacobi) matrices DΦRR(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = 1 2d � det(A) (A⊤a)d �1/(2d)−1 � − 2d det(A) a⊤ (a⊤a)d+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' det(A) � vec(A−1) �⊤ (a⊤a)d � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' DΦVV(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = 1 2 �tr(A) a⊤a �−1/2 � −2 tr(A) a⊤ (a⊤a)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 1 a⊤a (vec(Id))⊤ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' DΦAZ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A) = 1 2 � a⊤Aa (a⊤a)2 �−1/2 � −4a⊤Aa a⊤ (a⊤a)3 + 2 a⊤A (a⊤a)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' a⊤ ⊗ a⊤ (a⊤a)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Proof of Lemma 3: We first consider the maps �ΦRR(a, A) = det(A) (a⊤a)d , �ΦVV(a, A) = tr(A) a⊤a and �ΦAZ(a, A) = a⊤Aa (a⊤a)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let An → A ∈ GLsym(Rd), Bn → B ∈ Rd×d, an → a ∈ Rd \\ {0}, bn → b ∈ Rd and tn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then we first observe that t−1 n � det(An + tnBn) − det(An) � → det(A)tr(A−1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the fixed choice An = A, this follows from the differentiability of the determinant, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' shown in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 from Magnus and Neudecker (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For arbitrary An, we recall that the determinant can be expressed as a polynomial of the matrix’s entries and polynomials are uniformly differentiable in the sense of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Secondly, we obtain from the linearity of the trace t−1 n � tr(An + tnBn) − tr(An) � = tr(Bn) → tr(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thirdly, we recall that A is symmetric and, thus, t−1 n � (an + tnbn)⊤(An + tnBn)(an + tnbn) − a⊤ n Anan � = 2b⊤ n Anan + anBnan + O(tn) → 2b⊤Aa + aBa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, setting An = Id and Bn = 0d×d we obtain t−1 n � (an + tnbn)⊤(an + tnbn) − a⊤ n an � → 2b⊤a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, applying the chain rule yields the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In detail, we get t−1 n � �ΦRR[an + tnbn, An + tnBn] − �ΦRR[an, An] � → det(A)tr(A−1B)(a⊤a)d − d(a⊤a)d−1(2b⊤a) det(A) (a⊤a)2d = det(A)tr(A−1B)(a⊤a)d − d(a⊤a)d−1(2b⊤a) det(A) (a⊤a)2d = −2d det(A) a⊤b (a⊤a)d+1 + det(A)tr(A−1B) (a⊤a)d = � −2d det(A) a⊤ (a⊤a)d+1 , det(A) � vec(A−1) �⊤ (a⊤a)d � � b vec(B) � , 19 where we used the (easy-to-see) equation tr(E⊤F ) = (vec(E))⊤ vec(F ) for respective matrices E, F in the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, t−1 n � �ΦVV[an + tnbn, An + tnBn] − �ΦVV[an, An] � → tr(B)(a⊤a) − 2(b⊤a)tr(A) (a⊤a)2 = � −2tr(A) a⊤ (a⊤a)2 , 1 a⊤a (vec(Id))⊤ � � b vec(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Finally, t−1 n � �ΦAZ[an + tnbn, An + tnBn] − �ΦAZ[an, An] � → (2b⊤Aa + aBa)(a⊤a)2 − 2(a⊤a)(2b⊤a)a⊤Aa (a⊤a)4 = � −4a⊤Aa a⊤ (a⊤a)3 + 2 a⊤A (a⊤a)2 , a⊤ ⊗ a⊤ (a⊤a)2 � � b vec(B) � , where we used the following equality for general matrices E, F and G with appropriate dimensions such that the respective multiplications are well-defined: vec(EF G) = (G⊤ ⊗ E)vec(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The statements regarding the maps ΦRR, ΦVV and ΦAZ follow easily from the chain rule, when the (clearly uniformly differentiable) maps ϕ1, ϕ2 : (0, ∞) → R defined by ϕ1(x) = x1/(2d) and ϕ2(x) = x1/2 are applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' □ Combining Lemma 3, (10), the δ-method and the definition of Av(µ, Σ) yields n1/2 1 � �Cv − Cv� d −→ DΦv(µ, Σ)Dψ(µ)G ∼ N(0, σ2 Cv), The results for Bv = 1/Cv follows by the chain rule with the map ϕ3(x) = x−1, x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 Proof of Lemma 2 We first like to note that the proof for the case v = VN can be found in the appendix of Ditzhaus and Smaga (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In fact, we follow their arguments and adopt them to the different variants v = RR, VV, AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' First, we define, for abbreviation, � X = (X11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , X1d, X11X11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , X11X1d, X12X11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , X1dX1d)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (12) It is easy to see that the covariance matrix of � Xi equals Σ� X = � Σ Ψ⊤ 3 Ψ3 Ψ4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We observe that σ2 Cv = 0 implies degeneracy of Av(µ, Σ)� X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For v = RR, the latter translates to: There exists a constant �c ∈ R such that � −2d det(Σ) µ⊤ i (µ⊤µ)d+1 + det(Σ)(vec(Σ−1)) ⊤ (µ⊤µ)d � D(µ), det(Σ)(vec(Σ−1)) ⊤ (µ⊤µ)d � � X = �c (13) with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Define for m, s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d} as = � − 2d det(Σ) µ⊤ (µ⊤µ)d+1 + det(Σ) � vec(Σ−1) �⊤ (µ⊤µ)d � D(µ) � s ∈ R, bsm = �det(Σ) � vec(Σ−1) �⊤ (µ⊤µ)d � sd−d+m = det(Σ)[Σ−1]m,s (µ⊤µ)d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (14) 20 Now, we can simplify (13) to d � m=1 (amX1m + bmmX2 1m) + d � s,m=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='s̸=m bsmX1sX1m = �c (15) with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Since Σ−1 is nonsingular covariance matrix itself, the diagonal entries need to be positive and, thus, bmm ̸= 0 holds for all m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, fix some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Given the other components (X1s)s̸=m, the left hand side of (15) is a polynomial in X1m of degree two and, thus, X1m can take at most two different values to solve (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This contradicts Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This proof strategy can be used, in the same way, for the other two variants v ∈ {VV, AZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For this purpose, we just need to consider the respective counterparts of bsm from (14) and show bmm ̸= 0 for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Let us start with v = VV: bVV mm = � 1 µ⊤µ (vec(Id))⊤ � md−d+m = 1 µ⊤µ ̸= 0 for all m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Finally, we consider v = AZ: bAZ mm = �µ⊤ ⊗ µ⊤ (µ⊤µ)2 � md−d+m = [µ]2 m (µ⊤µ)2 ̸= 0 for all m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , d with [µ]m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Assumption 1, we have µ ̸= 0 and, thus, bAZ mm ̸= 0 for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 Proof of Theorem 2 As already mentioned in the paper, it remains to show that (HCv)⊤(HΣCvH⊤)+HCv as well as (HBv)⊤(HΣBvH⊤)+HBv are positive under any alternative H1,Cv : HCv ̸= 0 or H1,Bv : HBv ̸= 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We just give the proof for C and note that the statement for B follows by simply interchanging the letters C and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For the proof, we need some (well-known) properties of the Moore–Penrose inverse for a matrix A, which can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' in Rao and Mitra (1971): (I) (A⊤)+ = (A+)⊤, (II) (A⊤A)+ = A+(A⊤)+, and (III) AA+A = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, suppose that the alternative H1,Cv : HCv ̸= 0 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Lemma 2, the root Σ1/2 Cv = diag(κ−1/2 1 σ1,Cv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , κ−1/2 k σk,Cv) of the covariance matrix is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, there is some v ∈ Rk \\ {0k×1} such that Cv = Σ1/2 Cv v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' From this and (I)–(III) we obtain 0k×1 ̸= HCv = HΣ1/2 Cv v = HΣ1/2 Cv (HΣ1/2 Cv )+HΣ1/2 C v = HΣ1/2 Cv � (HΣ1/2 Cv )+HCv� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This implies (HΣ1/2 Cv )+HCv ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, (HCv)⊤(HΣCvH⊤)+HCv = (HCv)⊤(Σ1/2 Cv H⊤)+(HΣ1/2 Cv )+HCv = � (HΣ1/2 Cv )+HCv�⊤� (HΣ1/2 Cv )+HCv� > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 Proof of Theorem 3(a) One difficulty and significant difference to the asymptotic approach is that the permutation sample is dependent and, thus, the groups need to be considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The reason for the latter is that we pool the data and then draw without (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=') replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The action of pooling the data is also important for deriving the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Thus, we introduce, on the one hand, the pooled estimators �µ0, �Σ0, �Cv 0 and �Bv 0 depending on all observations and not just observations from a specific group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' On the other hand, let Y ∼ P0 be a random, d-dimensional vector following the asymptotic pooled distribution P0 = �k i=1 κiP Xi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, we denote by µY , ΣY , ΨY 3, ΨY 4 the respective theoretical quantities from the main paper but for Y instead of Xi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In particular, ΣY is the covariance matrix of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 21 Again, we benefit from the preparatory work of Ditzhaus and Smaga (2022), who already verified (see their (12)) that given the data almost surely n1/2 � � � � � � � �µπ 1 − �µ0 vec(�Σ π 1) − vec(�Σ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' �µπ k − �µ0 vec(�Σ π k) − vec(�Σ0) � � � � � � � d −→ Gπ = � � � Gπ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Gπ k � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (16) Furthermore, they showed that Gπ is centered, kd(d+1)-dimensional normal distributed with covariance structure � � � γ(1, 1)Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' γ(1, k)Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' γ(k, 1)Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' γ(k, k)Σπ � � � = � � � � κ−1 1 Σπ 0d′×(k−2)d′ 0d′×d′ 0(k−2)d′×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0(k−2)d′×d′ 0d′×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' κ−1 k Σπ � � � � − � � � Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Σπ � � � , (17) where d′ = d(d + 1), Σπ = � ΣY Ψ⊤ Y 3 ΨY 3 ΨY 4 � and γ(i, i′) = κ−1 i 1{i = i′} − 1 (i, i′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To obtain the asymptotic normality of the permuted MCVs, we apply the δ-method, similar to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, we need to respect that we center the permutation quantities �Cvπ i and �Bvπ i by �Cv 0 and �Bv 0, respectively, which both change with growing sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' That is why we need a stronger form of the δ-method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' van der Vaart and Wellner, 1996, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5) which requires uniform differentiability in the sense of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By Lemma 3 the maps ΦRR, ΦVV, ΦAZ fulfil this requirement and, hence, we get that given the observations almost surely n1/2� �Cvπ 1 − �Cv 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , �Cvπ k − �Cv 0 �⊤ d −→ � � � � DΦv(µY , ΣY ) 01×(k−2)d′ 01×d′ 0(k−2)×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0(k−2)×d′ 01×d′ 01×(k−2)d′ DΦv(µY , ΣY ) � � � � Gπ = Gπ Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To get the analogue results for �Bvπ i (v ∈ {RR, VV, AZ}) we apply the (uniform) δ-method to ϕ3 ◦ Φv instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' We leave the additional writing effort to the interested readers and proceed just with the C’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Clearly, Gπ Cv follows a centered, multidimensional normal distribution and, in regard to (17), we can simplify its covariance matrix to �ΣCv − (DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤)1k×k, where �ΣCv = diag(κ−1 1 DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , κ−1 k DΦv(µY , ΣY )ΣπDΦv(µY , ΣY )⊤) = diag(σ2 1,Cv,Y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , σ2 k,Cv,Y ) and σ2 1,Cv,Y is the pooled counterpart of σ2 1,Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, it becomes clear why we require the matrix H to be a contrast matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Namely, this implies H( �Cv 0 · 1k×1) = 0 as well as H1k×k = 0, and, consequently, it follows n1/2H �C π = n1/2H � �Cvπ 1 − �Cv 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , �Cvπ k − �Cv 0 �⊤ d −→ HGπ Cv ∼ N(0, H �ΣCvH⊤) given the observations almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2, this is the first important ingredient for the asymptotic convergence of the Wald-type statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The second is the convergence of (H �Σ π CvH⊤)+ to (H �ΣCvH⊤)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefore, it remains to argue (1) σ2 i,Cv,Y > 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k which implies the 22 regularity of �ΣCv, and (2) �Σ π Cv converges (conditionally) in probability to �ΣCv given the observations almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Hereby, (1) follows directly from Lemma 2 and the simple observation that Assumption 2 can directly be transferred to the pooled distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' For (2), we recall from Ditzhaus and Smaga (2022) that � �µπ i vec(�Σ π i ) � p −→ � µY vec(ΣY ) � , i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , k}, given the observations almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then (conditional) convergence in probability of �Σ π Cv to �ΣCv follows from the continuous mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 Proof of Theorem 3(b) In principle, the proof follows the same argumentation as for Theorem 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' However, the bootstrap procedure is slightly easier to handle because we draw with (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=') replacement and, thus, the groups are still independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The proof of Ditzhaus and Smaga (2022) for our (16) can be adopted to get an analogous result for the bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Here, one just need to replace Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 of van der Vaart and Wellner (1996) for the permutation procedure by the bootstrap counterparts Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 in their argumentation via empirical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' All these results originally cover only the two-sample case (k = 2) but can be directly extended to k ≥ 3, as argued e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' by Ditzhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (2021a) in their Lemma 9 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Finally, we can obtain that given the observations almost surely n1/2 � � � � � � � � �µb 1 − �µ0 vec(�Σ b 1) − vec(�Σ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' �µb k − �µ0 vec(�Σ b k) − vec(�Σ0) � � � � � � � � d −→ Gb = � � � Gb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Gb k � � � , (18) where Gb is centered, kd(d + 1)-dimensional normal distributed with covariance structure � � � � κ−1 1 Σπ 0d′×(k−2)d′ 0d′×d′ 0(k−2)d′×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0(k−2)d′×d′ 0d′×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' κ−1 k Σπ � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (19) Applying the (uniform) δ-method we get, given the observations almost surely, n1/2� �Cvb 1 − �Cv 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , �Cvb k − �Cv 0 �⊤ d −→ � � � � DΦv(µY , ΣY ) 01×(k−2)d′ 01×d′ 0(k−2)×d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 0(k−2)×d′ 01×d′ 01×(k−2)d′ DΦv(µY , ΣY ) � � � � Gb = Gb Cv ∼ N(0k×1, �ΣCv), (20) where �ΣCv = diag(σ2 1,Cv,Y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , σ2 k,Cv,Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The rest of the proof follows the arguments from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 and from the proof’s end of Theorem 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' To avoid unnecessary repetition, we leave the details to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' As an intermediate result, we just like to mentioned that given the data almost surely �Σ b Cv p→ �ΣCv (21) while �ΣCv is regular as already argued in the proof of Theorem 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 Proof of Theorem 4 The statements in (a) and (c) follow immediately from Theorem 1 as discussed briefly before Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The key step is to follow from Theorem 1 that �√nh⊤ 1 �C v − h⊤ 1 Cv � h⊤ 1 �ΣCvh1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , √nh⊤ r �C v − h⊤ r Cv � h⊤ r �ΣCvhr �⊤ = √ndiag((h⊤ 1 �ΣCvh1)−1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , (h⊤ r �ΣCvhr)−1/2)H( �C v − Cv) d→ Z ∼ N(0r×1, RCv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' (22) Then (c) follows by a simple inversion of this convergence statement and for (a) we just need to remind ourselves that h⊤ ℓ Cv = 0 for all ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , r under H0,Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, suppose H1,ℓ,Cv : h⊤ ℓ Cv ̸= 0 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then we can deduce from Slutzky’s Lemma and (22) |T v ℓ,n| ≥ √n |h⊤ ℓ Cv| � h⊤ ℓ �ΣCvhℓ − ��� √nh⊤ ℓ �C v − h⊤ ℓ Cv � h⊤ ℓ �ΣCvhℓ ��� p→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' From this, we obtain the second statement of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Now, let H0,1,Cv, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' , H0,r′,Cv be true for r′ ≤ r and define �R ′ = ([ �R]ℓm)ℓ,m=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' By (a) Pr � max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r′ |T v ℓ,n| > q1−α( �R ′) � → α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, it is easy to see that q1−α,max,Cv( �R) ≥ q1−α( �R ′), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' by noting Sn,max,Cv(H) ≥ maxℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r′ |T v ℓ,n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' In summary, we obtain Pr � r′ � ℓ=1 {|T v ℓ,n| > q1−α,max,Cv( �R)} � = Pr � max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r′ |T v ℓ,n| > q1−α,max,Cv( �R) � ≤ Pr � max ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=',r′ |T v ℓ,n| > q1−α( �R ′) � →α proving the first statement of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Clearly, an analogue of (22) is true for B instead of C, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Consequently, (d) follows from the same arguments as used for (a)–(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 Proof of Theorem 5 To prove the statement, we need a bootstrap analogue of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Therefore, we repeat from Lemma 1 that �Σ 1/2 Cv converges in probability to Σ1/2 Cv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Moreover, the assumptions ensure that Σ1/2 Cv and �ΣCv are regular, for the later we refer to the proof of Theorem 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Having classical subsequence arguments in mind, we fix the data and assume without a loss of generality that the aforementioned convergence as well as (20) and (21) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Then we obtain from Slutzky’s Lemma, (20) and (21) that n1/2 �Σ 1/2 Cv (�Σ b Cv)−1/2� �C vb − �C v 0 �⊤ d −→ Σ1/2 Cv (�ΣCv)−1/2Gb Cv ∼ N(0k×1, ΣCv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' This is the desired bootstrap analogue of (22) and the rest follows by the same arguments as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' References Aerts S, Haesbroeck G (2017) Robust asymptotic tests for the equality of multivariate coefficients of variation.' metadata={'source': 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electrophoresis techniques in External Quality Assessment schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' Accreditation and Quality Assurance 15:351–357 27 Supplementary materials to Inference for all variants of the multivariate coefficient of variation in factorial designs Marc Ditzhaus1 and �Lukasz Smaga2,∗ 1Faculty of Mathematics, Otto-von-Guericke University Magdeburg, Germany 2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poland January 31, 2023 This supplement contains all results of simulation studies of Sections 5 and 6 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' They are summarized in Figures S1-S13 and presented in Tables S1-S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' List of Figures S1 Box-and-whisker plots for the empirical sizes of the asymptotic tests obtained from all cases considered .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 3 S2 Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests obtained under normal distribution .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 4 S3 Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests obtained under t5-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 6 S5 Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 7 S6 Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 8 S7 Box-and-whisker plots for the empirical sizes of the permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 9 S8 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests obtained under normal distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 10 S9 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests obtained under t5-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 10 S10 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests obtained under χ2 10-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 11 S11 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 12 S13 Box-and-whisker plots for the empirical powers of the permutation and bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 12 List of Tables S1 Empirical sizes of all tests obtained in simulation study based on real data example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 13 S2 Empirical powers of the multiple contrast tests obtained in simulation based on real data example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 14 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='12009v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='ME] 27 Jan 2023 S3 Empirical sizes of all tests obtained under normal distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 15 S4 Empirical sizes of all tests obtained under t5-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 23 S5 Empirical sizes of all tests obtained under χ2 10-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 31 S6 Empirical powers of the permutation and bootstrap tests under normal distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 39 S7 Empirical powers of the permutation and bootstrap tests under t5-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 41 S8 Empirical powers of the permutation and bootstrap tests under χ2 10-distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical sizes (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR BRR CVV BVV CVN BVN CAZ BAZ CRR BRR CVV BVV CVN BVN CAZ BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Asymptotic MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S1: Box-and-whisker plots for the empirical sizes (as percentages) of the asymptotic tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='obtained from all cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S2: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained under normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S3: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained under t5-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S4: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained under χ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='10-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S5: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S6: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 ' metadata={'source': 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' metadata={'source': 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tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S7: Box-and-whisker plots for the empirical sizes (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' The solid, dashed, and dotted lines represent the significance level α = 5% and the 95% and 99% binomial proportion confidence intervals [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4%] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='2%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='8%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical powers (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical powers (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S8: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained under normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical powers (%) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} 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+page_content='Empirical powers (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S12: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical powers (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BRR ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Empirical powers (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CRR ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='CAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='BAZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Permutation Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap Wald−type tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Bootstrap MCT tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='n1 = n2 = n3 = n4 = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='Figure S13: Box-and-whisker plots for the empirical powers (as percentages) of the permutation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='bootstrap tests obtained for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' 12 Table S1: Empirical sizes (as percentages) of all tests obtained in simulation study based on real data example of Section 6 in the main paper (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a - asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method N 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 7.' 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tests obtained under normal distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 5.' 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tests obtained under normal distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='9 4.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 a 19 Table S3: Empirical sizes (as percentages) of all tests obtained under normal distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N 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tests obtained under t5-distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 1.' 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tests obtained under χ2 10-distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 4.' 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tests obtained under χ2 10-distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 4.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='9 a 35 Table S5: Empirical sizes (as percentages) of all tests obtained under χ2 10-distribution (Method: a - asymptotic Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, a asymptotic multiple contrast test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 4.' 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the permutation and bootstrap tests under t5-distribution (Method: π - permutation Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' b - bootstrap Wald-type test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content=' max, b - bootstrap multiple contrast test) (continued) ρ (C1, C2, C3, C4) ni ϕCRR ϕBRR ϕCV V ϕBV V ϕCV N ϕBV N ϕCAZ ϕBAZ Method 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFLT4oBgHgl3EQfKy8R/content/2301.12009v1.pdf'} +page_content='6 43.' metadata={'source': 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b/MdE4T4oBgHgl3EQfKQwG/content/tmp_files/2301.04927v1.pdf.txt @@ -0,0 +1,330 @@ +Planar fiber-chip-coupling using angle-polished +polarization maintaining fibers +M. Schneidera,1, L. A. Garcia Herreraa, B. Burgera, L. Eisenblättera, T. Kühnera +a Karlsruhe Institute of Technology, Institute for Data Processing and Electronics, 76131 Karlsruhe, +Germany +E-mail: marc.schneider@kit.edu +ABSTRACT: We report on our latest developments of a planar fiber-chip-coupling scheme, using +angle polished, polarization maintaining (PM) fibers. Most integrated photonic chip components +are polarization sensitive and a suitable way to launch several wavelength channels with the same +polarization to the chip is the use of PM fibers. Those impose several challenges at processing +and handling to achieve a stable, permanent, and low-loss coupling. We present the processing of +the fibers in detail and experimental results for our planar and compact fiber-chip-coupling +technique. +KEYWORDS: Glass fibers, polarization maintaining fibers, fiber processing, fiber-chip-coupling, +angle-polishing, grating couplers. + + + + + +– 1 – +Contents +1. Introduction +1  +2. Fiber Processing +2  +3. Results +4  +4. Conclusions +5  + + + +1. Introduction +High performance optical links using wavelength division multiplexing (WDM) are the future in +detector instrumentation to increase data transmission bandwidth and reduce fiber count. A key +issue of such a system is a compact and efficient fiber-chip-coupling, connecting optical glass +fibers to the photonic chip. As most components of integrated photonic chips are polarization +sensitive, one can use lossy on-chip polarization-insensitive couplers and polarization controllers +for each wavelength or use polarization maintaining (PM) fibers to feed the required polarization +directly from the lasers without the need of further manipulation. +Often used on-chip components for light-coupling are grating couplers, which offer the ad- +vantage to couple light in to and out of the chip-integrated waveguides almost anywhere on the +chip and not just at the edges. For coupling, grating couplers are irradiated with light under a +certain angle to the chip’s normal. Commonly used angles are around 10°, but they depend on the +specific design of the grating coupler and the used wavelength. Launching light with a cleaved +fiber leads to a rather high buildup, as the minimum bending radius of standard glass fibers is in +the range of several tens of millimeters [1]. For smaller, more planar buildups, other techniques +are required. +Our fiber-chip-coupling uses optical single mode glass fibers, whose tip is polished to a +certain angle, so that light is reflected radially out of the fiber by total internal reflection at a +defined angle (Figure 1). The fiber is positioned parallel to the chip surface with the tip above an + +Figure 1. Scheme of fiber-chip-coupling using grating couplers on chip and angle-polished glass fibers. + +Angle-polished +Angle-polished +glass fiber +glass fiber +Grating coupler +Grating coupler +Photonic chip + +– 2 – +on-chip grating coupler, so that the radially emitted light +hits the grating coupler, which diffracts the light into an +on-chip waveguide for further routing [2, 3]. +PM fibers, however, are not axially symmetric due +to the strain rods close to the fiber core introducing bire- +fringence (Figure 2). Therefore fiber-chip-coupling be- +comes more challenging, as the angle of in-axis rotation +has to be very well defined. +We will present the setup to initially adjust the PM +fibers for polishing, the optimized polishing process, a +recipe to reproduce the results, as well as measurement +results of fiber-chip-coupling experiments. + +2. Fiber Processing +Based on the work published in [3], we refined and enhanced the fiber processing for polarization +maintaining fibers. For a low-loss fiber chip coupling, the position and angular errors have to be +kept as small as possible. In our process the angle of axial rotation  is adjusted for the polishing +process with an error of less than 0.3°. According to our simulations, this value should keep the +additional coupling losses as low as 0.02 dB, which is negligible compared to other sources like +deviation from the ideal polishing angle and wavelength dependence of the grating couplers. To +achieve this, a special setup was developed (Figure 3), which allows a precise fiber rotation before +gluing the fibers to a fiber holder for polishing. The fiber holder consists of a grooved microscope +slide. A cleaved fiber with clean front facet is placed into one of the grooves and, further back, in +a fiber rotator from Piezosystem Jena. For alignment, the image of the front facet of a cleaved PM +fiber with its core, cladding, and, most important, strain rods is captured by a camera with a mi- +croscope objective, which is placed on a XYZ-linear stage. The camera picture is analyzed by a +machine vision system, that extracts the contours of the fiber and the strain rods, calculates the +rotation angle and indicates, if the angle is in an acceptable range. To display the glass strain rods +inside the cladding glass, special lighting with gracing incidence from behind, stray light shading, +and digital contrast enhancement is required. +If the rotation angle is correct, a small drop of UV-curing glue (Delo Photobond GB368) is +applied to fix the fiber. Additional glue is applied afterwards to improve fixation. The process is +repeated for several more fibers until all grooves are occupied. +The following polishing process is essential for a low-loss coupling, as the angle of the pol- +ished surface to the fiber axis and the surface quality are equally important. To maintain the pol- +ishing angle, a special polishing fixture with a parallelogram guidance was developed and printed +(Figure 4) using a Makerbot Replicator Dual 3D-printer. The polishing angle is defined by the +angle of the main, wedge shaped part of the chuck. +Several silicon carbide lapping films and diamond suspensions with decreasing grain sizes +were used for polishing. Table 1 lists the used materials with grain sizes, polishing durations, and +rotation speeds of the polishing machine used for each step. Between each step, the fiber chuck +and the machine were thoroughly cleaned to avoid contamination with coarser grains. Figure 5 +shows the polishing chuck on the disk of our polishing machine. The chuck was placed on the rim +and the moving part was pushed onto the rotating disk by hand. Here it is important that the + +Figure 2. Microscope image of tip of +cleaved polarization maintaining fiber. + +Strain rods + +– 3 – +applied pressure is roughly similar for each batch of fibers. +As there was, due to the applied pressure, a slight discrep- +ancy of 1.3° between the desired angle set by the polishing +chuck and the angle of the polished surface obtained, the +angle of the chuck was optimized in several iterations to ob- +tain fibers with a coupling angle of 12°, which is the optimal +angle for a center wavelength of 1550 nm of the grating cou- +plers on our photonic chips. +After polishing, the fibers were removed from their +holder by immersion in acetone for 30 minutes. The used +glue swells in acetone and loses most of its adhesion, but +removal has to be careful anyway, not to break the fibers. +Residues were wiped away with clean isopropanol. The final step to achieve a really smooth +surface was melting the glass surface. This was done by installing the angle polished fiber in a +fusion splicer and starting a standard program for fusing single mode fibers. Most fusion splicers +clean the surface of the respective fibers first by a short electric arc, burning residues and melting +the front facet to achieve good splicing results. For the angle polished fiber, the splicing program +was cancelled after this first step and the fiber removed. + +Figure 3. Setup for axial alignment of PM fibers in grooved holder for polishing. + +Figure 4. Fiber polishing chuck. +Table 1. Polishing sequence and parameters. +Material +Grain size +Duration +Rotation speed +Lapping film +15 µm +1 min. +200 rpm +Lapping film +8 µm +5 min. +200 rpm +Diamond emulsion +3 µm +8 min. +150 rpm +Diamond emulsion +1 µm +8 min. +150 rpm +Diamond emulsion +0.25 µm +8 min. +150 rpm + + +Light guide +Fiber rotator +Glass fiber +Grooved fixture +Microscope objectiveCamera: + +– 4 – +The result of this procedure is shown +for one fiber in Figure 6. The front facet has +a smooth mirror finish without scratches or +debris. + +3. Results +After processing, the fibers were character- +ized by comparative coupling experiments. +For this, reference measurements were +made with straight waveguides, terminated +with grating couplers on a silicon photonic +chip. The used fibers were cleaved single +mode glass fibers, mounted at the optimal +coupling angle to achieve a minimum cou- +pling loss of 9.8 dB at a center wavelength +of 1550 nm (Figure 7). The waveguide loss +was found to be 0.6 dB and the coupling +loss for a single grating coupler to be +4.6 dB. As the reference spectrum in Figure +7 belongs to a pair of couplers with a super- +position of both transmission spectra, the 3dB-bandwidth of a single coupler can be found 6 dB +below the maximum, indicated by a horizontal dotted line, with 76.2 nm. With this reference, one +fiber was replaced by an angle polished fiber to be characterized. The angle polished fiber was +placed into a fiber rotator similar to the one used in the fiber preparation setup. Special care was +spent to align the fiber axis parallel to the chip surface. In several iterations, the optimal rotation +angle was determined and, for standard fibers, the optical polarization optimized by a polarization +controller. With the optimum angle, a transmission spectrum was measured. The results were +corrected by the reference spectrum to determine the additional coupling loss for the angle pol- +ished glass fibers. +Figure 8a) shows the additional loss for four angle- +polished standard glass fibers over the wavelength. The +dots show the processed measurement data, the lines a +polynomial fit to guide the eye. As can be seen, the meas- +urements scatter up to 4 dB, which is a result of some +back reflections and interference in the waveguide as +well as between grating couplers and fibers. The line fits +reveal a much clearer picture and show that there is in +general no additional loss with respect to coupling with +cleaved fibers. +Similar measurements for PM fibers are shown in +Figure 8b). Again, the points are the processed measure- +ment data, the lines polynomial fits. Here the spread is +much larger and the wave-like shape of the fit curves, +most prominently for fiber 1, shows a transmission max- + +Figure 7. Reference transmission +spectrum of two cleaved fibers above +grating couplers on photonic chip. + + +Figure 6. Angle polished PM fiber. + +Figure 5. Fiber polishing on polishing machine with +fiber chuck. + +Cleaved +-10 +fibers +(dB) +.11Transmissi +-13 +-14-16 +1520 +15401560 +1580 +Wavelength (nm) + +– 5 – +imum other than 1550 nm (for fiber 1 +the transmission maximum was +measured to be 1545 nm). These fi- +bers show a larger deviation in cou- +pling angles as well as a much higher +loss. On average the additional cou- +pling loss for PM fibers is 4.2 dB +with a standard deviation of 1.2 dB, +while the absolute spread is +3.4 dB +and -2.4 dB. The higher loss was not +unexpected, as we observed a similar +penalty over standard fibers using +cleaved PM fibers for coupling. A +possible reason for the higher loss +might be a significant mode mis- +match, but that needs further investi- +gation. + +4. Conclusions +We presented a method to adjust PM +fibers for angle-polishing and a rec- +ipe to polish them with high quality +for a planar fiber-chip-coupling using +on-chip grating couplers. The polish- +ing process contains several mechan- +ical polishing steps, followed by +smoothing the polished surface by +melting with an electric arc. Consec- +utive measurements of such pro- +cessed fibers show that they exhibit no additional coupling loss, but coupling with PM fibers still +suffers from a 4.2 dB higher loss compared to standard fibers. To reduce it, further investigations +and optimizations are required. + +References +[1] Datasheet +of +Corning +SMF-28e+ +single +mode +glass +fiber, +viewed +at +2022-10-17, +https://www.corning.com/media/worldwide/coc/documents/Fiber/PI-1463-AEN.pdf +[2] D. Karnick et al., Optical Links for Detector Instrumentation: On-Detector Multi-Wavelength Silicon +Photonic Transmitters, JINST, DOI 10.1088/1748-0221/12/03/C03078. +[3] D. Karnick et al., Efficient, easy-to-use, planar fiber-to-chip coupling process with angle-polished +fibers, 67th ECTC, DOI 10.1109/ECTC.2017.245. + + + +Figure 8. Measurement results of additional coupling loss for +a) angle-polished standard fibers and b) angle-polished PM +fibers. + +Angle-polished standard glass fibers +2 +dB) +1Loss +nal LAdditi +-2 +Fiber 1 +Fiber 2 +-3 +Fiber 3Fiber 4 +1530 +1540 +1550 +1560 +1570 +a) +Wavelength (nm)8 +Fiber 1 +Angle-polished PM fibers +Fiber 2 +7 +:Fiber 3ILosS +6 +onal +5Additi +4 +32 +1530 +1540 +1550 +1560 +1570 +b) +Wavelength (nm) \ No newline at end of file diff --git a/MdE4T4oBgHgl3EQfKQwG/content/tmp_files/load_file.txt b/MdE4T4oBgHgl3EQfKQwG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..163fbc27fc07df394d6132ebf7d93db8167aa829 --- /dev/null +++ b/MdE4T4oBgHgl3EQfKQwG/content/tmp_files/load_file.txt @@ -0,0 +1,144 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf,len=143 +page_content='Planar fiber-chip-coupling using angle-polished polarization maintaining fibers M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Schneidera,1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Garcia Herreraa, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Burgera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Eisenblättera, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Kühnera a Karlsruhe Institute of Technology, Institute for Data Processing and Electronics, 76131 Karlsruhe, Germany E-mail: marc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='schneider@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='edu ABSTRACT: We report on our latest developments of a planar fiber-chip-coupling scheme, using angle polished, polarization maintaining (PM) fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Most integrated photonic chip components are polarization sensitive and a suitable way to launch several wavelength channels with the same polarization to the chip is the use of PM fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Those impose several challenges at processing and handling to achieve a stable, permanent, and low-loss coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' We present the processing of the fibers in detail and experimental results for our planar and compact fiber-chip-coupling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' KEYWORDS: Glass fibers, polarization maintaining fibers, fiber processing, fiber-chip-coupling, angle-polishing, grating couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' – 1 – Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Fiber Processing 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Results 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Conclusions 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Introduction High performance optical links using wavelength division multiplexing (WDM) are the future in detector instrumentation to increase data transmission bandwidth and reduce fiber count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' A key issue of such a system is a compact and efficient fiber-chip-coupling, connecting optical glass fibers to the photonic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' As most components of integrated photonic chips are polarization sensitive, one can use lossy on-chip polarization-insensitive couplers and polarization controllers for each wavelength or use polarization maintaining (PM) fibers to feed the required polarization directly from the lasers without the need of further manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Often used on-chip components for light-coupling are grating couplers, which offer the ad- vantage to couple light in to and out of the chip-integrated waveguides almost anywhere on the chip and not just at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For coupling, grating couplers are irradiated with light under a certain angle to the chip’s normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Commonly used angles are around 10°, but they depend on the specific design of the grating coupler and the used wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Launching light with a cleaved fiber leads to a rather high buildup, as the minimum bending radius of standard glass fibers is in the range of several tens of millimeters [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For smaller, more planar buildups, other techniques are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Our fiber-chip-coupling uses optical single mode glass fibers, whose tip is polished to a certain angle, so that light is reflected radially out of the fiber by total internal reflection at a defined angle (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The fiber is positioned parallel to the chip surface with the tip above an Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Scheme of fiber-chip-coupling using grating couplers on chip and angle-polished glass fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Angle polished Angle polished glass fiber glass fiber Grating coupler Grating coupler Photonic chip – 2 – on-chip grating coupler, so that the radially emitted light hits the grating coupler, which diffracts the light into an on-chip waveguide for further routing [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' PM fibers, however, are not axially symmetric due to the strain rods close to the fiber core introducing bire- fringence (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Therefore fiber-chip-coupling be- comes more challenging, as the angle of in-axis rotation has to be very well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' We will present the setup to initially adjust the PM fibers for polishing, the optimized polishing process, a recipe to reproduce the results, as well as measurement results of fiber-chip-coupling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Fiber Processing Based on the work published in [3], we refined and enhanced the fiber processing for polarization maintaining fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For a low-loss fiber chip coupling, the position and angular errors have to be kept as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' In our process the angle of axial rotation \uf066 is adjusted for the polishing process with an error of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='3°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' According to our simulations, this value should keep the additional coupling losses as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='02 dB, which is negligible compared to other sources like deviation from the ideal polishing angle and wavelength dependence of the grating couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' To achieve this, a special setup was developed (Figure 3), which allows a precise fiber rotation before gluing the fibers to a fiber holder for polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The fiber holder consists of a grooved microscope slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' A cleaved fiber with clean front facet is placed into one of the grooves and, further back, in a fiber rotator from Piezosystem Jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For alignment, the image of the front facet of a cleaved PM fiber with its core, cladding, and, most important, strain rods is captured by a camera with a mi- croscope objective, which is placed on a XYZ-linear stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The camera picture is analyzed by a machine vision system, that extracts the contours of the fiber and the strain rods, calculates the rotation angle and indicates, if the angle is in an acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' To display the glass strain rods inside the cladding glass, special lighting with gracing incidence from behind, stray light shading, and digital contrast enhancement is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' If the rotation angle is correct, a small drop of UV-curing glue (Delo Photobond GB368) is applied to fix the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Additional glue is applied afterwards to improve fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The process is repeated for several more fibers until all grooves are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The following polishing process is essential for a low-loss coupling, as the angle of the pol- ished surface to the fiber axis and the surface quality are equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' To maintain the pol- ishing angle, a special polishing fixture with a parallelogram guidance was developed and printed (Figure 4) using a Makerbot Replicator Dual 3D-printer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The polishing angle is defined by the angle of the main, wedge shaped part of the chuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Several silicon carbide lapping films and diamond suspensions with decreasing grain sizes were used for polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Table 1 lists the used materials with grain sizes, polishing durations, and rotation speeds of the polishing machine used for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Between each step, the fiber chuck and the machine were thoroughly cleaned to avoid contamination with coarser grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 5 shows the polishing chuck on the disk of our polishing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The chuck was placed on the rim and the moving part was pushed onto the rotating disk by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Here it is important that the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Microscope image of tip of cleaved polarization maintaining fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Strain rods – 3 – applied pressure is roughly similar for each batch of fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' As there was, due to the applied pressure, a slight discrep- ancy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='3° between the desired angle set by the polishing chuck and the angle of the polished surface obtained, the angle of the chuck was optimized in several iterations to ob- tain fibers with a coupling angle of 12°, which is the optimal angle for a center wavelength of 1550 nm of the grating cou- plers on our photonic chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' After polishing, the fibers were removed from their holder by immersion in acetone for 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The used glue swells in acetone and loses most of its adhesion, but removal has to be careful anyway, not to break the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Residues were wiped away with clean isopropanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The final step to achieve a really smooth surface was melting the glass surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' This was done by installing the angle polished fiber in a fusion splicer and starting a standard program for fusing single mode fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Most fusion splicers clean the surface of the respective fibers first by a short electric arc, burning residues and melting the front facet to achieve good splicing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For the angle polished fiber, the splicing program was cancelled after this first step and the fiber removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Setup for axial alignment of PM fibers in grooved holder for polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Fiber polishing chuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Polishing sequence and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Material Grain size Duration Rotation speed Lapping film 15 µm 1 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 200 rpm Lapping film 8 µm 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 200 rpm Diamond emulsion 3 µm 8 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 150 rpm Diamond emulsion 1 µm 8 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 150 rpm Diamond emulsion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='25 µm 8 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 150 rpm Light guide Fiber rotator Glass fiber Grooved fixture Microscope objectiveCamera: – 4 – The result of this procedure is shown for one fiber in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The front facet has a smooth mirror finish without scratches or debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Results After processing, the fibers were character- ized by comparative coupling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' For this, reference measurements were made with straight waveguides, terminated with grating couplers on a silicon photonic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The used fibers were cleaved single mode glass fibers, mounted at the optimal coupling angle to achieve a minimum cou- pling loss of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='8 dB at a center wavelength of 1550 nm (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The waveguide loss was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='6 dB and the coupling loss for a single grating coupler to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='6 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' As the reference spectrum in Figure 7 belongs to a pair of couplers with a super- position of both transmission spectra, the 3dB-bandwidth of a single coupler can be found 6 dB below the maximum, indicated by a horizontal dotted line, with 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' With this reference, one fiber was replaced by an angle polished fiber to be characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The angle polished fiber was placed into a fiber rotator similar to the one used in the fiber preparation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Special care was spent to align the fiber axis parallel to the chip surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' In several iterations, the optimal rotation angle was determined and, for standard fibers, the optical polarization optimized by a polarization controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' With the optimum angle, a transmission spectrum was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The results were corrected by the reference spectrum to determine the additional coupling loss for the angle pol- ished glass fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 8a) shows the additional loss for four angle- polished standard glass fibers over the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The dots show the processed measurement data, the lines a polynomial fit to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' As can be seen, the meas- urements scatter up to 4 dB, which is a result of some back reflections and interference in the waveguide as well as between grating couplers and fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The line fits reveal a much clearer picture and show that there is in general no additional loss with respect to coupling with cleaved fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Similar measurements for PM fibers are shown in Figure 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Again, the points are the processed measure- ment data, the lines polynomial fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Here the spread is much larger and the wave-like shape of the fit curves, most prominently for fiber 1, shows a transmission max- Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Reference transmission spectrum of two cleaved fibers above grating couplers on photonic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Angle polished PM fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Fiber polishing on polishing machine with fiber chuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Cleaved 10 fibers (dB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='11Transmissi 13 14 16 1520 15401560 1580 Wavelength (nm) – 5 – imum other than 1550 nm (for fiber 1 the transmission maximum was measured to be 1545 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' These fi- bers show a larger deviation in cou- pling angles as well as a much higher loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' On average the additional cou- pling loss for PM fibers is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='2 dB with a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='2 dB, while the absolute spread is +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='4 dB and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='4 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The higher loss was not unexpected, as we observed a similar penalty over standard fibers using cleaved PM fibers for coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' A possible reason for the higher loss might be a significant mode mis- match, but that needs further investi- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Conclusions We presented a method to adjust PM fibers for angle-polishing and a rec- ipe to polish them with high quality for a planar fiber-chip-coupling using on-chip grating couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' The polish- ing process contains several mechan- ical polishing steps, followed by smoothing the polished surface by melting with an electric arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Consec- utive measurements of such pro- cessed fibers show that they exhibit no additional coupling loss, but coupling with PM fibers still suffers from a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='2 dB higher loss compared to standard fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' To reduce it, further investigations and optimizations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' References [1] Datasheet of Corning SMF-28e+ single mode glass fiber, viewed at 2022-10-17, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='corning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='com/media/worldwide/coc/documents/Fiber/PI-1463-AEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='pdf [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Karnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=', Optical Links for Detector Instrumentation: On-Detector Multi-Wavelength Silicon Photonic Transmitters, JINST, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='1088/1748-0221/12/03/C03078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Karnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=', Efficient, easy-to-use, planar fiber-to-chip coupling process with angle-polished fibers, 67th ECTC, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='1109/ECTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content='245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Measurement results of additional coupling loss for a) angle-polished standard fibers and b) angle-polished PM fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} +page_content=' Angle polished standard glass fibers 2 dB) 1Loss nal LAdditi 2 Fiber 1 Fiber 2 3 Fiber 3Fiber 4 1530 1540 1550 1560 1570 a) Wavelength (nm)8 Fiber 1 Angle polished PM fibers Fiber 2 7 :Fiber 3ILosS 6 onal 5Additi 4 32 1530 1540 1550 1560 1570 b) Wavelength (nm)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfKQwG/content/2301.04927v1.pdf'} diff --git a/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/2301.00360v1.pdf.txt b/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/2301.00360v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d74f974b7b1a38a4be140726c5736ec333bf2309 --- /dev/null +++ b/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/2301.00360v1.pdf.txt @@ -0,0 +1,9644 @@ +arXiv:2301.00360v1 [stat.ME] 1 Jan 2023 +Iterative Least Squares Algorithm for Large-dimensional Matrix +Factor Model by Random Projection +Yong He∗, Ran Zhao∗, and Wen-Xin Zhou† +January 3, 2023 +The matrix factor model has drawn growing attention for its advantage in achieving two-directional +dimension reduction simultaneously for matrix-structured observations. In this paper, we propose a simple +iterative least squares algorithm for matrix factor models, in contrast to the Principal Component Analysis +(PCA)-based methods in the literature. In detail, we first propose to estimate the latent factor matrices by +projecting the observations with two deterministic weight matrices, which are chosen to diversify away the +idiosyncratic components. We show that the inferences on factors are still asymptotically valid even if we +overestimate both the row/column factor numbers. We then estimate the row/column loading matrices by +minimizing the squared loss function under certain identifiability conditions. The resultant estimators of the +loading matrices are treated as the new weight/projection matrices and thus the above update procedure can +be iteratively performed until convergence. Theoretically, given the true dimensions of the factor matrices, we +derive the convergence rates of the estimators for loading matrices and common components at any s-th step +iteration. Thorough numerical simulations are conducted to investigate the finite-sample performance of the +proposed methods and two real datasets associated with financial portfolios and multinational macroeconomic +indices are used to illustrate practical usefulness. +Keyword: Latent low rank; Least squares; Matrix factor model; Random Projection. +1 +Introduction +Factor modeling is an extremely popular approach for dimension reduction in large-dimensional time se- +ries analysis, which has been successfully applied to large panels of time series for forecasting macroe- +conomic variables (Stock and Watson, 2002a), building low-dimensional indicators of the whole economic +∗Institute of Financial Studies, Shandong University, China. E-mail: heyong@sdu.edu.cn, Zhaoran@mail.sdu.edu.cn +†Department of Mathematical Sciences, University of California, San Diego, USA. E-mail: wez243@ucsd.edu +1 + +activity (Stock and Watson, 2002b). +In the last two decades, there has been a flourish of literature on +large-dimensional factor models; see, for example, Bai (2003),Onatski (2009), Ahn and Horenstein (2013), +Fan et al. (2013), Trapani (2018), A¨ıt-Sahalia and Xiu (2017), Kong (2017), Barigozzi et al. (2018), Yu et al. +(2019), Barigozzi and Cho (2020), Chen et al. (2021), He et al. (2022a) and Fan and Liao (2022). +In economics and finance, observations are usually well structured to be an array/matrix, such as a +time list of tables recording several macroeconomic variables across a number of countries or a series of +customers’ ratings on a large number of items in an online platform. In the last few years, the literature +has paid increasing attention to factor analysis for matrix time series. Wang et al. (2019) for the first time +proposed the following factor model for matrix time series: +Xt = RFtC⊤ + Et, t = 1, . . . , T, +(1.1) +where R is the p1×k1 row factor loading matrix exploiting the variations of Xt across the rows, C is the p2×k2 +column factor loading matrix reflecting the differences in the columns of Xt, Ft is the k1 ×k2 common factor +matrix and Et is the idiosyncratic component. Wang et al. (2019) proposed estimators of the factor loading +matrices by an eigen-analysis of the auto-cross-covariance matrix; Chen and Fan (2021) proposed an α-PCA +method by exploiting an eigen-analysis of a weighted average of the mean and the column (row) covariance +matrix of the data; Yu et al. (2022) proposed a Projection Estimation (PE) method that further improved +the estimation efficiency of the factor loading matrices. He et al. (2021a) established the equivalence between +minimizing the squared loss and the PE method by Yu et al. (2022) and further proposed a robust method +by replacing the squared loss with the Huber loss. +The resultant estimators of factor loading matrices +can be simply obtained by an eigen-analysis of weighted sample covariance matrices of the projected data. +He et al. (2022b) proposed to recover the loading spaces of the matrix elliptical factor model by an eigen- +analysis of the generalized row/column matrix Kendall’s tau, which generalizes the multivariate Kendall’s +tau to the random matrix setting. However, to our knowledge, all the theoretical studies of the matrix +factor model in the literature crucially rely on the assumption that the pair of factor numbers k1 and k2 is +consistently estimated, which typically requires that the factors are relatively strong, data have weak serial +correlation or the number of observations is large. In practical applications, these requirements may fail to +hold due to weak signal-to-noise ratio or non-stationarity, making the first top eigenvalues of the row/column +covariance matrix less separated from the remaining ones; see also the discussions in Fan and Liao (2022) +for vector factor models. Over-estimating the number of factors would be a promising remedy as discussed +in Moon and Weidner (2015); Westerlund and Urbain (2015); Barigozzi and Cho (2020) for classical vector +factor models. The impact of over-estimating the pair of factor numbers for matrix factors models remains +unknown. +In this article, we propose a simple iterative least squares algorithm for matrix factor models, in contrast +2 + +to the Principal Component Analysis (PCA)-based methods in the literature. In detail, in the first step, we +propose to estimate the latent factor matrices by projecting the matrix observations with two deterministic +weight matrices, which are chosen to diversify away the idiosyncratic components. This idea is similar in +spirit to that in Fan and Liao (2022) and the estimator does not rely on eigenvectors. In the second step, we +update the row/column loading matrices by minimizing the squared loss function under the identifiability +condition. +Then the estimators of the loading matrices are treated as the new weight matrices and we +iteratively proceed with the above two steps until a convergence criterion is reached. The contributions of +the current work lie in the following aspects. Firstly, to our knowledge, this is the first work on matrix +factor analysis that does not involve any eigen-decomposition of large matrices. The proposed iterative least +squares algorithm is quite simple and computationally efficient, with computational complexity O (T p1p2) in +contrast to the typical O +� +T p2 +1 + T p2 +2 +� +complexity of the eigen-decomposition based methods. Secondly, we +show that even if both numbers of row and column factors are over-estimated, the inferences on factors are +still asymptotically valid, which is new to the literature on matrix factor models. At last, given the factor +numbers are correctly specified, we establish the convergence rates of the estimators for loading matrices and +common components at the s-th iteration for any s ≥ 1 under some strong factor identifiability condition. +Compared to one-step estimation (Yu et al., 2022), the multi-step estimator is proven to be less sensitive to +the initial estimator. In addition, the iterative least squares algorithm also reduces the magnitudes of the +idiosyncratic error components in each step, thereby increasing the signal-to-noise ratio and enjoying the +same advantage as the projection estimation method by Yu et al. (2022). +The rest of the article is organized as follows. In Section 2, we introduce the factor estimation method via +two-directional diversified projections. We derive the consistency of the vectorized factor space even when +the factor numbers are over-estimated. In Section 3, we propose the iterative least squares estimators for +the loading spaces and derive the convergence rates of the estimators at any s-th step iteration. In Section +4, we conduct thorough numerical studies to illustrate the advantages of the proposed methods over the +state-of-the-art methods. In Section 5, we analyze a financial dataset and a multinational macroeconomic +indices dataset to illustrate the empirical usefulness of the proposed methods. We discuss possible future +research directions and conclude the paper in Section 6. The proofs of the main theorems and additional +details are collected in the supplementary materials. +We end this section by introducing some notations that will be used throughout the paper. For any +vector µ = (µ1, . . . , µp)⊤ ∈ Rp, let ∥µ∥2 = (�p +i=1 µ2 +i )1/2, ∥µ∥∞ = maxi |µi|. For a real number a, denote [a] +as the largest integer smaller than or equal to a, let sgn(a) = 1 if a ≥ 0 and sgn(a) = −1 if a < 0. Let I(·) be +the indicator function and diag(a1, . . . , ap) be a p × p diagonal matrix, whose diagonal entries are a1 . . . , ap. +For a matrix A, let Aij (or Ai,j) be the (i, j)-th entry of A, A⊤ the transpose of A, tr(A) the trace of A, +rank(A) the rank of A, A+ the Moore-Penrose generalized inverse of A and diag(A) a vector composed of +3 + +the diagonal elements of A. Denote PA as the projection matrix PA = A(A⊤A)−1A⊤ and span(A) as the +space spanned by the columns of A. Denote λj(A) as the j-th largest eigenvalue of a nonnegative definitive +matrix A, and let ∥A∥ be the spectral norm of matrix A and ∥A∥F be the Frobenius norm of A. For two +series of random variables, Xn and Yn, Xn ≍ Yn means Xn = Op(Yn) and Yn = Op(Xn). For two random +variables (vectors) X and Y , Var(X) denotes the variance (covariance matrix) of X, X +d= Y means the +distributions of X and Y are identical. The constants c, C1, C2 in different lines can be different. +2 +Factor Estimation via Two-directional Diversified Projections +In this section, we introduce a simple two-directional diversified projection method to estimate the factor +score matrices. We also investigate the theoretical properties of the estimators under cases of finite samples +and overestimation of the factor numbers. +2.1 +The estimation of factors +In this section, we propose a simple two-directional diversified projection method to estimate the factor +score matrices, which does not involve eigen-decomposition. Let Wi = (wi,.1, wi,.2, . . . , wi,.mi) be a given +exogenous (or deterministic) pi × mi matrix i = 1, 2, where wi,.j, the j-th column of the matrix Wi, is a +vector of “diversified weights” in the sense that its strength should be approximately equally distributed +across most of its components. We call W1 and W2 the “left projection matrix” and “right projection +matrix”, respectively. We propose to estimate Ft simply by +�Ft = +1 +p1p2 +W⊤ +1 XtW2. +(2.1) +By the matrix factor model (1.1), we have +�Ft = +1 +p1p2 +W⊤ +1 RFtC⊤W2 + +1 +p1p2 +W⊤ +1 EtW2 := H1FtH⊤ +2 + Et, +(2.2) +where H1 = W⊤ +1 R/p1, H2 = W⊤ +2 C/p2 and Et = W⊤ +1 EtW2/(p1p2). Thus �Ft estimates Ft up to two +affine transformation with Et as the estimation error. The assumption that W1, W2 should be diversified +guarantees that as min(p1, p2) → ∞, Et is diversified away (converging to zero in probability). We call the +new factor matrix estimator “bi-diversified factors”, which reduces the dimensions of Xt from p1 × p2 to +m1 × m2 by two-directional diversified projections. Due to the clean expansion (2.2), the mathematics for +theoretical analysis is much simpler than the most benchmark estimators. Intuitively, �Ft would lead to valid +inferences in factor-augmented models so long as mi ≥ ki, i = 1, 2, in the same spirit as Fan and Liao (2022) +for vector factor model, and we leave this to future work as the matrix factor-augmented model is still in its +4 + +infancy. +2.2 +Theoretical analysis for the estimators of Factors +In this section, we investigate the theoretical properties of the estimators of factors under finite sample cases +and the overestimation of the factor numbers. We assume that the predetermined constants mi, i = 1, 2 do +not grow with pi, which are named as “the working (pseudo) numbers of row (column) factors”. Since in +practice we do not know the true number of factors, we often take slightly large numbers mi such that mi ≥ ki +are likely to hold. Let Wi be either deterministic or random but independent of the σ-algebra generated by +{Et : t ∈ [T ]}. We further assume that the projection matrices Wi, i = 1, 2 satisfy the following: +Assumption 2.1. There are positive constants c1 and c2, such that (almost surely if W1, W2 are random) +as p1, p2 → ∞, +(1) max +i≤p1 |w1,ij| < c1, max +i≤p2 |w2,ij| < c2; +(2) the m1 × m1 matrix W⊤ +1 W1/p1 and m2 × m2 matrix W⊤ +2 W2/p2 satisfy λmin(W⊤ +1 W1/p1) ≥ c1, +λmin(W⊤ +2 W2/p2) ≥ c2; +(3) W1, W2 are independent of Et, t ∈ [T ]. +Vectorizing the matrix �Ft in (2.2), we have +Vec(�Ft) = (H2 ⊗ H1)Vec(Ft) + +1 +p1p2 +(W2 ⊗ W1)⊤Vec(Et) := HVec(Ft) + +1 +p1p2 +W⊤Vec(Et), +where H = H2 ⊗ H1 and W = W2 ⊗ W1. Therefore, Vec(�Ft) can be treated as an estimate of Vec(Ft) +up to a transformation matrix, where H ∈ Rm1m2×k1k2, with estimation error equal to W⊤Vec(Et)/(p1p2). +When each element of Et is cross-sectional weakly dependent, Assumption 2.1 guarantees the cross-sectional +central limit theorem of W⊤Vec(Et)/(p1p2). For example, assume each element of Et is cross-sectional +independent, under Assumption 2.1, as p1p2 → ∞, we get +1 +√p1p2 +W⊤Vec(Et) +d→ N(0, V), +(2.3) +where V = limp1p2→∞ W⊤Var(Vec(Et))W/(p1p2), assuming it exists. The convergence (2.3) shows that for +each t ≤ T , √p1p2(Vec(�Ft) − HVec(Ft)) is asymptotically normal regardless of whether T goes to infinity, +m1 = k1, m2 = k2 or not. It only requires p1p2 → ∞, which is particularly useful for analyzing short matrix +time series. +In addition, the factor components should not be diversified away, which entails the following conditions +on transformation matrix Hi, i = 1, 2. Let νmin(Hi), νmax(Hi) denote the minimum and maximum nonzero +singular values of Hi, respectively. +5 + +Assumption 2.2. Suppose m1 ≥ k1, m2 ≥ k2. Almost surely +(1) rank(H1) = k1, rank(H2) = k2; +(2) there exist constants c1 and c2 such that +ν2 +min(Hi) ≫ 1 +pi +, νmax(Hi) ≤ c1νmin(Hi), +i = 1, 2. +Assumption 2.2 (1) requires W1 to have at least k1 columns that are not orthogonal to R and W2 to have +at least k2 columns that are not orthogonal to C so that R and C are not diversified away. It also ensures +that the space spanned by Vec(�Ft) is asymptotically equal to the space spanned by Vec(Ft). This is the key +assumption imposed, but not stringent in the context of over-estimating factors (Barigozzi and Cho, 2020; +Fan and Liao, 2022). Assumption 2.2 (2) determines the rate of convergence in recovering the space spanned +by the factors and ensures that the weight matrix and the loading matrix are not orthogonal. Assume +νmin(W⊤ +1 R) = pα1 +1 , νmin(W⊤ +2 C) = pα2 +2 , then Assumption 2.2 (2) entails that α1, α2 ≥ 1/2. Assumption 2.1 +and Assumption 2.2 are direct generalizations of Assumptions 2.1 and 2.2 in Fan and Liao (2022) to the +matrix factor models. +For the matrix factor model, once we vectorize the observations, the model reduces to a vector factor +model +X = F(C ⊗ R)⊤ + E, +where F = (Vec(F1), Vec(F2), . . . , Vec(FT ))⊤, X = (Vec(X1), Vec(X2), . . . , Vec(XT ))⊤. E = (Vec(E1), Vec(E2), +. . . , Vec(ET ))⊤. By (2.2), we have +�F = FH⊤ + E, +where �F = (Vec(�F1), Vec(�F2), . . . , Vec(�FT ))⊤, E = (Vec(E1), Vec(E2), . . . , Vec(ET ))⊤. To derive the theo- +retical properties of the factor estimators, we further impose the following assumptions. +Assumption 2.3. There are constants c, C such that +(1) c < λmin(�T +t=1 Vec(Ft)Vec(Ft)⊤/T ) ≤ λmax(�T +t=1 Vec(Ft)Vec(Ft)⊤/T ) < C, a.s.; +(2) (2i) 1/T �T +s=1 +�T +T =1 E∥Ft∥F ∥Fs∥F tr(E(EtE⊤ +s |F)) < c; (2ii) ∥E(EtE⊤ +t )∥2 ≤ c; +(3) λmin +� +E +� +E⊤E/T +�� +≥ c0, E∥E +� +Vec(Et)Vec(Et)⊤|F +� +∥2 < c; +(4) for any t ∈ [T ], i1, i1 ∈ [p1], j1, j2 ∈ [p2], �T +s=1 +�p1 +u1,v1=1 +�p2 +u2,v2=1 |Cov(et,i1i2et,j1j2, es,u1u2es,v1v2)| ≤ c. +Assumption 2.3 requires weak dependence among idiosyncratic errors, which is common in the literature; +see, for example, He et al. (2021b), Chen and Fan (2021) and Yu et al. (2022). +The following theorem +establishes the convergence of �Ft to the true Ft up to transformation matrices, regardless of whether T → ∞ +or overestimating the factor numbers. +6 + +Theorem 2.1. Suppose Assumptions 2.1 (1), 2.1 (2) and 2.3 (2i) hold. As min{p1, p2} → ∞ and T is finite +or T → ∞. Then for any bounded m1 ≥ k1, m2 ≥ k2, for any t ∈ [T ], +��M⊤ +1 �FtM2 − Ft +�� +2 = Op +� +1 +√p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +, +where M1 = (H1H⊤ +1 )+H1 ∈ Rm1×k1, M2 = (H2H⊤ +2 )+H2 ∈ Rm2×k2. +The following theorem shows that the linear space spanned by �F equals to the linear space spanned by +F asymptotically. +Theorem 2.2. Suppose Assumptions 2.1-2.3 hold. +For any bounded m1 ≥ k1, m2 ≥ k2, we have as +min{p1, p2} → ∞ that +∥P�FPF − PF∥2 = Op +� +1 +√p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +, +∥P�FM − PF∥2 = Op +� +1 +√p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +, +where M = (HH⊤)+H and H = H2 ⊗ H1. +Theorem 2.2 establishes that when m1 ≥ k1, m2 ≥ k2, the linear space spanned by �F is asymptotically +the same as the linear space spanned by F. Theorem 2.2 also shows that a particular subspace of span(�F) +is consistent for span(F). In particular, when m1 = k1, m2 = k2, we have P�FM = P�F as M is invertible. It +then degenerates to the usual space consistency. +Consistent estimation of the number of factors k1, k2 typically requires strong conditions, which are +difficult to fulfill in finite samples case. One advantage of the proposed method for estimating the factor +matrices is that it is still robust against overestimating the number of factors in many statistical inference +problems. As for the choices of weight matrices, one can select W1 and W2 following the same strategies +by Fan and Liao (2022), such as the Hadamard Projection. By Theorem 2.2, {�Ft, t = 1, . . . , T } would lead +to valid inferences in factor-augmented models so long as mi ≥ ki, i = 1, 2 and we leave this to our future +work as the matrix factor-augmented model is still in its infancy. +3 +Iterative Least Squares Estimators for Loading Spaces +In this section, we introduce the iterative least squares estimators for the column/row loading spaces. We +also derive the convergence rates of the estimators for loading matrices at the s-th iteration (for any s ≥ 1) +provided that the pair of factor numbers are correctly specified. In case that the pair of factor numbers are +unknown, many methods have been proposed in the literature to estimate them consistently; see for example +the α-PCA-ER in Chen and Fan (2021) and the Iter-ER in Yu et al. (2022). +7 + +3.1 +The Estimation of Loading Spaces +In Section 2, we introduce the way to estimate the factor matrices with two diversified projection matrices +W1, W2 and denote the estimators as {�Ft, t = 1, . . . , T }, which are of dimension m1 × m2 with m1 ≥ +k1, m2 ≥ k2. Given {�Ft}, it is straightforward to estimate the row factor loading matrix R by minimizing +the squared Frobenius loss under the identifiability condition: +min +R L1(R) = min +R +1 +T +T +� +t=1 +∥Xt − R�FtW⊤ +2 ∥2 +F , +s.t +1 +p1 +R⊤R = Im1, m1 ≥ k1, +(3.1) +where W2 is the column projection matrix. The objective function in (3.1) can be simplified as +L1(R) = 1 +T +T +� +t=1 +� +tr +� +X⊤ +t Xt +� +− 2tr +� +X⊤ +t R�FtW⊤ +2 +� ++ p1tr +� +W2�F⊤ +t �FtW⊤ +2 +�� +, +and the Lagrangian function is +min +R,Θ L1(R, Θ) = min +R,Θ +� +L1(R) + tr +� +Θ +� 1 +p1 +R⊤R − Im1 +��� +, +where the Lagrangian multipliers Θ is a symmetric matrix. Taking ∂L1(R, Θ)/∂R = 0 and ∂L1(R, Θ)/∂Θ = +0, we obtain +∂L1(R, Θ) +∂R += 1 +T +T +� +t=1 +� +− 2XtW2�F⊤ +t + 2 +p1 +RΘ +� += 0, +∂L1(R, Θ) +∂Θ += 1 +p1 +R⊤R − Im1 = 0. +(3.2) +Further, we can derive the explicit expression for �Θ and �R satisfying (3.2), that is, +�Θ = +√p1 +T +�� T +� +t=1 +�FtW⊤ +2 X⊤ +t +� � T +� +t=1 +XtW2�F⊤ +t +��1/2 +, +�R = √p1 +� T +� +t=1 +XtW2�F⊤ +t +� �� T +� +t=1 +�FtW⊤ +2 X⊤ +t +� � T +� +t=1 +XtW2�F⊤ +t +��−1/2 +, +(3.3) +8 + +and we set �R as the one-step estimator of the row loading matrix. Similarly, by minimizing the following +loss function under the identifiability condition: +min +C L2(C) = min +C +1 +T +T +� +t=1 +��Xt − W1�FtC⊤��2 +F , +s.t 1 +p2 +C⊤C = Im2, m2 ≥ k2, +with the Lagrangian function +min +C,Λ L2(C, Λ) = min +C,Λ +� +1 +T +T +� +t=1 +��Xt − W1�FtC⊤��2 +F + tr +� +Λ +� 1 +p2 +C⊤C − Im2 +��� +, +we get the following estimator of the column factor loading matrix +�C = √p2 +� T +� +t=1 +X⊤ +t W1�Ft +� �� T +� +t=1 +�F⊤ +t W⊤ +1 Xt +� � T +� +t=1 +X⊤ +t W1�Ft +��−1/2 +. +(3.4) +To be consistent with the following iterative algorithm, we also denote �Ft in (2.2) as �F(1) +t +and �R in (3.3) +as �R(1). Once we get the estimator �R(1), we can obtain �C(1) by replacing W1 in (3.4) with �R(1), i.e., +�C(1) = √p2 +� T +� +t=1 +X⊤ +t �R(1)�Ft +� �� T +� +t=1 +�F⊤ +t �R(1)⊤Xt +� � T +� +t=1 +X⊤ +t �R(1)�Ft +��−1/2 +. +(3.5) +Given �C(1), we update the estimator of Ft as �F(2) +t += �R(1)⊤Xt �C(1)/(p1p2), thereby updating the estimators +of R and C. In detail, update the estimator of R as �R(2) by replacing �Ft, W2 in (3.3) with �F(2) +t , �C(1), +respectively; update the estimator of C as �C(2) by replacing �Ft, W1 in (3.4) with �F(2) +t , �R(2), respectively; +we repeat the above steps until a convergence criterion is met. At the (s + 1)-th iteration, the estimators +�F(s+1) +t +, �R(s+1) and �C(s+1) have the following expressions: +�F(s+1) +t += +1 +p1p2 +�R(s)⊤Xt �C(s), +�R(s+1) = √p1 +� T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +� �� T +� +t=1 +�F(s+1) +t +�C(s)⊤X⊤ +t +� � T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +��−1/2 +, +�C(s+1) = √p2 +� T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +� �� T +� +t=1 +�F(s+1)⊤ +t +�R(s+1)⊤Xt +� � T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +��−1/2 +, +where �R(s) and �C(s) are the estimators from the s-th iteration. The Random Projection-based Iterative +Least Squares (RPILS) procedure for the matrix factor model is summarized in Algorithm 1 below and the +9 + +theoretical analysis is presented in the following section. +As for the convergence criterion, denote the common component matrix at the (s+1)-th step as �S(s+1) = +�R(s)�F(s+1) +t +�C(s)⊤. In our simulation studies, the iterative procedure is terminated either when a pre-specified +maximum iteration number (maxiter = 100) is reached or when +∥�S(s+1) − �S(s)∥F ≤ ǫ, +where ǫ is a small constant (10−6) given in advance. +Algorithm 1 Random Projection based Iterative Least Squares (RPILS) procedure for matrix factor model +Input: Data matrices {Xt}, t ≤ T , the pair of pseudo row and column factor numbers m1(m1 ≥ k1) and +m2(m2 ≥ k2), the diversified projection matrices W1, W2 +Output: Factor loading matrices �R ∈ Rp1×m1, �C ∈ Rp2×m2 and factor matrix �Ft ∈ Rm1×m2, t ≤ T +1: obtain the initial estimator �F(1) +t +by �F(1) +t += W⊤ +1 XtW2/(p1p2); +2: given �F(1) +t +and W2, get an estimator �R(1) by Equation (3.3); +3: given �F(1) +t +and �R(1), get an estimator �C(1) by replacing W1 in the Equation (3.4) with �R(1); +4: update �F(2) +t +by �F(2) +t += �R(1)⊤Xt �C(1)/(p1p2); +5: update �R(2) by replacing �Ft, W2 in the Equation (3.3) with �F(2) +t , �C(1), respectively; +6: update �C(2) by replacing �Ft, W1 in the Equation (3.4) with �F(2) +t , �R(2), respectively; +7: repeat steps 4-6 until convergence and output the estimators from the last step denoted as �R, �C and +{�Ft, t ≤ T }, respectively. +3.2 +Convergence rates +In this section, we establish the convergence rates of the estimators of loading matrices at s-th iteration for +any s ≥ 1. To this end, we first impose some additional conditions that are common in the literature. +Assumption 3.1. The factor matrix satisfies E(Ft) = 0, E∥Ft∥4 ≤ c < ∞ for some constant c > 0, and +1 +T +T +� +t=1 +FtF⊤ +t +p→ Σ1 +and +1 +T +T +� +t=1 +F⊤ +t Ft +p→ Σ2, +where Σi, i = 1, 2 is a ki × ki positive definite matrix with bounded eigenvalues. +Assumption 3.2. There exist positive constants c, c0 < ∞ such that (1) E(et,ij) = 0. (2) for any t ∈ +[T ], i ∈ [p1], j ∈ [p2], �T +s=1 +�p1 +l=1 +�p2 +h=1 |E(et,ijes,lh)| ≤ c and �p1 +l=1 +�p2 +h=1 |E(et,ljet,ih)| ≤ c. (3) for any +T ∈ [T ], i1, i1 ∈ [p1], j1, j2 ∈ [p2], �T +s=1 +�p1 +u1,v1=1 +�p2 +u2,v2=1 |Cov(et,i1i2et,j1j2, es,u1u2es,v1v2)| ≤ c. +Assumption 3.3. There exist positive constants c1, c2 such that ∥R∥max ≤ c1, ∥C∥max ≤ c2. +Assumption 3.4. There exists a constant c > 0 such that (1) for any deterministic vectors v and w satisfying +∥v∥ = 1 and ∥w∥ = 1, E +�� 1 +√ +T +�T +t=1(Ftv⊤Etw) +��2 ≤ c; (2) for any i ∈ [p1], j ∈ [p2], ∥ �p1 +i′=1 +�p2 +j′=1 E(ξi,j ⊗ +10 + +ξi′,j′)∥max ≤ c, where ξi,j = Vec(T −1/2 �T +t=1 Ftet,ij); (3) for any i1, i2 ∈ [p1], j1, j2 ∈ [p2], ∥ �p1 +i′ +1,i′ +1=1 +�p2 +j′ +1,j′ +2=1 +Cov(ξi1,j1 ⊗ ξi2,j2, ξi′ +1,j′ +1 ⊗ ξi′ +2,j′ +2)∥max ≤ c. +Assumptions 3.1–3.4 are standard in the literature on matrix factor models; see for example Chen and Fan +(2021), Yu et al. (2022) and He et al. (2021b). Assumption 3.2 ensures the (cross-sectional and time series) +summability of the idiosyncratic terms Et, which allows for (weak) dependence in both space and time +domains. +Assumption 3.3 requires that the common factors are pervasive. +Assumption 3.4 allows the +common factors Ft and errors Et to be weakly correlated, which is satisfied, e.g., when {Ft} and {Et} +are two mutually independent groups. One may refer to the detailed discussions on these assumptions by +He et al. (2021b). In the following theorem, we first establish the convergence rates of the one-step estimator +�R ( �R(1)) and �C defined in (3.3) and (3.4), respectively. +Theorem 3.1. Suppose that m1 = k1, m2 = k2 and the factor numbers k1, k2 are fixed. Further assume +that ν2 +min(H2) ≫ max(1/T, 1/p2). Under the Assumption 2.1 (1), Assumption 2.2, Assumptions 3.1-3.4, as +T, p1, p2 go to infinity, there exists an asymptotic orthogonal matrix �H(1) +r , such that +1 +p1 +∥ �R(1) − R �H(1) +r ∥2 +F = Op +� +1 +T p2ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν4 +min(H2) + +1 +p2 +1p2 +2ν2 +min(H1)ν4 +min(H2) +� +, +(3.6) +where H1 = W⊤ +1 R/p2, H2 = W⊤ +2 C/p2; Symmetrically, assume ν2 +min(H1) ≫ max(1/T, 1/p1) hold, then +there exists an asymptotic orthogonal matrix �Hc, such that +1 +p2 +∥�C − C �Hc∥2 +F = Op +� +1 +T p1ν2 +min(H1) + +1 +T p1p2ν4 +min(H1)ν2 +min(H2) + +1 +p2 +1p2 +2ν4 +min(H1)ν2 +min(H2) +� +. +The condition ν2 +min(H2) ≫ max(1/T, 1/p2) guarantees that the matrices �Hr are asymptotic orthogonal. +This condition in essence requires that the space spanned by the columns of the initial projection matrix +W2 does not deviate far from that spanned by the columns of C, and fails to hold if the two spaces are +orthogonal. This is conceivable that the iterative algorithm would never converge to the true space if we +start from its orthogonal space. Theorem 3.1 shows that the closer the initial projection directions (space +spanned by the columns of W2) are to the true loading directions (space spanned by the columns of C), the +faster the estimated loading matrix �R(1) converges to the true loading matrix R up to an orthogonal matrix. +In particular, if νmin(H1) = νmin(H2) = Op(1) (as long as we start from the consistent α-PCA estimators in +Chen and Fan (2021) as projection directions), then we have +1 +p1 +∥ �R(1) − R �H(1) +r ∥2 +F = Op +� 1 +T p2 ++ +1 +p2 +1p2 +2 +� +, +1 +p2 +∥�C − C �Hc∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2 +� +. +In the following theorem, we establish the convergence rate of the one-step estimator �C(1) defined in +(3.5). +11 + +Theorem 3.2. Under the same conditions as in Theorem 3.1, there exists an asymptotic orthogonal matrix +�H(1) +c , such that +w(1) +c += 1 +p2 +∥�C(1)−C �H(1) +c ∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2ν2 +min(H1)ν2 +min(H2) + +1 +T 2p2 +2ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν2 +min(H2) +� +. +In addition, w(1) +r += 1 +p1 +∥ �R(1) − R �H(1) +r ∥2 +F is the rate derived in (3.6). +Assume that νmin(H1) = νmin(H2) = Op(1), then the derived convergence rate of �C(1) is +1 +p2 +∥�C(1) − C �H(1) +c ∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2 ++ +1 +T 2p2 +2 +� +, +which is the same as that derived in Corollary 3.1 of Yu et al. (2022). +As long as we get the estimators of row/column loading matrices, i.e., �R(1) and �C(1), the update of the +factor matrix can be obtained by �F(2) +t += �R(1)⊤Xt �C(1)/(p1p2) and the corresponding common-components +matrix is then updated by �S(2) = �R(1)�F(2) +t +�C(1)⊤ = (�S(2) +t,ij). The following theorem provides the convergence +rates of the estimated factors and common components after one iteration. +Theorem 3.3. Under the same conditions as in Theorem 3.1, as min{T, p1, p2} → ∞, for any t ∈ [T ], +i ∈ [p1] and j ∈ [p2], we have +�����F(2) +t +− ( �H(1) +r )−1Ft +� +( �H(1) +c )−1�⊤���� +F += Op +� +1 +√ +Tp1 ++ +1 +√ +Tp2νmin(H2) ++ +1 +√p1p2 ++ +1 +p1p2νmin(H1)ν2 +min(H2) + γf +� +, +where +γf = +1 +√T p1p2νmin(H1)νmin(H2) + +1 +T p1√p2νmin(H1)ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν3 +min(H2), +and for the common components, we have +|�S(2) +t,ij−St,ij| = Op +� +1 +√T p1 ++ +1 +√p1p2 ++ +1 +√T p2νmin(H2) + +1 +√T p1p2νmin(H1)ν2 +min(H2) + +1 +p1p2νmin(H1)ν2 +min(H2) +� +, +where St,ij is the (i, j)-th entry of S = RFtC⊤. +The derived convergence rates in Theorem 3.3 are the same as those derived in Theorem 3.5 of Yu et al. +(2022) when νmin(H1) = νmin(H2) = Op (1). In the following theorem, we establish the recurrence formula +of the convergence rate for the estimators �R(s+1) and �C(s+1) given any integer s ≥ 1. +Theorem 3.4. Under the same conditions stated in Theorem 3.1, there exist asymptotic orthogonal matrices +12 + +�H(s+1) +r +and �H(s+1) +c +such that for any integer s ≥ 1, we have +w(s+1) +r += 1 +p1 +∥ �R(s+1) − R �H(s+1) +r +∥2 +F = Op +� 1 +T p2 ++ +1 +p2 +1p2 +2 ++ γ(s+1) +r +� +, +w(s+1) +c += 1 +p2 +∥�C(s+1) − C �H(s+1) +c +∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2 ++ γ(s+1) +c +� +, +where +γ(s+1) +r += 1 +p2 +2 +w(s) +r ++ 1 +T w(s) +c ++ w(s) +r w(s) +c +p2 ++ w(s) +r w(s)2 +c ++ 1 +p2 +1 +w(s)2 +c +, +γ(s+1) +c += +1 +T p2 +w(s) +r ++ 1 +T w(s+1) +r ++ 1 +p2 +1 +w(s) +c ++ 1 +p1 +w(s) +r w(s) +c ++ 1 +T w(s) +r w(s) +c ++ w(s) +r w(s+1) +r +w(s) +c ++ 1 +p2 +2 +w(s) +r w(s+1) +r +. +In the following theorem, we also establish the recurrence formula of the convergence rate for the esti- +mators �F(s+1) and �S(s+1) given any integer s ≥ 1. +Theorem 3.5. Under the same conditions stated in Theorem 3.1, as min{T, p1, p2} → ∞, for any t ∈ [T ], +i ∈ [p1] and j ∈ [p2] and any integer s > 1, we have +�����F(s+1) +t +− ( �H(s) +r )−1Ft +� +( �H(s) +c )−1�⊤���� +F += Op + + +� +w(s) +r +p2 ++ +� +w(s−1) +r +T p2 ++ +� +w(s) +c +p1 ++ +� +w(s−1) +c +T p1 ++ +1 +√p1p2 ++ γ(s+1) +f + + , +where +γ(s+1) +f += +� +w(s) +r w(s) +c ++ +� +w(s−1) +r +w(s−1) +c +T ++ +� +w(s−1) +r +w(s−1) +c +p1 ++ +� +w(s−1) +r +√p1p2 +, +and for the common components, we have +|�S(s+1) +t,ij +− St,ij| = Op + + +� +w(s) +r ++ +� +w(s−1) +r +T p2 ++ +� +w(s) +c ++ +� +w(s−1) +c +T p1 ++ +1 +√p1p2 ++ γ(s+1) + + , +where γ(s+1) = +� +w(s−1) +r +w(s−1) +c +T ++ +� +w(s−1) +r +w(s−1) +c +p1 ++ +� +w(s−1) +r +√p1p2 +and St,ij is the (i, j)-th entry of S = RFtC⊤. +Theoretical analysis for the estimators of loading matrices above relies on the correct specification of +factor numbers (note that we suppose m1 = k1, m2 = k2 in Theorem 3.1) and the strong factor conditions +R⊤R/p1 = Ik1, C⊤C/p2 = Ik2, which means that the row and column factors are pervasive along both +dimensions and is an extension of the pervasive assumption in Stock and Watson (2002a) to the matrix +regime. +13 + +4 +Simulation Study +In this section, we investigate the empirical performance of the Random Projection-based Iterative Least +Squares (RPILS) procedure in terms of estimating the loading and factor spaces. We first introduce the data +generation mechanism of the synthetic dataset, which is similar to He et al. (2022b). We set k1 = 3, k2 = 3, +draw the entries of R and C independently from uniform distribution U(−1, 1), and let +Ft = φFt−1 + +� +1 − φ2ǫt, +Et = ψEt−1 + +� +1 − ψ2Ut, +where Vec(ǫt) +i.i.d +∼ N(0, Ik1×k2), Ut +i.i.d +∼ MN(0, UE, VE), i.e., Vec(Ut) +i.i.d +∼ N(0, VE ⊗UE). The parameters +φ and ψ control the temporal correlations, and UE and VE are matrices with ones on the diagonal, and the +off-diagonal entries are 1/p1 and 1/p2, respectively. +We compare the performances of our Random Projection based Iterative Least Squares (RPILS) method +with the α-PCA method (α = 0) by Chen and Fan (2021) and the PE method by Yu et al. (2022) in terms +of estimating the loading and factor spaces. For the RPILS method, the initial weight matrices W1, W2 +are randomly chosen and all their entries from a standard normal distribution and the column dimensions +m1, m2 are set as the true dimensions of factor matrices, i.e, m1 = k1 = 3, m2 = k2 = 3. To show the impact +of the initial weight matrices, we also compare with the One-Step Estimators (OSE), either with completely +random standard normal elements as entries of the initial weight matrices or with α-PCA estimators as initial +weight matrices, denoted as OSE1 and OSE2 respectively. We point out that in unreported simulations, we +have tried to use the Walsh-Hadamard matrices (Fan and Liao, 2022) as the initial weight matrices for the +RPILS method, and find that the performances are almost the same as using initial weight matrices with +standard normal entries. We consider the following two scenarios of parameter settings: +Scenario A: p1 = 20, T = p2 ∈ {20, 50, 100, 150, 200}, φ = 0.1, ψ = 0.1. +Scenario B: p2 = 20, T = p1 ∈ {20, 50, 100, 150, 200}, φ = 0.1, ψ = 0.1. +To measure the performances of various methods in terms of estimating loading/factor spaces, we adopt +a metric between linear spaces which was also utilized in Yu et al. (2022); He et al. (2021a). For two column- +wise orthogonal matrices (Q1)p×q1 and (Q2)p×q2, we define +D(Q1, Q2) = +� +1 − +1 +max (q1, q2)Tr +� +Q1Q⊤ +1 Q2Q⊤ +2 +��1/2 +. +By the definition of D(Q1, Q2), we can easily see that 0 ≤ D(Q1, Q2) ≤ 1, which measures the distance +between the column spaces spanned by Q1 and Q2, i.e., span(Q1) and span(Q2). In particular, span(Q1) +and span(Q2) are the same when D(Q1, Q2) = 0, while span(Q1) and span(Q2) are orthogonal when +D(Q1, Q2) = 1. The Gram-Schmidt orthogonalization can be used to make Q1 and Q2 column-orthogonal +14 + +Table 1: Averaged estimation errors (standard errors in parentheses) in terms of D( �R, R), +D(�C, C) and D(Vec(Ft), Vec(�Ft)) for Scenarios A and B under Matrix Normal distribu- +tion over 500 replications. +Evaluation +T +p1 +p2 +OSE1 +OSE2 +RPILS +α-PCA +PE +Setting A: p1 = 20, p2 = T +D( �R, R) +20 +20 +20 +0.6178(0.1205) +0.0954(0.0173) +0.0938(0.0158) +0.1151(0.0308) +0.0947(0.0162) +50 +20 +50 +0.5992(0.1197) +0.0357(0.0053) +0.0355(0.0052) +0.0588(0.0219) +0.0355(0.0052) +100 +20 +100 +0.5797(0.1330) +0.0177(0.0025) +0.0176(0.0025) +0.0459(0.0221) +0.0176(0.0025) +150 +20 +150 +0.5835(0.1315) +0.0117(0.0017) +0.0117(0.0016) +0.0426(0.0199) +0.0117(0.0016) +200 +20 +200 +0.5762(0.1305) +0.0088(0.0013) +0.0088(0.0012) +0.0415(0.0203) +0.0088(0.0012) +D(�C, C) +20 +20 +20 +0.6138(0.1182) +0.0947(0.0166) +0.0933(0.0156) +0.1148(0.0290) +0.0942(0.0160) +50 +20 +50 +0.5458(0.1446) +0.0572(0.0063) +0.0568(0.0062) +0.0594(0.0068) +0.0572(0.0063) +100 +20 +100 +0.4785(0.1508) +0.0402(0.0034) +0.0399(0.0033) +0.0407(0.0035) +0.0402(0.0034) +150 +20 +150 +0.4400(0.1643) +0.0326(0.0026) +0.0324(0.0025) +0.0328(0.0025) +0.0326(0.0026) +200 +20 +200 +0.4217(0.1619) +0.0282(0.0021) +0.0280(0.0021) +0.0282(0.0021) +0.0282(0.0021) +D(Vec(Ft), Vec(�Ft)) +20 +20 +20 +0.6740(0.0380) +0.1837(0.0371) +0.1783(0.0338) +0.1837(0.0371) +0.1784(0.0339) +50 +20 +50 +0.7928(0.0440) +0.1074(0.0122) +0.1059(0.0111) +0.1074(0.0122) +0.1059(0.0111) +100 +20 +100 +0.8213(0.0494) +0.0747(0.0079) +0.0737(0.0070) +0.0747(0.0079) +0.0738(0.0070) +150 +20 +150 +0.8342(0.0503) +0.0606(0.0059) +0.0599(0.0053) +0.0606(0.0059) +0.0599(0.0053) +200 +20 +200 +0.8394(0.0499) +0.0521(0.0049) +0.0515(0.0044) +0.0521(0.0049) +0.0515(0.0044) +Setting B: p2 = 20, p1 = T +D( �R, R) +20 +20 +20 +0.6178(0.1205) +0.0954(0.0173) +0.0938(0.0158) +0.1151(0.0308) +0.0947(0.0162) +50 +50 +20 +0.5514(0.1440) +0.0574(0.0058) +0.0569(0.0056) +0.0595(0.0061) +0.0574(0.0058) +100 +100 +20 +0.4945(0.1526) +0.0402(0.0037) +0.0399(0.0036) +0.0406(0.0038) +0.0402(0.0037) +150 +150 +20 +0.4481(0.1573) +0.0327(0.0027) +0.0325(0.0027) +0.0328(0.0027) +0.0327(0.0027) +200 +200 +20 +0.4193(0.1671) +0.0281(0.0021) +0.0279(0.0021) +0.0281(0.0021) +0.0281(0.0021) +D(�C, C) +20 +20 +20 +0.6138(0.1182) +0.0947(0.0166) +0.0933(0.0156) +0.1148(0.0290) +0.0942(0.0160) +50 +50 +20 +0.5903(0.1269) +0.0359(0.0054) +0.0357(0.0053) +0.0599(0.0194) +0.0357(0.0053) +100 +100 +20 +0.5797(0.1291) +0.0177(0.0025) +0.0177(0.0025) +0.0469(0.0206) +0.0177(0.0025) +150 +150 +20 +0.5728(0.1292) +0.0116(0.0017) +0.0116(0.0016) +0.0424(0.0185) +0.0116(0.0016) +200 +200 +20 +0.5761(0.1287) +0.0088(0.0012) +0.0088(0.0012) +0.0425(0.0204) +0.0088(0.0012) +D(Vec(Ft), Vec(�Ft)) +20 +20 +20 +0.6740(0.0380) +0.1837(0.0371) +0.1783(0.0338) +0.1837(0.0371) +0.1784(0.0339) +50 +50 +20 +0.7925(0.0454) +0.1080(0.0123) +0.1064(0.0113) +0.1080(0.0123) +0.1064(0.0113) +100 +100 +20 +0.8253(0.0480) +0.0751(0.0073) +0.0742(0.0066) +0.0751(0.0073) +0.0742(0.0066) +150 +150 +20 +0.8349(0.0466) +0.0604(0.0061) +0.0597(0.0057) +0.0604(0.0061) +0.0597(0.0057) +200 +200 +20 +0.8382(0.0504) +0.0524(0.0051) +0.0518(0.0047) +0.0524(0.0051) +0.0518(0.0047) +15 + +matrices. +Table 1 reported the averaged estimation errors (standard errors in parentheses) in terms of D( �R, R), +D(�C, C) and D(Vec(Ft), Vec(�Ft)) for Scenarios A and B over 500 replications. All methods benefit from +large dimensions in terms of estimating loading spaces. By comparing the results for OSE1 and OSE2, we +conclude that the better the initial projection directions, the faster the loading/factor spaces converge to the +corresponding true ones. As for RPILS, the results indicate that even if we start from a random guess of the +projection directions, we can finally get satisfactory estimators via the iterative procedure in Algorithm 1. In +addition, the RPILS performs comparably with the PE method and better than the α-PCA method, which +is consistent with our theoretical analysis. In other words, the RPILS method also reduces the magnitudes +of the idiosyncratic error components, thereby increasing the signal-to-noise ratio and enjoying the same +advantage as PE. However, compared with the eigen-decomposition-based PE method, the RPILS method +is computationally simpler. +5 +Real Data Example +5.1 +Fama-French 10 × 10 portfolios +In this section, we study a financial portfolio dataset studied in Wang et al. (2019) and Yu et al. (2022). +The dataset is composed of monthly returns of 100 portfolios, well structured into a 10 × 10 matrix at each +time point, with rows corresponding to 10 levels of market capital size (denoted as S1-S10) and columns +corresponding to 10 levels of book-to-equity ratio (denoted as BE1-BE10). The dataset collects monthly +returns from January 1964 to December 2019 covering a total of 672 months. The details are available at +the website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Follow- +ing the same preprocessing as in Yu et al. (2022) and Wang et al. (2019), we adjusted the return series by +first subtracting the corresponding monthly market excess returns and then standardizing each of the series. +We imputed the missing values by the factor-model-based method introduced in Xiong and Pelger (2022). +All augmented Dickey-Fuller tests reject the null hypothesis, which indicates the stationarity of all the series. +As for the factor numbers, we take (k1, k2) = (2, 2) as in Yu et al. (2022). +Table 2 shows the estimated row and column loading matrices after varimax rotation and scaling. From +the table, we can see the proposed RPILS method performs similarly to PE, α-PCA and ACCE methods in +terms of the estimated loading matrices. From the perspective of Size, small Size portfolios load heavily on +the first factor while large Size portfolios load on the second. From the perspective of Book-to-Equity, small +BE portfolios load heavily on the second factor while large BE portfolios load mainly on the first factor. +We also use a rolling-validation scheme as in Yu et al. (2022) and Wang et al. (2019) to further compare +the methods. For each year t from 1996 to 2019, we repeatedly use n (bandwidth) years before t to fit the +16 + +Table 2: Loading matrices for Fama-French data set after varimax rotation and scaling +by 30. +Size +Method +Factor +S1 +S2 +S3 +S4 +S5 +S6 +S7 +S8 +S9 +S10 +RPILS +1 +-16 +-15 +-12 +-10 +-8 +-5 +-3 +-1 +4 +7 +2 +5 +1 +-3 +-5 +-8 +-10 +-12 +-13 +-15 +-11 +PE +1 +-16 +-15 +-12 +-10 +-8 +-5 +-3 +-1 +4 +7 +2 +-6 +-1 +3 +5 +8 +10 +12 +13 +15 +11 +α-PCA +1 +-14 +-14 +-13 +-11 +-9 +-7 +-4 +-2 +3 +7 +2 +-4 +-2 +1 +3 +6 +9 +12 +13 +16 +14 +ACCE +1 +-12 +-14 +-12 +-13 +-10 +-6 +-3 +-1 +4 +9 +2 +-1 +-1 +-1 +2 +5 +10 +11 +18 +15 +11 +Book-to-Equity +Method +Factor +BE1 +BE2 +BE3 +BE4 +BE5 +BE6 +BE7 +BE8 +BE9 +BE10 +RPILS +1 +-6 +-1 +4 +7 +10 +11 +12 +12 +12 +10 +2 +-20 +-17 +-11 +-8 +-4 +-2 +0 +1 +1 +0 +PE +1 +6 +1 +-4 +-7 +-10 +-11 +-12 +-12 +-12 +-10 +2 +20 +17 +11 +8 +4 +2 +0 +-1 +-1 +0 +α-PCA +1 +6 +2 +-4 +-7 +-10 +-11 +-12 +-13 +-12 +-11 +2 +19 +18 +12 +8 +4 +2 +0 +-1 +-1 +-1 +ACCE +1 +6 +-1 +-4 +-8 +-8 +-9 +-10 +-13 +-15 +-12 +2 +21 +15 +11 +6 +5 +2 +1 +-2 +-3 +1 +Table 3: Rolling validation for the Fama-French portfolios. The sample size of the training +set is 12n and k1 = k2 = k. +¯ +MSE, ¯ρ, ¯v are the mean pricing error, mean unexplained +proportion of total variances and mean variation of the estimated loading space. +MSE +¯ρ +¯v +n +k +RPILS +PE +α-PCA +ACCE +RPILS +PE +α-PCA +ACCE +RPILS +PE +α-PCA +ACCE +5 +1 +0.8766 +0.8703 +0.8624 +0.8846 +0.8001 +0.8022 +0.7960 +0.8284 +0.2653 +0.1757 +0.2412 +0.3032 +10 +1 +0.8735 +0.8548 +0.8596 +0.8797 +0.7948 +0.7836 +0.7913 +0.8149 +0.2341 +0.0847 +0.2027 +0.1654 +15 +1 +0.8570 +0.8530 +0.8599 +0.8844 +0.7828 +0.7822 +0.7918 +0.8118 +0.0606 +0.0636 +0.2335 +0.1520 +5 +2 +0.5954 +0.5965 +0.6010 +0.6673 +0.6231 +0.6248 +0.6284 +0.6727 +0.2331 +0.2390 +0.3497 +0.4605 +10 +2 +0.6014 +0.6013 +0.6108 +0.6545 +0.6273 +0.6276 +0.6364 +0.6684 +0.0892 +0.0924 +0.2606 +0.2568 +15 +2 +0.6027 +0.6025 +0.6115 +0.6375 +0.6261 +0.6262 +0.6302 +0.6516 +0.0564 +0.0573 +0.1735 +0.1894 +5 +3 +0.5204 +0.5216 +0.5291 +0.5639 +0.5473 +0.5495 +0.5558 +0.5900 +0.2781 +0.2865 +0.4321 +0.4974 +10 +3 +0.5181 +0.5193 +0.5262 +0.5728 +0.5465 +0.5481 +0.5549 +0.5936 +0.1074 +0.1142 +0.3532 +0.3029 +15 +3 +0.5166 +0.5172 +0.5220 +0.5601 +0.5438 +0.5446 +0.5444 +0.5825 +0.0783 +0.0839 +0.3082 +0.2986 +matrix-variate factor model and estimate the loading matrices. The estimated loadings are then used to +estimate the factors and corresponding residuals of the 12 months in the current year. In detail, let Yi +t and +17 + +�Yi +t be the observed and estimated price matrix of month i in year t, denote ¯Yt as the mean price matrix, +and define +MSEt = +1 +12 × 10 × 10 +12 +� +i=1 +∥ �Yi +t − Yi +t∥2 +F, +ρt = +�12 +i=1 ∥ �Yi +t − Yi +t∥2 +F +�12 +i=1 ∥Yi +t − ¯Yt∥2 +F +, +as the mean squared pricing error and unexplained proportion of total variances, respectively. The variation +of loading space is measured by +vt = D(�Ct ⊗ �Rt, �Ct−1 ⊗ �Rt−1), +during the rolling-validation procedure. +Table 3 reports the results of the means of MSE, ρ, v by the proposed RPILS method and the competitors. +For different bandwidth n and the number of factors k1, k2, our RPILS method is comparable to the PE +method and better than the other methods in terms of the averaged MSE, ρ, v. +5.2 +Multinational macroeconomic indices +In this section, we analyze a multinational macroeconomic index dataset collected from Organization for Eco- +nomic Co-operation and Development (OECD), which contains 10 macroeconomic indices across 8 countries +over 130 quarters from 1988-Q1 to 2020-Q2. The 8 countries are the United States, the United Kingdom, +Canada, France, Germany, Norway, Australia and New Zealand. +The indices are from 4 major groups, +namely consumer price, interest rate, production, and international trade. For the preprocessing proce- +dure of the dataset, we refer to Yu et al. (2022) for details. As for the factor numbers, we take the advice +(k1, k2) = (3, 4) by Yu et al. (2022) for better illustration. The estimated loading matrices are shown in Table +4 and Table 3.3. The proposed RPILS method behaves almost the same as the PE method. As concluded in +Yu et al. (2022), the countries excluding Germany naturally divide into 3 groups, Oceania, North American +and European. On the other hand, the macroeconomic indices divide into 4 groups, consumer price, interest +rate, production and international trade, which coincide with economic interpretations. +We also adopt a rolling prediction procedure to further investigate the practical use of different methods. +First, we consider the change of inflation (second-order difference of the log level of the total consumer price +index–CPI:Tot) of a selected country at time t, denoted as yt. Let xt be the vector of all the other 9 indices of +the selected country at time t, and Zt be the 8×10 panel at time t, with rows corresponding to the countries +and column corresponding to all macroeconomic indices. We predict yt+1 by the following Auto-Regression +(AR) model (Model 1) and Factor-Augmented-Auto-Regression (FAAR) models (Models 2–4), similar to the +Diffusion Index forecasting by Stock and Watson (2002b). +Model 1 yt+1 = a + byt + ǫt+1, +Model 2 yt+1 = a + byt + β⊤f1t + ǫt+1, where f1t’s are estimated from the vector factor model with +18 + +Table 4: Row loading matrices by different methods for multinational macroeconomic +index dataset, varimax rotated and multiplied by 10. +Method +Factor +AUS +NZL +USA +CAN +NOR +DEU +FRA +GBR +RPILS +1 +5 +-4 +1 +-4 +-20 +-7 +-15 +-12 +2 +0 +1 +-20 +-16 +11 +-8 +-5 +-4 +3 +24 +16 +1 +-3 +0 +6 +-1 +2 +PE +1 +0 +1 +-7 +-6 +3 +-3 +-2 +-1 +2 +2 +-2 +1 +-1 +-7 +-2 +-5 +-5 +3 +8 +6 +0 +-1 +0 +2 +-1 +1 +α-PCA +1 +-1 +1 +-7 +-5 +3 +-3 +-2 +-1 +2 +1 +-1 +0 +-1 +-7 +-2 +-5 +-4 +3 +-7 +-7 +0 +1 +0 +-1 +1 +-1 +ACCE +1 +2 +-2 +1 +-2 +-6 +0 +-6 +-5 +2 +7 +5 +0 +0 +0 +5 +0 +0 +3 +0 +-2 +8 +4 +-2 +1 +1 +2 +Table 5: Column loading matrices by different methods for multinational macroeconomic +index dataset, varimax rotated and multiplied by 10. +Method +Factor +CPI:Tot +CPI:Enter +CPI:NFNE +IR:3-Mon +IR:Long +P:TIEC +P:TM +GDP +IT:Ex +IT:Im +RPILS +1 +-2 +4 +-7 +-3 +3 +-20 +-20 +-5 +2 +0 +2 +1 +1 +-2 +-18 +-24 +1 +-1 +2 +1 +-3 +3 +-2 +6 +-7 +1 +0 +1 +1 +16 +19 +14 +4 +-19 +-20 +-10 +4 +-4 +1 +-1 +-1 +-1 +1 +PE +1 +1 +-2 +3 +1 +-1 +6 +7 +2 +-1 +0 +2 +6 +7 +3 +-1 +1 +0 +0 +0 +0 +0 +3 +0 +0 +-1 +-6 +-8 +0 +0 +1 +0 +-1 +4 +1 +-2 +3 +0 +0 +-1 +0 +-5 +-6 +-5 +α-PCA +1 +0 +-1 +1 +1 +-1 +7 +6 +4 +0 +0 +2 +7 +5 +5 +-1 +1 +0 +1 +0 +0 +0 +3 +0 +0 +0 +-7 +-7 +1 +0 +-1 +1 +0 +4 +0 +2 +-2 +0 +0 +0 +0 +2 +7 +6 +ACCE +1 +0 +0 +0 +0 +1 +-7 +-7 +0 +0 +0 +2 +1 +0 +0 +-5 +-4 +1 +-1 +-2 +-4 +-6 +3 +-4 +2 +-9 +0 +0 +2 +-1 +2 +0 +0 +4 +6 +7 +0 +-1 +3 +0 +0 +2 +1 +-1 +observations {xt}. +Model 3 yt+1 = a + byt + β⊤f2t + ǫt+1, where f2t’s are estimated from the vector factor model with +observations {Vec(Zt)}. +Model 4 yt+1 = a + byt + β⊤Vec(Ft) + ǫt+1, where Ft’s are estimated from the matrix factor model with +observations {Zt}, by the RPILS, PE, ACCE, and α-PCA, respectively. +The models for comparison here are exactly the same with Yu et al. (2022), we explain these models +here again for completeness. +First, Model 1 is a simple auto-regression model. +Model 2 adds common +19 + +index factors of the selected country into the auto-regression model in Model 1. In Model 3 and Model 4, +both index and country factors are taken into account. The difference is that Model 4 considers the more +parsimonious matrix factor structure while Model 3 vectorizes the matrix time series and considers the vector +factor structure. To avoid possible over-fitting in prediction, we also use the LASSO (Tibshirani, 1996) to +select factors and estimate the coefficients for Models 2-4. +Table 6: MAPEs for inflation and the growth rate of GDP (both at an annual rate) for +different countries with different methods, (k1, k2) = (3, 4). +Model +AUS +NZL +USA +CAN +NOR +DEU +FRA +GBR +MAPEs for inflation rates +Model 1 +1.5880 +1.8866 +2.8346 +2.4605 +1.9359 +1.8751 +1.8060 +1.5776 +Model 2 +1.6258 +1.9086 +2.5835 +2.2349 +2.2607 +1.9574 +1.8322 +1.5019 +Model 3 +1.5989 +2.0507 +2.4709 +1.9653 +2.3865 +1.7279 +1.2411 +1.1980 +Model 4 (RPILS) +1.5607 +1.9319 +2.6829 +2.2736 +2.1377 +1.8756 +1.7236 +1.3863 +Model 4 (PE) +1.5886 +2.0038 +2.3527 +1.8515 +2.1550 +1.7330 +1.4074 +1.3241 +Model 4 (ACCE) +1.5853 +1.9292 +2.2466 +1.8273 +2.4442 +1.8040 +1.3822 +1.3348 +Model 4 (α-PCA) +1.5981 +1.9177 +2.3777 +1.9023 +2.3132 +1.7402 +1.4335 +1.3490 +MAPEs for the growth rate of GDP +Model 1 +1.8897 +2.9211 +2.4317 +2.8279 +3.7794 +3.2970 +2.6777 +3.2515 +Model 2 +1.8709 +2.9117 +2.4683 +2.8584 +3.8977 +3.3413 +2.8395 +3.3231 +Model 3 +1.8669 +2.8391 +2.5664 +2.6364 +3.7375 +3.9167 +2.8535 +3.3285 +Model 4 (RPILS) +1.8904 +3.0531 +2.5087 +2.7084 +3.7467 +3.3315 +2.7672 +3.3198 +Model 4 (PE) +1.8731 +2.9268 +2.5819 +2.5505 +3.9012 +3.4485 +2.8079 +3.3523 +Model 4 (ACCE) +1.9118 +3.0370 +2.5067 +2.6869 +3.6739 +3.5838 +2.7446 +3.4251 +Model 4 (α-PCA) +1.8709 +3.0364 +2.6078 +2.6544 +3.8955 +3.5226 +2.8701 +3.3843 +For each quarter t from 2008-Q1 to 2020-Q2, we use the 80 neighboring observations before t to train +the models and predict yt+1 (denoted as �yt+1). As yt was standardized in preprocessing, we transformed +the predicted yt+1 to match the change of inflation rate by multiplying the standard deviation and adding +back the sample mean. For simplicity of notation, we still denote the transformed predictor as �yt+1. The +inflation It+1 is then predicted by integrating �yt+1 and It, i.e., �It+1 = �yt+1 + It. In Model 2 and Model +3, the factor numbers before model selection are set as k2 and k1 × k2, respectively. We also focus on the +case that (k1, k2) = (3, 4). The top panel of Table 6 shows the mean absolute prediction errors (MAPEs) +20 + +for the annualized inflation rates. For the largest Oceania country, Australia, Model 4 with RPILS has the +best prediction performance in terms of MAPE. For Norway and New Zealand, Model 1 performs the best, +indicating that the index and country factors act as noises in Model 2-4. For the USA, Canada, France, +Great Britain and Germany, both index and country factors are useful for improving prediction performance. +Note that the results also show that the matrix factor structure can further improve the prediction for two +American countries. We also consider the rolling prediction of the GDP growth rate (first-order difference of +the log level of GDP) for all countries. The results shown in the bottom panel of Table 6 demonstrate that +for the strong manufacturing American and European countries, USA, Germany, France and Great Britain, +the simple AR model suffices to predict the GDP growth rates well. For the other countries, the country +and the index factors contribute to improving the prediction performance of the GDP growth rates, while +for Canada and Norway, the advantage of the matrix factor structure is more obvious. +6 +Discussion +We propose a simple iterative least squares algorithm for the matrix factor model. In the first step, we +estimate the latent factor matrices by projecting the observations with two deterministic weight matrices. +We show that the inferences on factors are still asymptotically valid under some regularity conditions, even +if both row and column factor numbers are overestimated. In the second step, we estimate the row/column +loading matrices by minimizing the squared Frobenius loss function under some identifiability conditions. +The resultant estimators of the loading matrices are further treated as the new weight/projection matrices +and we iteratively perform the above two steps until convergence. Given the true dimensions of the factor +matrices, we establish the convergence rates of the estimators for loading matrices and common components +at the s-th iteration for any s ≥ 1. +As a future direction, our methodology could be generalized to tensor-valued time series. Intuitively, if +one substitutes the squared loss in (3.1) with the Huber loss, it would lead to a more robust estimator, which +is of independent interest because real-world financial returns and macroeconomic indexes often exhibit +heavy tails. Since a significant amount of additional work is still needed, we leave this to future work. +References +Ahn, S.C., Horenstein, A.R., 2013. +Eigenvalue ratio test for the number of factors. +Econometrica 81, +1203–1227. +A¨ıt-Sahalia, Y., Xiu, D., 2017. Using principal component analysis to estimate a high dimensional factor +model with high frequency data. Journal of Econometrics 201, 388–399. +21 + +Bai, J., 2003. Inferential theory for factor models of large dimensions. Econometrica 71, 135–171. +Barigozzi, M., Cho, H., 2020. Consistent estimation of high-dimensional factor models when the factor +number is over-estimated. Electronic Journal of Statistics 14, 2892–2921. +Barigozzi, M., Cho, H., Fryzlewicz, P., 2018. 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Projected estimation for large-dimensional matrix factor models. +Journal of Econometrics 229, 201–217. +Yu, L., He, Y., Zhang, X., 2019. Robust factor number specification for large-dimensional elliptical factor +model. Journal of Multivariate analysis 174, 104543. +23 + +Supplementary Materials for “Iterative Least Squares Algorithm for +Large-dimensional Matrix Factor Model by Random Projection” +Yong He ∗, Ran Zhao∗, Wen-Xin Zhou†, +This document provides the detailed proofs of the main theorems and additional lemmas and propositions. +S1 +Proofs of the main theorems +S1.1 +Proof of Theorem 2.1 +Proof. By the fact that H⊤ +1 (H1H⊤ +1 )+H1 = Ik1, H⊤ +2 (H2H⊤ +2 )+H2 = Ik2, then �Ft = H1FtH⊤ +2 + Et im- +plies M⊤ +1 �FtM2 − Ft = M⊤ +1 EtM2 with M1 = (H1H⊤ +1 )+H1, M2 = (H2H⊤ +2 )+H2. As ∥(H1H⊤ +1 )+H1∥2 = +Op(ν−1 +min(H1)) and (H2H⊤ +2 )+H2 = Op(ν−1 +min(H2)), by Lemma S2.2 (1), we have +∥M⊤ +1 �FtM2 − Ft∥2 = Op +� +1 +√p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +. +S1.2 +Proof of Theorem 2.2 +Proof. By Proposition S2.1, λmin( 1 +T M⊤�F⊤�FM) ≥ λmin( 1 +T +�F⊤�F)λmin(M⊤M) ≥ c(p1p2)−1λmin(D−2 +H ) with +large probability. By the SVD of H⊤, i.e., H⊤ = UH(DH, 0)E⊤ +H, we conclude that P�FM is well defined. +As �F = FH⊤ + E, then we have �FM − F = E(HH⊤)+H with M = (HH⊤)+H. Further by the fact that +∥(HH⊤)+H∥2 = Op +� +ν−1 +min +� +and Lemma S2.2 (2), (3), we have +1 +√ +T +∥�FM − F∥2 = +1 +√ +T +∥E(HH⊤)+H∥2 ≤ +1 +√ +T +∥E∥2∥(HH⊤)+H∥2 = Op +� +1 +√p1p2 +ν−1 +min +� +, +1 +T ∥F⊤(�FM − F)∥2 ≤ 1 +T ∥F⊤E∥2∥(HH⊤)+H∥2 = Op +� +1 +√T p1p2 +ν−1 +min +� +. +Further by +��� 1 +T M⊤�F⊤�FM − 1 +T F⊤F +��� +2 = +��� 1 +T (�FM − F)⊤(�FM − F) + 1 +T F⊤(�FM − F) + 1 +T (�FM − F)⊤F +��� +2 +≤ 1 +T ∥�FM − F∥2 +2 + 2∥F⊤(�FM − F)∥2 = Op +� +1 +√T p1p2 +ν−1 +min + +1 +p1p2 +ν−2 +min +� +, +we have +���( 1 +T M⊤�F⊤�FM)−1��� +2 = Op (1) +and +��� +� 1 +T M⊤�F⊤�FM +�−1 − +� 1 +T F⊤F +�−1��� +2 = Op +� +1 +√T p1p2 +ν−1 +min + +1 +p1p2 +ν−2 +min +� +. +∗Institute of Financial Studies, Shandong University, China. E-mail:heyong@sdu.edu.cn, Zhaoran@mail.sdu.edu.cn +†Department of Mathematical Sciences, University of California, San Diego, USA. E-mail:wez243@ucsd.edu +1 + +As a result, +P�FM − PF = +1 +√ +T +�FM +� +( 1 +T M⊤�F⊤�FM)−1 − ( 1 +T F⊤F)−1 +� +( 1 +√ +T +�FM)⊤ ++ +1 +√ +T +�FM( 1 +T F⊤F)−1 1 +√ +T +(�FM − F)⊤ + +1 +√ +T +(�FM − F)( 1 +T F⊤F)−1 1 +√ +T +F⊤, +∥P�FM − PF∥2 = Op +� +1 +√p1p2 +ν−1 +min +� +. +Finally, noting that P�FP�FM = P�FM, we have +∥P�FPF − PF∥2 ≤ ∥P�F(PF − P�FM)∥2 + ∥P�FM − PF∥2 = Op +� +1 +√p1p2 +ν−1 +min +� +. +S1.3 +Proof of Theorem 3.1 +Proof. Without loss of generality, we assume in the following that m1 = k1 = 1, m2 = k2 = 1. +1 +T +T +� +t=1 +XtW2�F⊤ +t = +1 +T p1p2 +T +� +t=1 +XtW2W⊤ +2 X⊤ +t W1 += +1 +T p1p2 +T +� +t=1 +(RFtC⊤ + Et)W2W⊤ +2 (RFtC⊤ + Et)⊤W1 += +1 +T p1p2 +T +� +t=1 +RFtC⊤W2W⊤ +2 CF⊤ +t R⊤W1 + +1 +T p1p2 +T +� +t=1 +RFtC⊤W2W⊤ +2 E⊤ +t W1 ++ +1 +T p1p2 +T +� +t=1 +EtW2W⊤ +2 CF⊤ +t R⊤W1 + +1 +T p1p2 +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1 +:= δ1 + δ2 + δ3 + δ4. +In the following, we analyze δ1, δ2, δ3, δ4 term by term. For the first term δ1, we have +∥δ1∥F = ∥ +1 +T p1p2 +T +� +t=1 +RFtC⊤W2W⊤ +2 CF⊤ +t R⊤W1∥F = ∥p2 +T +T +� +t=1 +RFtH⊤ +2 H2F⊤ +t H1∥F +≤ p2∥R∥F∥ 1 +T +T +� +t=1 +FtF⊤ +t ∥F ∥∥H2∥2 +2∥H1∥2 = Op +�√p1p2νmax(H1)ν2 +max(H2) +� += Op +�√p1p2νmin(H1)ν2 +min(H2) +� +, +For the first term δ2, we have +∥δ2∥F = ∥ +1 +T p1p2 +T +� +t=1 +RFtC⊤W2W⊤ +2 E⊤ +t W1∥F ≤ ∥ +1 +T p1p2 +T +� +t=1 +RFtW⊤ +2 E⊤ +t W1∥F∥C⊤W2∥2 +≤ +1 +T p1 +∥R∥F∥ +T +� +t=1 +FtW⊤ +2 E⊤ +t W1∥F ∥H2∥2 = Op +��p2 +T νmin(H2) +� +, +where the last equation is derived according to Lemma S2.3 (3). +2 + +By Lemma S2.3 (3), we have E∥ �T +t=1 FtW⊤ +2 E⊤ +t ∥2 +F = O(T p1p2), thus, +∥δ3∥F = ∥ +1 +T p1p2 +T +� +t=1 +EtW2W⊤ +2 CF⊤ +t R⊤W1∥F ≤ +1 +T p1p2 +∥ +T +� +t=1 +EtW2F⊤ +t ∥F ∥W⊤ +2 C∥2∥R⊤W1∥2 += Op +��p1p2 +T +νmin(H1)νmin(H2) +� +. +By Lemma S2.3 (4), +∥δ4∥F = ∥ +1 +T p1p2 +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1∥F = Op +��p2 +T + +1 +√p1 +� +. +Let Z = 1 +T 2 +��T +t=1 �FtW⊤ +2 X⊤ +t +� ��T +t=1 XtW2�F⊤ +t +� +, then +Z = +1 +p2 +1p2 +2T 2 ( +T +� +t=1 +W⊤ +1 XtW2W⊤ +2 X⊤ +t )( +T +� +t=1 +XtW2W⊤ +2 X⊤ +t W1) += (δ1 + δ2 + δ3 + δ4)⊤(δ1 + δ2 + δ3 + δ4) = +4 +� +i=1 +4 +� +j=1 +δ⊤ +i δj. +We can prove that when ν2 +min(H2) ≫ max( 1 +T , 1 +p2 +), the leading term of Z is δ⊤ +1 δ1, which is of the order +Op +� +p1p2 +2ν2 +min(H1)ν4 +min(H2) +� +, thus +∥Z−1/2∥F = Op +� +1 +√p1p2νmin(H1)ν2 +min(H2) +� +. +�R(1) = √p1 +� T +� +t=1 +XtW2�F⊤ +t +� �� T +� +t=1 +�FtW⊤ +2 X⊤ +t +� � T +� +t=1 +XtW2�F⊤ +t +��−1/2 += √p1 +� +1 +T +T +� +t=1 +XtW2�F⊤ +t +� +Z−1/2 += R +� +1 +√p1p2 +� +1 +T +T +� +t=1 +FtC⊤W2W⊤ +2 CF⊤ +t R⊤W1 +� +Z−1/2 +� ++ +1 +√p1p2 +� +1 +T +T +� +t=1 +RFtC⊤W2W⊤ +2 E⊤ +t W1 +� +Z−1/2 ++ +1 +√p1p2 +� +1 +T +T +� +t=1 +EtW2W⊤ +2 CF⊤ +t R⊤W1 +� +Z−1/2 + +1 +√p1p2 +� +1 +T +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1 +� +Z−1/2 +:= I + II + III + IV. +Let �H(1) +r += +1 +√p1p2T +��T +t=1 FtC⊤W2W⊤ +2 CF⊤ +t R⊤W1 +� +Z−1/2 then, +�R(1) − R �H(1) +r += II + III + IV. +(S1.1) +3 + +As +��� 1 +T +T +� +t=1 +FtC⊤W2W⊤ +2 CF⊤ +t R⊤W1 +��� +2 ≤ +��� 1 +T +T +� +t=1 +FtC⊤W2W⊤ +2 CF⊤ +t +��� +2∥R⊤W1∥2 += p1p2 +2∥ 1 +T +T +� +t=1 +FtH⊤ +2 H2F⊤ +t ∥2∥H1∥2 ≤ p1p2 +2∥ 1 +T +T +� +t=1 +F⊤ +t Ft∥2∥H2∥2 +2∥H1∥2 = Op +� +p1p2 +2νmin(H1)ν2 +min(H2) +� +, +thus we have the following results: +∥ �H(1) +r ∥F = ∥ +1 +√p1p2 +( 1 +T +T +� +t=1 +FtC⊤W2W⊤ +2 CF⊤ +t R⊤W1)Z−1/2∥F +≤ +1 +√p1p2 +∥ 1 +T +T +� +t=1 +FtC⊤W2W⊤ +2 CF⊤ +t R⊤W1∥2∥Z−1/2∥F = Op (1) ; +∥II∥F = ∥ +1 +√p1p2 +� +1 +T +T +� +t=1 +RFtC⊤W2W⊤ +2 E⊤ +t W1 +� +Z−1/2∥F = Op +� +1 +√T p2 +ν−1 +min(H1)ν−1 +min(H2) +� +; +∥III∥F = ∥ +1 +√p1p2 +� +1 +T +T +� +t=1 +EtW2W⊤ +2 CF⊤ +t R⊤W1 +� +Z−1/2∥F = Op +�� p1 +T p2 +ν−1 +min(H2) +� +; +∥IV∥F = ∥ +1 +√p1p2 +� +1 +T +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1 +� +Z−1/2∥F = Op +� +1 +√T p2 +ν−1 +min(H1)ν−2 +min(H2) + +1 +√p1p2 +ν−1 +min(H1)ν−2 +min(H2) +� +; +Finally, we can get +1 +p1 +∥ �R(1) − R �H(1) +r ∥2 +F = Op +� +1 +T p2ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν4 +min(H2) + +1 +p2 +1p2 +2ν2 +min(H1)ν4 +min(H2) +� +. +It remains to show that �H(1)⊤ +r +�H(1) +r +p→ Ik1. Under the condition that ν2 +min(H2) ≫ max( 1 +T , 1 +p2 +), we can +obtain +1 +p1 +∥ �R(1) − R �H(1) +r ∥2 +F = op (1) , +∥ 1 +p1 +R⊤( �R(1) − R �H(1) +r )∥F ≤ (∥R∥2 +F +p1 +∥ �R(1) − R �H(1) +r ∥2 +F +p1 +)1/2 = op(1), ∥ 1 +p1 +�R(1)⊤( �R(1) − R �H(1) +r )∥F = op(1). +Note that p−1 +1 +�R(1)⊤ �R(1) = Ik1, while p−1 +1 R⊤R = Ik1, then +Ik1 = 1 +p1 +�R(1)⊤R �H(1) +r ++ op(1) = �H(1)⊤ +r +�H(1) +r ++ op(1). +4 + +In the following we show the row-wise consistency of �R(1). By equation (S1.1), we have +� +R(1) +i· − �H(1)⊤ +r +Ri· = +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 EtW2W⊤ +2 CF⊤ +t Ri· +� ++ +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 RFtC⊤W2W⊤ +2 et,i· +� ++ +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 EtW2W⊤ +2 et,i· +� +. +By Lemma S2.3 (3), we have +����� +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 EtW2W⊤ +2 CF⊤ +t Ri· +������ +2 +≤ +1 +T √p1p2 +���Z−1/2��� +F +����� +T +� +t=1 +W⊤ +1 EtW2F⊤ +t +����� +F +��W⊤ +2 C +�� +2 += Op +� +1 +√T p1p2νmin(H1)νmin(H2) +� +. +Similar to the proof of Lemma S2.3 (3), we can get +����T +t=1 FtW⊤ +2 et,i· +��� +2 +F = Op (T p2), then +����� +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 RFtC⊤W2W⊤ +2 et,i· +������ +2 +≤ +1 +T √p1p2 +���Z−1/2��� +F +��W⊤ +1 R +�� +2 +��C⊤W2 +�� +2 +����� +T +� +t=1 +FtW⊤ +2 et,i· +����� +F += Op +� +1 +√T p2νmin(H2) +� +. +Similar to the proof of Lemma S2.3 (5), we can also get +��� +�T +t=1 W⊤ +1 EtW2W⊤ +2 et,i· +��� +2 +F = Op +� +T p1p2 +2 + T 2p2 +� +, +����� +1 +T √p1p2 +Z−1/2 +� T +� +t=1 +W⊤ +1 EtW2W⊤ +2 et,i· +������ +2 +≤ +1 +T √p1p2 +���Z−1/2��� +F ∥W2∥F +����� +T +� +t=1 +W⊤ +1 EtW2W⊤ +2 et,i· +����� +F += Op +� +1 +√T p1p2νmin(H1)ν2 +min(H2) + +1 +p1p2νmin(H1)ν2 +min(H2) +� +. +Symmetrically, the convergence rate of the �C can be similarly derived by equation (3.4) and we omit the +proof for saving space. +5 + +S1.4 +Proof of Theorem 3.2 +Proof. First we decompose �T +t=1 X⊤ +t �R(1)�Ft into four terms: +T +� +t=1 +X⊤ +t �R(1)�Ft = +1 +p1p2 +T +� +t=1 +X⊤ +t �R(1)W⊤ +1 XtW2 += +1 +p1p2 +T +� +t=1 +(RFtC⊤ + Et)⊤ �R(1)W⊤ +1 (RFtC⊤ + Et)W2 += +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 + +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 RFtC⊤W2 ++ +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 EtW2 + +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 EtW2 += δ(1) +1 ++ δ(1) +2 ++ δ(1) +3 ++ δ(1) +4 . +As +���� +1 +T δ(1) +1 +���� +F += +����� +1 +T p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 +����� +F +≍ +����� +1 +T p2 +T +� +t=1 +CF⊤ +t W⊤ +1 RFtC⊤W2 +����� +F += p1∥C∥F +����� +1 +T +T +� +t=1 +F⊤ +t Ft +����� +F +∥H1∥2 ∥H2∥2 = Op (p1 +√p2νmin(H1)νmin(H2)) , +thus, we have ∥δ(1) +1 ∥F = Op +� +T p1√p2νmin(H1)νmin(H2) +� +. +By Lemma S2.4, we get +����� +T +� +s=1 +E⊤ +s ( �R(1) − R �H(1) +r )Fs +����� +2 +F += Op +� +p2 +1 +p2ν2 +min(H2) + +p1 +ν2 +min(H1)ν4 +min(H2) + +T +p2ν2 +min(H1)ν4 +min(H2) +� +, +then +∥δ(1) +2 ∥F = +����� +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 RFtC⊤W2 +����� +F += +����� +1 +p1p2 +T +� +t=1 +E⊤ +t ( �R(1) − R �H(1) +r ++ R �H(1) +r )W⊤ +1 RFtC⊤W2 +����� +F +≤ +����� +1 +p1p2 +T +� +t=1 +E⊤ +t ( �R(1) − R �H(1) +r )W⊤ +1 RFtC⊤W2 +����� +F ++ +����� +1 +p1p2 +T +� +t=1 +E⊤ +t R �H(1) +r W⊤ +1 RFtC⊤W2 +����� +F +≤ +����� +T +� +t=1 +E⊤ +t ( �R(1) − R �H(1) +r )Ft +����� +F +∥H1∥2 ∥H2∥2 + +����� +T +� +t=1 +E⊤ +t RFt +����� +F +∥H1∥2 ∥H2∥2 +��� �H(1) +r +��� +F += Op +�� +T p1p2νmin(H1)νmin(H2) + p1 +√p2 +νmin(H1) + +√p1 +νmin(H2) +� +. +By Lemma S2.3 (3), we also have E∥ �T +t=1 E⊤ +t W1Ft∥2 +F = O (T p1p2), thus +6 + +���δ(1) +3 +��� +F = +����� +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 EtW2 +����� +F +≍ 1 +p2 +����� +T +� +t=1 +CF⊤ +t W⊤ +1 EtW2 +����� +F +≤ 1 +p2 +∥C∥F +����� +T +� +t=1 +F⊤ +t W⊤ +1 EtW2 +����� +F += Op +�� +T p1 +� +. +���δ(1) +4 +��� +F = +����� +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 EtW2 +����� +F += +����� +1 +p1p2 +T +� +t=1 +E⊤ +t ( �R(1) − R �H(1) +r ++ R �H(1) +r )W⊤ +1 EtW2 +����� +F +≤ +����� +1 +p1p2 +T +� +t=1 +E⊤ +t ( �R(1) − R �H(1) +r )W⊤ +1 EtW2 +����� +F ++ +����� +1 +p1p2 +T +� +t=1 +E⊤ +t R �H(1) +r W⊤ +1 EtW2 +����� +F +≤ +1 +p1p2 +����� +T +� +t=1 +E⊤ +t W⊤ +1 EtW2 +����� +F +��� �R(1) − R �H(1) +r +��� +F + +1 +p1p2 +����� +T +� +t=1 +E⊤ +t W⊤ +1 EtW2 +����� +F +∥R∥F ∥ �H(1) +r ∥F += Op +�� +T p1 + +T +√p2 +� +. +Let Z(1) = (�T +t=1 �F⊤ +t �R(1)⊤Xt)(�T +t=1 X⊤ +t �R(1)�Ft), then Z(1) = �4 +i=1 +�4 +j=1 δ(1)⊤ +i +δ(1) +j . We can prove the +leading term in Z(1) is δ(1)⊤ +1 +δ(1) +1 , hence ∥(Z(1))−1/2∥F = Op +� +1 +T p1√p2νmin(H1)νmin(H2) +� +. +�C(1) = √p2 +� T +� +t=1 +X⊤ +t �R(1)�Ft +� �� T +� +t=1 +�F⊤ +t �R(1)⊤Xt +� � T +� +t=1 +X⊤ +t �R(1)�Ft +��−1/2 += √p2 +� +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 +� � +Z(1)�−1/2 ++ √p2 +� +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 RFtC⊤W2 +� � +Z(1)�−1/2 ++ √p2 +� +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(1)W⊤ +1 EtW2 +� � +Z(1)�−1/2 ++ √p2 +� +1 +p1p2 +T +� +t=1 +E⊤ +t �R(1)W⊤ +1 EtW2 +� � +Z(1)�−1/2 +. += √p2 +� +δ(1) +1 (Z(1))−1/2 + δ(1) +2 (Z(1))−1/2 + δ(1) +3 (Z(1))−1/2 + δ(1) +4 (Z(1))−1/2� +. +Let �H(1) +c += +� +1 +p1√p2 +�T +t=1 F⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 +� � +Z(1)�−1/2, then +�C(1) − C �H(1) +c += √p2δ(1) +2 (Z(1))−1/2 + √p2δ(1) +3 (Z(1))−1/2 + √p2δ(1) +4 (Z(1))−1/2. +7 + +∥ �H(1) +c ∥F = +����� +� +1 +p1√p2 +T +� +t=1 +F⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 +� � +Z(1)�−1/2 +����� +F +≤ +����� +1 +p1√p2 +T +� +t=1 +F⊤ +t R⊤ �R(1)W⊤ +1 RFtC⊤W2 +����� +F +���� +� +Z(1)�−1/2���� +F += Op (1) . +∥δ(1) +2 (Z(1))−1/2∥F = Op +� +1 +√T p1 ++ +1 +T p2 +ν−1 +min(H2) + +1 +T √p1p2 +ν−1 +min(H1)ν−2 +min(H2) +� +. +∥δ(1) +3 (Z(1))−1/2∥F = Op +� +1 +√T p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +. +∥δ(1) +4 (Z(1))−1/2∥F = Op +� +1 +√T p1p2 +ν−1 +min(H1)ν−1 +min(H2) + +1 +p1p2 +ν−1 +min(H1)ν−1 +min(H2) +� +. +As a result +1 +p2 +∥�C(1) − C �H(1) +c ∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2ν2 +min(H1)ν2 +min(H2) + +1 +T 2p2 +2ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν2 +min(H2) +� +. +��� 1 +p2 +C⊤(�C(1) − C �H(1) +c ) +��� +F ≤ (∥C∥2 +F +p2 +∥�C(1) − C �H(1) +c ∥2 +F +p2 +)1/2 = op (1) , +��� 1 +p2 +�C(1)⊤(�C(1) − C �H(1) +c ) +��� +F = op (1) . +Note that p−1 +2 C⊤C = Ik2, p−1 +2 +�C(1)⊤ �C(1) = Ik2, then +Ik2 = 1 +p2 +�C(1)⊤C �H(1) +c ++ op (1) = �H(1)⊤ +c +�H(1) +c ++ op (1) , �H(1)⊤ +c +�H(1) +c +p→ Ik2. +In the following, we shows the row-wise consistency of �C(1). +For j ≤ p2, +� +C(1) +j· − �H(1)⊤ +c +Cj· = +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 CF⊤ +t R⊤W1 �R(1)⊤et,·j +� ++ +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤RFtCj· +� ++ +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤et,·j +� +. +Firstly, similar to the proof of Lemma S2.4, we can obtain +����� +T +� +t=1 +F⊤ +t ( �R(1) − R �H(1) +r )⊤et,·j +����� +2 +F += Op +� +p2 +1 +p2 +2ν2 +min(H2) + +p1 +p2ν2 +min(H1)ν4 +min(H2) + +T +p2 +2ν2 +min(H1)ν4 +min(H2) +� +, +8 + +����� +1 +p1√p2 +T +� +t=1 +W⊤ +2 CF⊤ +t R⊤W1 �R(1)⊤et,·j +����� +2 +≤ +1 +p1√p2 +����� +T +� +t=1 +F⊤ +t �R(1)⊤et,·j +����� +F +∥W⊤ +2 C∥2∥R⊤W1∥2 +≤ +1 +p1√p2 +����� +T +� +t=1 +F⊤ +t ( �R(1) − Rt �H(1) +r )⊤et,·j +����� +F +∥W⊤ +2 C∥2∥R⊤W1∥2 ++ +1 +p1√p2 +����� +T +� +t=1 +F⊤ +t R⊤ +t et,·j +����� +F +∥W⊤ +2 C∥2∥R⊤W1∥2∥ �H(1) +c ∥F , +then, +����� +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 CF⊤ +t R⊤W1 �R(1)⊤et,·j +������ +2 +≤ +1 +p1√p2 +����� +T +� +t=1 +W⊤ +2 CF⊤ +t R⊤W1 �R(1)⊤et,·j +����� +F +���(Z(1))−1/2��� +F += Op +� +1 +T p2νmin(H2) + +1 +T √p1p2νmin(H1)ν2 +min(H2) + +1 +√T p1 +� +. +Secondly, +����� +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤RFtCj· +������ +2 +≍ +����� +1 +√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1Ft +������ +2 += Op +� +1 +√T p1p2νmin(H1)νmin(H2) +� +. +Thirdly, +����� +T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤et,·j +����� +F += +����� +T +� +t=1 +W⊤ +2 E⊤ +t W1( �R(1) − R �H(1) +r ++ R �H(1) +r )⊤et,·j +����� +F +≤ +����� +T +� +t=1 +W⊤ +2 E⊤ +t W1e⊤ +t,·j +����� +F +��� �R(1) − R �H(1) +r +��� +F + +����� +T +� +t=1 +W⊤ +2 E⊤ +t W1e⊤ +t,·j +����� +F +∥R∥F , +����� +1 +p1√p2 +(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤et,·j +������ +F +≤ +1 +p1√p2 +�����(Z(1))−1/2 +� T +� +t=1 +W⊤ +2 E⊤ +t W1 �R(1)⊤et,·j +������ +F += Op +� +1 +p1p2νmin(H1)νmin(H2) + +1 +√T p1p2νmin(H1)νmin(H2) +� +. +The conclusion is obtained by summarizing the above three terms. +9 + +S1.5 +Proof of Theorem 3.3 +Proof. By definition, +�F(2) +t += +1 +p1p2 +�R(1)⊤Xt �C(1) = +1 +p1p2 +�R(1)⊤RFtC⊤ �C(1) + +1 +p1p2 +�R(1)⊤Et �C(1) += +1 +p1p2 +�R(1)⊤ � +R − �R(1)( �H(1) +r )−1 + �R(1)( �H(1) +r )−1� +Ft +� +C − �C(1)( �H(1) +c )−1 + �C(1)( �H(1) +c )−1�⊤ �C(1) ++ +1 +p1p2 +� +�R(1) − R �H(1) +r ++ R �H(1) +r +�⊤ +Et +� +�C(1) − C �H(1) +c ++ C �H(1) +c +� +. +Note that �R(1)⊤ �R(1) = p1Ik1, �C(1)⊤ �C(1) = p2Ik2, then +�F(2) +t +− ( �H(1) +r )−1Ft +� +( �H(1) +c )−1�⊤ += +1 +p1p2 +�R(1)⊤ � +R − �R(1)( �H(1) +r )−1� +Ft +� +C − �C( �H(1) +c )−1�⊤ �C(1) ++ 1 +p1 +�R(1)⊤ � +R − �R(1)( �H(1) +r )−1� +Ft +� +( �H(1) +c )−1�⊤ ++ 1 +p2 +( �H(1) +r )−1Ft +� +C − �C( �H(1) +c )−1�⊤ �C(1) ++ +1 +p1p2 +� +�R(1) − R �H(1) +r +�⊤ +Et +� +�C(1) − C �H(1) +c +� ++ +1 +p1p2 +� +�R(1) − R �H(1) +r +�⊤ +EtC �H(1) +c ++ +1 +p1p2 +�H(1)⊤ +r +R⊤Et +� +�C(1) − C �H(1) +c +� ++ +1 +p1p2 +�H(1)⊤ +r +R⊤EtC �H(1) +c . +In Lemma S2.5, we prove that +���R⊤( �R(1) − R �H(1) +r ) +��� +F = Op +� +√p1 +√T p2νmin(H1)νmin(H2) + +1 +p2νmin(H1)ν2 +min(H2) +� +, +���C⊤(�C(1) − C �H(1) +c ) +��� +F = Op +� +1 +T νmin(H2) + +√p2 +√T p1νmin(H1)νmin(H2) + +1 +p1νmin(H1)νmin(H2) +� +. +Therefore, by Cauchy-Schwartz inequality, Theorem 3.1, Theorem 3.2, and the bounds for ∥R⊤Et∥2 +F , +∥EtC∥2 +F and ∥R⊤EtC∥2 +F , we have +�����F(2) +t +− ( �H(1) +r )−1Ft +� +( �H(1) +c )−1�⊤���� +F += Op +� +1 +√ +Tp1 ++ +1 +√ +Tp2νmin(H2) ++ +1 +√p1p2 ++ +1 +p1p2νmin(H1)ν2 +min(H2) + γf +� +, +where +γf = +1 +√T p1p2νmin(H1)νmin(H2) + +1 +T p1√p2νmin(H1)ν2 +min(H2) + +1 +T p1p2ν2 +min(H1)ν3 +min(H2). +Next, for any t, i, j +�S(2) +t,ij − St,ij = � +R(1)⊤ +i· +�F(2) +t +� +C(1) +j· − R⊤ +i· FtCj· += ( � +R(1) +i· − �H(1)⊤ +r +Ri· + �H(1)⊤ +r +Ri·)⊤�F(2) +t ( � +C(1) +j· − �H(1)⊤ +c +Cj· + �H(1)⊤ +c +Cj·) − R⊤ +i· FtCj· += ( � +R(1) +i· − �H(1)⊤ +r +Ri·)⊤�F(2) +t ( � +C(1) +j· − �H(1)⊤ +c +Cj·) + R⊤ +i· �H(1) +r �F(2) +t ( � +C(1) +j· − �H(1)⊤ +c +Cj·) ++ ( � +R(1) +i· − �H(1)⊤ +r +Ri·)⊤�F(2) +t +�H(1)⊤ +c +Cj· + R⊤ +i· ( �H(1) +r �F(2) +t +�H(1)⊤ +c +− Ft)Cj·. +10 + +Then, by Cauchy-Schwartz inequality, Theorem 3.1, Theorem 3.2 and the consistency of the estimators for +the factor matrix, we have +|�S(2) +t,ij−St,ij| = Op +� +1 +√T p1 ++ +1 +√p1p2 ++ +1 +√T p2νmin(H2) + +1 +√T p1p2νmin(H1)ν2 +min(H2) + +1 +p1p2νmin(H1)ν2 +min(H2) +� +. +S1.6 +Proof of Theorem 3.4 +Proof. Recall first that +�R(s+1) = √p1 +� T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +� �� T +� +t=1 +�F(s+1) +t +�C(s)⊤X⊤ +t +� � T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +��−1/2 +. +T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t += +1 +p1p2 +T +� +t=1 +Xt �C(s) �C(s)⊤X⊤ +t �R(s) += +1 +p1p2 +T +� +t=1 +(RFtC⊤ + Et)�C(s) �C(s)⊤(RFtC⊤ + Et)⊤ �R(s) += +1 +p1p2 +T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s) + +1 +p1p2 +T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤E⊤ +t �R(s) ++ +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s) + +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t �R(s) += δ(s+1) +1 ++ δ(s+1) +2 ++ δ(s+1) +3 ++ δ(s+1) +4 +. +As +∥ 1 +T δ(s+1) +1 +∥2 +F = ∥ +1 +T p1p2 +T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s)∥2 +F +≍ p2 +2∥ 1 +T +T +� +t=1 +RFtF⊤ +t ∥2 +F ≤ p2 +2∥R∥2 +F∥ 1 +T +T +� +t=1 +FtF⊤ +t ∥2 +F = Op +� +p1p2 +2 +� +, +thus, ∥δ(s+1) +1 +∥2 +F = Op +� +T 2p1p2 +2 +� +. Similarly, +∥δ(s+1) +2 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤E⊤ +t �R(s)∥2 +F ≍ 1 +p2 +1 +∥ +T +� +t=1 +RFt �C(s)⊤E⊤ +t �R(s)∥2 +F += 1 +p2 +1 +∥ +T +� +t=1 +RFt �C(s)⊤E⊤ +t ( �R(s) − R �H(s) +r )∥2 +F + 1 +p2 +1 +∥ +T +� +t=1 +RFt �C(s)⊤E⊤ +t R �H(s) +r ∥2 +F +≤ 1 +p2 +1 +∥R∥2 +F∥ +T +� +t=1 +Ft �C(s)⊤E⊤ +t ∥2 +F ∥ �R(s) − R �H(s) +r ∥2 +F + 1 +p2 +1 +∥R∥2 +F∥ +T +� +t=1 +Ft �C(s)⊤E⊤ +t R∥2 +F∥ �H(s) +r ∥2 +F += Op +� +T p1p2 +2w(s) +r w(s) +c ++ T p1p2w(s) +r ++ T p2 +2w(s) +c ++ T p2 +� +, +11 + +the last equality is due to Lemma S2.9. +∥δ(s+1) +3 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s)∥2 +F ≍ ∥ +T +� +t=1 +Et �C(s)F⊤ +t ∥2 +F = Op +� +T p1p2 +2w(s) +c ++ T p1p2 +� +. +∥δ(s+1) +4 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t �R(s)∥2 +F +≤ ∥ +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t ( �R(s) − R �H(s) +r )∥2 +F + ∥ +1 +p1p2 +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t R �H(s) +r ∥2 +F +≤ +1 +p2 +1p2 +2 +T +� +t=1 +∥Et �C(s)∥4 +F ∥ �R(s) − R �H(s) +r ∥2 +F + +1 +p2 +1p2 +2 +∥ +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t R∥2 +F ∥ �H(s) +r ∥2 +F += Op +� +T p2 + T 2 +p1 ++ T 2p1w(s) +r ++ T 2p1p2w(s) +r w(s) +c ++ T 2p1p2 +2w(s) +r w(s)2 +c ++ T 2p2 +2 +p1 +w(s)2 +c ++ T p2 +2w(s)2 +c +� +, +the last equality is according to Lemma S2.7 and Lemma S2.8. +Let Z(s+1) = +��T +t=1 �F(s+1) +t +�C(s)⊤X⊤ +t +� ��T +t=1 Xt �C(s)�F(s+1)⊤ +t +� +, +Z(s+1) = +1 +p2 +1p2 +2 +� T +� +t=1 +�R(s)⊤Xt �C(s) �C(s)⊤X⊤ +t +� � T +� +t=1 +Xt �C(s) �C(s)⊤X⊤ +t �R(s) +� += +4 +� +i=1 +4 +� +j=1 +δ(s+1)⊤ +i +δ(s+1) +j +, +then we can prove +��(Z(s+1))−1/2��2 +F = Op +� +1 +T 2p1p2 +2 +� +. +�R(s+1) = √p1 +� T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +� �� T +� +t=1 +�F(s+1) +t +�C(s)⊤X⊤ +t +� � T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +��−1/2 += √p1 +� T +� +t=1 +Xt �C(s)�F(s+1)⊤ +t +� � +Z(s+1)�1/2 += R +� +1 +√p1p2 +� T +� +t=1 +FtC⊤ �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s) +� � +Z(s+1)�−1/2 +� ++ +1 +√p1p2 +� T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 += +1 +√p1p2 +� T +� +t=1 +Et �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s) +� � +Z(s+1)�−1/2 ++ +1 +√p1p2 +� T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 +. +12 + +Denote �H(s+1) +r += +1 +√p1p2 +��T +t=1 FtC⊤ �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s)� � +Z(s+1)�−1/2, we have +�R(s+1) − R �H(s+1) +r += +1 +√p1p2 +� T +� +t=1 +RFtC⊤ �C(s) �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 ++ +1 +√p1p2 +� T +� +t=1 +Et �C(s) �C(s)⊤CF⊤ +t R⊤ �R(s) +� � +Z(s+1)�−1/2 ++ +1 +√p1p2 +� T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 += √p1(δ(s+1) +2 ++ δ(s+1) +3 ++ δ(s+1) +4 +)(Z(s+1))−1/2, +thus, +1 +p1 +∥ �R(s+1) − R �H(s+1) +r +∥2 +F = Op +� +1 +T p2 ++ +1 +p2 +1p2 +2 ++ 1 +p2 +2 +w(s) +r ++ 1 +T w(s) +c ++ w(s) +r w(s) +c +p2 ++ w(s) +r w(s)2 +c ++ 1 +p2 +1 +w(s)2 +c +� +. +For the row-consistency of �R(s+1), we have +� +R(s+1) +i· +− �H(s+1)⊤ +r +Ri· = +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤CF⊤ +t Ri· +� ++ +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤RFtC⊤ �C(s) �C(s)⊤et,i· +� ++ +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤et,i· +� +. +First, by Lemma S2.9 (1) and (2), +����� +T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤CF⊤ +t Ri· +����� +2 +F +≍ p2 +2 +����� +T +� +t=1 +�R(s)⊤Et �C(s)F⊤ +t +����� +2 +F +≤ p2 +2 +����� +T +� +t=1 +Et �C(s)F⊤ +t +����� +2 +F +����R(s) − R �H(s) +r +��� +2 +F + p2 +2 +����� +T +� +t=1 +R⊤Et �C(s)F⊤ +t +����� +2 +F += Op +� +T p2 +1p4 +2w(s) +r w(s) +c ++ T p2 +1p3 +2w(s) +r ++ T p1p4 +2w(s) +c ++ T p1p3 +2 +� +, +then, +����� +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤CF⊤ +t Ri· +������ +2 +F +≤ +1 +p1p2 +2 +���(Z(s+1))−1/2��� +2 +F +����� +T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤CF⊤ +t Ri· +����� +2 +F += Op +� +w(s) +r w(s) +c +T ++ w(s) +r +T p2 ++ w(s) +c +T p1 ++ +1 +T p1p2 +� +. +13 + +Second, similar to the proof of Lemma S2.9 (1), we have +����� +T +� +t=1 +Ft �C(s)⊤et,i· +����� +2 +F += Op +� +T p2 +2w(s) +c ++ T p2 +� +, +����� +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤RFtC⊤ �C(s) �C(s)⊤et,i· +������ +2 +F +≲ p1 +���� +� +Z(s+1)�−1/2���� +2 +F +����� +T +� +t=1 +Ft �C(s)⊤et,i· +����� +2 +F += Op +� +w(s) +c +T ++ +1 +T p2 +� +. +Third, similar to the proof of Lemma S2.7 (1) and Lemma S2.8 (1), we have +����� +T +� +t=1 +�C(s)⊤et,i· +����� +2 +F += Op +� +T p2 +2w(s) +c ++ T p2 +� +, +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F += Op +� +T p1p3 +2 + T 2p2 +2 + (T 2p4 +2 + T p1p4 +2)w(s)2 +c +� +, +hence, +����� +T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F +≤ +����� +T +� +t=1 +( �R(s) − R �H(s) +r )⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F ++ +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F +≤ +��� �R(s) − R �H(s) +r +��� +2 +F +����� +T +� +t=1 +Et �C(s) +����� +2 +F +����� +T +� +t=1 +�C(s)⊤et,i· +����� +2 +F ++ +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F += Op +� +T p1p3 +2 + T 2p2 +2 + T 2p2 +1p2 +2w(s) +r ++ T p1p4 +2w(s)2 +c ++ T 2p4 +2w(s)2 +c ++ T 2p2 +1p4 +2w(s)2 +c +ws +r + T 2p2 +1p3 +2w(s) +r w(s) +c +� +����� +1 +√p1p2 +� +Z(s+1)�−1/2 +� T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤et,i· +������ +2 +F +≤ +1 +p1p2 +2 +���� +� +Z(s+1)�−1/2���� +2 +F +����� +T +� +t=1 +�R(s)⊤Et �C(s) �C(s)⊤et,i· +����� +2 +F += Op +� +1 +T p1p2 ++ +1 +p2 +1p2 +2 ++ w(s) +r +p2 +2 ++ w(s)2 +c +T p1 ++ w(s)2 +c +p2 +1 ++ w(s) +r w(s)2 +c ++ w(s) +r w(s) +c +p2 +� +. +As a result +∥ � +R(s+1) +i· +− �H(s+1)⊤ +r +Ri·∥2 +2 = Op +� +1 +T p2 ++ +1 +p2 +1p2 +2 ++ 1 +p2 +2 +w(s) +r ++ 1 +T w(s) +c ++ w(s) +r w(s) +c +p2 ++ w(s) +r w(s)2 +c ++ 1 +p2 +1 +w(s)2 +c +� +. +As for the �C(s+1), +�C(s+1) = √p2 +� T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +� �� T +� +t=1 +�F(s+1)⊤ +t +�R(s+1)⊤Xt +� � T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +��−1/2 +. +14 + +T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t += +1 +p1p2 +T +� +t=1 +X⊤ +t �R(s+1) �R(s)⊤Xt �C(s) += +1 +p1p2 +T +� +t=1 +(RFtC⊤ + Et)⊤ �R(s+1) �R(s)⊤(RFtC⊤ + Et)�C(s) += +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s) + +1 +p1p2 +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤RFtC⊤ �C(s) ++ +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤Et �C(s) + +1 +p1p2 +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤Et �C(s) += ∆(s+1) +1 ++ ∆(s+1) +2 ++ ∆(s+1) +3 ++ ∆(s+1) +4 +. +As +∥ 1 +T ∆(s+1) +1 +∥F = ∥ +1 +T p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s)∥2 +F +≍ p2 +1∥ 1 +T +T +� +t=1 +CF⊤ +t Ft∥2 +F = Op +� +p2 +1p2 +� +, +then ∥∆(s+1) +1 +∥2 +F = Op +� +T 2p2 +1p2 +� +. By Lemma S2.9, +∥∆(s+1) +2 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤RFtC⊤ �C(s)∥2 +F ≍ ∥ +T +� +t=1 +E⊤ +t �R(s+1)Ft∥F = Op +� +T p2 +1p2w(s+1) +r ++ T p1p2 +� +. +∥∆(s+1) +3 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤Et �C(s)∥2 +F ≍ 1 +p2 +2 +∥ +T +� +t=1 +CF⊤ +t �R(s)⊤Et �C(s)∥2 +F +≤ 1 +p2 +2 +∥ +T +� +t=1 +CF⊤ +t �R(s)⊤Et(�C(s) − C �H(s) +c )∥2 +F + 1 +p2 +2 +∥ +T +� +t=1 +CF⊤ +t �R(s)⊤EtC �H(s) +c ∥2 +F +≤ 1 +p2 +2 +∥C∥2 +F∥ +T +� +t=1 +F⊤ +t �R(s)⊤Et∥2 +F ∥�C(s) − C �H(s) +c ∥2 +F + 1 +p2 +2 +∥C∥2 +F∥ +T +� +t=1 +F⊤ +t �R(s)⊤EtC∥2 +F∥ �H(s) +c ∥2 +F += Op +� +T p2 +1p2w(s) +r w(s) +c ++ T p1p2w(s) +c ++ T p2 +1w(s) +r ++ T p1 +� +. +According to Lemma S2.7 and Lemma S2.8, we have +∥∆(s+1) +4 +∥2 +F = ∥ +1 +p1p2 +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤Et �C(s)∥2 +F +≤ +1 +p2 +1p2 +2 +∥ +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤Et(�C(s) − C �H(s) +c )∥2 +F + +1 +p2 +1p2 +2 +∥ +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤EtC �H(s) +c ∥2 +F +≤ +1 +p2 +1p2 +2 +T +� +t=1 +∥E⊤ +t �R(s+1)∥2 +F +T +� +t=1 +∥E⊤ +t �R(s)∥2 +F ∥�C(s) − C �H(s) +c ∥2 +F + +1 +p2 +1p2 +2 +∥ +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤EtC∥2 +F ∥ �H(s) +c ∥2 +F += Op +� +T p1 + T 2 +p2 ++ T 2p2w(s) +c ++ T 2p1p2w(s) +r w(s) +c ++ T 2p2 +1p2w(s) +r w(s+1) +r +w(s) +c ++ T p2 +1w(s) +r w(s+1) +r ++ T 2p2 +1 +p2 +w(s) +r w(s+1) +r +� +. +Let Y(s+1) = +��T +t=1 �F(s+1)⊤ +t +�R(s+1)⊤Xt +� ��T +t=1 X⊤ +t �R(s+1)�F(s+1) +t +� +, then Y(s+1) = �4 +i=1 +�4 +j=1 ∆(s+1) +i +∆(s+1) +j +, +15 + +we can prove +���� +� +Y(s+1)�−1/2���� = Op +� +1 +T 2p2 +1p2 +� +. +�C(s+1) = √p2 +� T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +� �� T +� +t=1 +�F(s+1)⊤ +t +�R(s+1)⊤Xt +� � T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +��−1/2 += √p2 +� T +� +t=1 +X⊤ +t �R(s+1)�F(s+1) +t +� � +Y(s+1)�−1/2 += C +� +1 +p1√p2 +� T +� +t=1 +F⊤ +t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s) +� � +Y(s+1)�−1/2 +� ++ +1 +p1√p2 +� T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤RFtC⊤ �C(s) +� � +Y(s+1)�−1/2 ++ +1 +p1√p2 +� T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤Et �C(s) +� � +Y(s+1)�−1/2 ++ +1 +p1√p2 +� T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤Et �C(s) +� � +Y(s+1)�−1/2 +. +Denote �H(s+1) +c += +1 +p1√p2 +��T +t=1 F⊤ +t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s)� � +Y(s+1)�−1/2, then +�C(s+1) − C �H(s+1) +c += +1 +p1√p2 +� T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤RFtC⊤ �C(s) +� � +Y(s+1)�−1/2 ++ +1 +p1√p2 +� T +� +t=1 +CF⊤ +t R⊤ �R(s+1) �R(s)⊤Et �C(s) +� � +Y(s+1)�−1/2 ++ +1 +p1√p2 +� T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤Et �C(s) +� � +Y(s+1)�−1/2 += √p2(∆(s+1) +2 ++ ∆(s+1) +3 ++ ∆(s+1) +4 +)(Y(s+1))−1/2. +thus +1 +p2 +∥�C(s+1) − C �H(s+1) +c +∥2 +F = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2 ++ γ(s+1) +c +� +, +where γ(s+1) +c += +1 +T p2 +w(s) +r ++ 1 +T w(s+1) +r ++ 1 +p2 +1 +w(s) +c ++ 1 +p1 +w(s) +r w(s) +c ++ 1 +T w(s) +r w(s) +c ++ w(s) +r w(s+1) +r +w(s) +c ++ 1 +p2 +2 +w(s) +r w(s+1) +r +. +Similar to the proof the row-consistency of �R(s+1), for any j ∈ [p2], we can obtain +∥ � +C(s+1) +j· +− �H(s+1)⊤ +c +Cj·∥2 +2 = Op +� 1 +T p1 ++ +1 +p2 +1p2 +2 ++ γ(s+1) +c +� +, +where γ(s+1) +c += +1 +T p2 +w(s) +r ++ 1 +T w(s+1) +r ++ 1 +p2 +1 +w(s) +c ++ 1 +p1 +w(s) +r w(s) +c ++ 1 +T w(s) +r w(s) +c ++ w(s) +r w(s+1) +r +w(s) +c ++ 1 +p2 +2 +w(s) +r w(s+1) +r +. +16 + +S1.7 +Proof of Theorem 3.5 +Proof. By definition, +�F(s+1) +t += +1 +p1p2 +�R(s)⊤Xt �C(s) = +1 +p1p2 +�R(s)⊤RFtC⊤ �C(s) + +1 +p1p2 +�R(s)⊤Et �C(s) += +1 +p1p2 +�R(s)⊤ � +R − �R(s)( �H(s) +r )−1 + �R(s)( �H(s) +r )−1� +Ft +� +C − �C(s)( �H(s) +c )−1 + �C(s)( �H(s) +c )−1�⊤ �C(s) ++ +1 +p1p2 +� +�R(s) − R �H(s) +r ++ R �H(s) +r +�⊤ +Et +� +�C(s) − C �H(s) +c ++ C �H(s) +c +� +. +Note that �R(s)⊤ �R(s) = p1Ik1, �C(s)⊤ �C(s) = p2Ik2, then +�F(s+1) +t +− ( �H(s) +r )−1Ft +� +( �H(s) +c )−1�⊤ += +1 +p1p2 +�R(s)⊤ � +R − �R(s)( �H(s) +r )−1� +Ft +� +C − �C( �H(s) +c )−1�⊤ �C(s) ++ 1 +p1 +�R(s)⊤ � +R − �R(s)( �H(s) +r )−1� +Ft +� +( �H(s) +c )−1�⊤ ++ 1 +p2 +( �H(s) +r )−1Ft +� +C − �C( �H(s) +c )−1�⊤ �C(s) ++ +1 +p1p2 +� +�R(s) − R �H(s) +r +�⊤ +Et +� +�C(s) − C �H(s) +c +� ++ +1 +p1p2 +� +�R(s) − R �H(s) +r +�⊤ +EtC �H(s) +c ++ +1 +p1p2 +�H(s)⊤ +r +R⊤Et +� +�C(s) − C �H(s) +c +� ++ +1 +p1p2 +�H(s)⊤ +r +R⊤EtC �H(s) +c . +Therefore, by Cauchy-Schwartz inequality, Lemma S2.6 and Theorem 3.4, we have +�����F(s+1) +t +− ( �H(s) +r )−1Ft +� +( �H(s) +c )−1�⊤���� +F += Op + + +� +w(s) +r +√p2 ++ +� +w(s−1) +r +√T p2 ++ +� +w(s) +c +√p1 ++ +� +w(s−1) +c +√T p1 ++ +1 +√p1p2 ++ γ(s+1) +f + + , +where γ(s+1) +f += +� +w(s) +r w(s) +c ++ +� +w(s−1) +r +w(s−1) +c +√ +T ++ +� +w(s−1) +r +w(s−1) +c +p1 ++ +� +w(s−1) +r +√p1p2 +. +Next, for any t, i, j +�S(s+1) +t,ij +− St,ij = � +R(s)⊤ +i· +�F(s+1) +t +� +C(s) +j· − R⊤ +i· FtCj· += ( � +R(s) +i· − �H(s)⊤ +r +Ri· + �H(s)⊤ +r +Ri·)⊤�F(s+1) +t +( � +C(s) +j· − �H(s)⊤ +c +Cj· + �H(s)⊤ +c +Cj·) − R⊤ +i· FtCj· += ( � +R(s) +i· − �H(s)⊤ +r +Ri·)⊤�F(s+1) +t +( � +C(s) +j· − �H(s)⊤ +c +Cj·) + R⊤ +i· �H(s) +r �F(s+1) +t +( � +C(s) +j· − �H(s)⊤ +c +Cj·) ++ ( � +R(s) +i· − �H(s)⊤ +r +Ri·)⊤�F(s+1) +t +�H(s)⊤ +c +Cj· + R⊤ +i· ( �H(s) +r �F(s+1) +t +�H(s)⊤ +c +− Ft)Cj·. +Then, by Cauchy-Schwartz inequality, Theorem 3.1, Theorem 3.2 and the consistency of the estimators for +the factor matrices, we have +|�S(s+1) +t,ij +− St,ij| = Op + + +� +w(s) +r ++ +� +w(s−1) +r +√T p2 ++ +� +w(s) +c ++ +� +w(s−1) +c +√T p1 ++ +1 +√p1p2 ++ γ(s+1) + + , +17 + +where γ(s+1) = +� +w(s−1) +r +w(s−1) +c +√ +T ++ +� +w(s−1) +r +w(s−1) +c +p1 ++ +� +w(s−1) +r +√p1p2 +. +S2 +Propositions and Lemmas +In this section, we first give some propositions and lemmas which are essential for the proofs of main theorems. +Proposition S2.1. Suppose m1 ≥ k1, m2 ≥ k2 and T, p1, p2 → ∞.Under Assumptions 2.1-2.3, we have +• λmin( 1 +T +�F⊤�F) ≥ c(p1p2)−1 with probability approaching one for some c > 0, +Proof. It is easy to get that +�F = FH⊤ + E, +where E = EW/(p1p2), W = W2 ⊗ W1. Let +∆ := 1 +T HF⊤E + 1 +T E⊤FH⊤ + 1 +T EE⊤E + 1 +T (E⊤E − EE⊤E). +Then we have +1 +T +�F⊤�F = 1 +T HF⊤FH⊤ + ∆. +By the assumption +λmin +� 1 +T EE⊤E +� += λmin +� +1 +T +T +� +t=1 +EVec(Et)Vec(Et)⊤ +� +≥ c0, +and the property of the Kronecker product, we can get +λmin +� 1 +T EE⊤E +� +≥ λmin +� 1 +T EE⊤E +� +λmin +� +1 +p2 +1p2 +2 +W⊤W +� +≥ c0(p1p2)−1 +for some c0 > 0. In addition, in the following Lemma S2.2 (4), we show that +∥ 1 +T (E⊤E − EE⊤E)∥2 = Op +� +1 +p1p2 +√ +T +� +. +Therefore, the inequality ∥ 1 +T (E⊤E − EE⊤E)∥2 ≤ 1 +2λmin( 1 +T EE⊤E) hold with large probability. +We now +continue the argument conditioning on this event. +Let v be the unit vector such that v⊤ 1 +T +�F⊤�Fv = λmin( 1 +T +�F⊤�F), as v⊤ 1 +T +�F⊤�Fv = v⊤ 1 +T HF⊤FH⊤v + +v⊤∆v, then we have +λmin( 1 +T +�F⊤�F) ≥ 1 +T v⊤HF⊤FH⊤v + 2 +T v⊤HF⊤Ev + +c0 +2p1p2 +. +If v⊤H = 0, then λmin( 1 +T +�F⊤�F) ≥ +c0 +2p1p2 +. If v⊤H ̸= 0, we have 1 +T v⊤HF⊤FH⊤v > 0 with large probability +because 1 +T F⊤F is positive definite. Let +α2 +v = 1 +T v⊤HF⊤FH⊤v, +X = ( +α2 +v +T p1p2 +)−1/22v⊤ 1 +T HF⊤Ev, +2v⊤ 1 +T HF⊤Ev = X +� +α2 +v +T p1p2 +. +18 + +Then +λmin( 1 +T +�F⊤�F) ≥ α2 +v + X +� +α2 +v +T p1p2 ++ +c0 +2p1p2 +. +First, we prove that X = Op(1). By the Assumption 2.3, it holds that λmin( 1 +T F⊤F) > c > 0, then we +have +α2 +v ≥ λmin( 1 +T F⊤F)v⊤HH⊤v > c∥v⊤H∥2 +2. +By Lemma S2.2 (3) in the following, we have ∥ 1 +T F⊤E∥2 +2 = Op +� +1 +T p1p2 +� +and as a result, +|X|2 ≤ 4T p1p2α−2 +v ∥v⊤H∥2 +2∥ 1 +T F⊤E∥2 +2 ≤ Op(1)α−2 +v ∥v⊤H∥2 +2 ≤ Op(1)c−1∥v⊤H∥−2 +2 ∥v⊤H∥2 +2 = Op (1) . +Then we consider two cases. +Case one: α2 +v ≤ 4|X| +� +α2 +v +T p1p2 +. Then |αv| ≤ 4|X| +1 +√T p1p2 +and +λmin( 1 +T +�F⊤�F) ≥ +c0 +2p1p2 +− |X||αv| +1 +√T p1p2 +≥ +c0 +2p1p2 +− 4|X|2 +1 +T p1p2 +≥ +c0 +4p1p2 +, +where the last inequality holds with probability approaching to 1 by the fact that X = Op(1) and T → ∞. +Case two: α2 +v > 4|X| +� +α2 +v +T p1p2 +, then +λmin( 1 +T +�F⊤�F) ≥ α2 +v − |X| +� +α2 +v +T p1p2 ++ +c0 +2p1p2 +≥ 3 +4α2 +v + +c0 +2p1p2 +≥ +c0 +2p1p2 +. +From what has been discussed above, we conclude that with probability approaching one, we have +λmin( 1 +T +�F⊤�F) > c0/(p1p2). +Lemma S2.2. For any m1 ≥ 1 and m2 ≥ 1, (note that m1, m2 can be either smaller, equal to or larger +than k1, k2), +(1) ∥E(E⊤ +t Et)∥2 = O +� +1 +p1p2 +� +and ∥Et∥2 = Op +� +1 +√p1p2 +� +, t ∈ [T ], +(2) ∥E∥2 = Op +�� T +p1p2 +� +, +(3) E∥ 1 +T F⊤E∥2 +2 ≤ O +� +1 +T p1p2 +� +, +(4) ∥ 1 +T (E⊤E − EE⊤E)∥2 ≤ Op +� +1 +p1p2 +√ +T +� +. +Proof. (1) By Assumption 2.3, we have +∥E(EtE⊤ +t )∥2 ≤ c. +19 + +Thus, +∥E(E⊤ +t W1W⊤ +1 Et)∥2 ≤ tr(EE⊤ +t W1W⊤ +1 Et) ≤ E(tr(E⊤ +t W1W⊤ +1 Et)) += tr(W⊤ +1 E(EtE⊤ +t )W1) ≤ m1∥W1∥2 +2∥E(EtE⊤ +t )∥2 ≤ cm1p1, +where the penultimate inequality is derived as follows: let vi be the i-th eigenvector of W⊤ +1 E(EtE⊤ +t )W1, +then +tr(W⊤ +1 E(EtE⊤ +t )W1) = +m1 +� +i=1 +v⊤ +i W⊤ +1 E(EtE⊤ +t )W1vi ≤ ∥E(EtE⊤ +t )∥2 +m1 +� +i=1 +∥W1vi∥2 +2 ≤ m1∥E(EtE⊤ +t )∥2∥W1∥2 +2. +∥E(E⊤ +t Et)∥2 = +1 +p2 +1p2 +2 +∥E(W⊤ +2 E⊤ +t W1W⊤ +1 EtW2)∥2 += +1 +p2 +1p2 +2 +∥W⊤ +2 E(E⊤ +t W1W⊤ +1 Et)W2∥2 +≤ +1 +p2 +1p2 +2 +∥W2∥2 +2∥E(E⊤ +t W1W⊤ +1 Et)∥2, +which imply +∥E(E⊤ +t Et)∥2 ≤ +c +p1p2 +. +Thus, we have E∥E2 +t∥2 ≤ tr(EE⊤ +t Et) ≤ m2∥EE⊤ +t Et∥2 ≤ cm1m2 +p1p2 +. +(2) It holds that E = +1 +p1p2 +EW, W = W2 ⊗ W1. By the Assumption 2.3, +��� 1 +T E(E⊤E) +��� +2 = +��� 1 +T +T +� +t=1 +EVec(Et)Vec(Et)⊤��� +2 += ∥EVec(Et)Vec(Et)⊤∥2 +≤ E∥E +� +Vec(Et)Vec(Et)⊤|F +� +∥2 ≤ c. +Thus +E +��� +1 +p1p2 +EW +��� +2 +2 ≤ +1 +p2 +1p2 +2 +trE(W⊤E⊤EW) ≤ +1 +p2 +1p2 +2 +∥W∥2 +F∥EE⊤E∥2 ≤ cT +p1p2 +. +(3) By the Assumption 2.3, we have +1 +T +T +� +s=1 +T +� +T =1 +E(∥Ft∥F ∥Fs∥F )tr(E(EtE⊤ +s |F)) < c, +20 + +then we can obtain +E∥ 1 +T F⊤E∥2 +2 = +1 +T 2p2 +1p2 +2 +E∥F⊤EW∥2 +2 = +1 +T 2p2 +1p2 +2 +E∥ +T +� +t=1 +W⊤Vec(Et)Vec(Ft)⊤∥2 +2 += +1 +T 2p2 +1p2 +2 +E +� +E +� +∥ +T +� +t=1 +W⊤Vec(Et)Vec(Ft)⊤∥2 +2 +� �����F +� +≤ +1 +T 2p2 +1p2 +2 +E +� +E +� +tr( +T +� +t=1 +T +� +s=1 +Vec(Ft)Vec(Et)⊤WW⊤Vec(Es)Vec(Fs)⊤) +� �����F +� += +1 +T 2p2 +1p2 +2 +E +� +tr +� T +� +t=1 +T +� +s=1 +Vec(Ft)E +� +Vec(Et)⊤WW⊤Vec(Es) +���F +� +Vec(Fs)⊤ +�� += +1 +T 2p2 +1p2 +2 +E +� +tr +� T +� +t=1 +T +� +s=1 +Vec(Ft)Vec(Fs)⊤E +� +Vec(Et)⊤WW⊤Vec(Es) +���F +��� +≤ +1 +T 2p2 +1p2 +2 +k1 +� +i=1 +k2 +� +j=1 +T +� +t=1 +T +� +s=1 +E +� +ft,ijfs,ijE +� +Vec(Et)⊤WW⊤Vec(Es) +���F +�� += +1 +T 2p2 +1p2 +2 +k1 +� +i=1 +k2 +� +j=1 +T +� +t=1 +T +� +s=1 +E +� +ft,ijfs,ijtr +� +W⊤E +� +Vec(Es)Vec(Et)⊤���F +� +W +�� +≤ +1 +T 2p2 +1p2 +2 +k1 +� +i=1 +k2 +� +j=1 +T +� +t=1 +T +� +s=1 +E +� +|ft,ijfs,ij|∥W∥2 +F∥E +� +Vec(Es)Vec(Et)⊤���F +�� +≤ +c +T 2p1p2 +T +� +t=1 +T +� +s=1 +E∥Ft∥F ∥∥Fs∥F E +� +Vec(Es)Vec(Et)⊤���F +� +∥ ≤ +C +T p1p2 +. +Thus, E∥ 1 +T F⊤E∥2 +2 ≤ O +� +1 +T p1p2 +� +. +(4) By the Assumption 2.3 (4), we have +E∥ 1 +T (E⊤E − EE⊤E)∥2 +2 ≤ E∥ 1 +T (E⊤E − EE⊤E)∥2 +F +≤ +m1m2 +� +k,q=1 +E + + +1 +T p2 +1p2 +2 +T +� +t=1 +p1p2 +� +i,j=1 +wikwjq(etietj − Eetietj) + + +2 += +m1m2 +� +k,q=1 +E + + +1 +T p2 +1p2 +2 +T +� +t=1 +p1p2 +� +i,j=1 +wikwjq (Vec(Et)iVec(Et)j − EVec(Et)iVec(Et)j) + + +2 +≤ +1 +T p2 +1p2 +2 +m1m2 +� +k,q=1 +1 +T p2 +1p2 +2 +T +� +t,s=1 +p1p2 +� +i,j,u,v=1 +��wikwjqwukwvq +����Cov (Vec(Et)iVec(Et)j, Vec(Es)uVec(Es)v) +�� +≤ +c +T p2 +1p2 +2 +1 +T p2 +1p2 +2 +T +� +t,s=1 +p1 +� +i1,j1,u1,v1=1 +p2 +� +i2,j2,u2,v2=1 +|Cov(et,i1i2et,j1j2, es,u1u2es,v1v2)| ≤ +c +T p2 +1p2 +2 +. +Lemma S2.3. Under Assumption 2.1 (1), Assumption 3.2-3.4, as min{T, p1, p2} → ∞, we have +21 + +(1) �T +t=1 E∥E⊤ +t R∥2 +F = O (T p1p2) , �T +t=1 E∥EtC∥2 +F = O (T p1p2) , E∥ �T +t=1 E⊤ +t Ft∥2 +F = O (T p1p2), +(2) E∥ �T +t=1 FtC⊤E⊤ +t ∥2 +F = O (T p1p2) , E∥ �T +t=1 F⊤ +t R⊤Et∥2 +F = O (T p1p2), +E∥ �T +t=1 FtC⊤E⊤ +t R∥2 +F = O (T p1p2) , E∥ �T +t=1 F⊤ +t R⊤EtC∥2 +F = O (T p1p2), +(3) E∥ �T +t=1 FtW⊤ +2 E⊤ +t W1∥2 +F = O (T p1p2) , E∥ �T +t=1 E⊤ +t W1Ft∥2 +F = O (T p1p2), +E∥ �T +t=1 FtW⊤ +2 E⊤ +t ∥2 +F = O (T p1p2) , E∥ �T +t=1 W⊤ +1 EtW2∥2 +F = O (T p1p2), +(4) E∥ �T +t=1 R⊤EtW2F⊤ +t ∥2 +F = O (T p1p2) , E∥ �T +t=1 R⊤EtW2∥2 +F = O (T p1p2) +(5) E∥ �T +t=1 EtW2W⊤ +2 E⊤ +t W1∥2 +F = O +� +T p2 +1p3 +2 + T 2p1p2 +2 +� +. +Proof. Assume m1 = k1 = 1, m2 = k2 = 1. +(1) By Assumption 3.2, we can get +E∥E⊤ +t R∥2 +F = +p2 +� +j=1 +E( +p1 +� +i=1 +et,ijri)2 = +p2 +� +j=1 +p1 +� +i1,i2=1 +E(ri1ri2et,i1jet,i2j) = O (p1p2) . +Similarly, �T +t=1 E∥EtC∥2 +F∥2 +F = O (T p1p2). By Assumption 3.4(2), +E∥ +T +� +t=1 +E⊤ +t Ft∥2 +F = +p1 +� +i=1 +p2 +� +j=1 +E( +T +� +t=1 +Ftet,ij)2 = T +p1 +� +i=1 +p2 +� +j=1 +E(ξi,j)2 = O (T p1p2) . +(2) The results hold directly by Assumption 3.4 (1). +(3) By Assumption 2.1 (1) and Assumption 3.4 (2), we have +E∥ +T +� +t=1 +FtW⊤ +2 E⊤ +t W1∥2 +F = E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +Ftw1,iet,ijw2,j)2 = T E( +p1 +� +i=1 +p2 +� +j=1 +ξi,jw1,iw2,j)2 += T +p1 +� +i1,i2=1 +p2 +� +j1,j2=1 +E(ξi1,j1ξi2,j2)w1,i1w1,i2w2,j1w2,j2 = O (T p1p2) . +E∥ +T +� +t=1 +E⊤ +t W1Ft∥2 +F = +p2 +� +j=1 +E( +T +� +t=1 +p1 +� +i=1 +et,ijw1,jFt)2 = T +p2 +� +j=1 +E( +p1 +� +i=1 +ξi,jw1,j)2 += T +p2 +� +j=1 +p1 +� +i,i′=1 +E(ξi,jξi′,j)w1,iw1,i′ = O (T p1p2) . +Analogously, E∥ �T +t=1 FtW⊤ +2 E⊤ +t ∥2 +F = O(T p1p2). By Assumption 3.2 (2), we can also obtain +E∥ +T +� +t=1 +W⊤ +1 EtW2∥2 +F = E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +w1,iw2,jet,ij)2 = +T +� +s,t=1 +p1 +� +i1,i2=1 +p2 +� +j1,j2=1 +w1,i1w1,i2w2,j1w2,j2E(et,i1j1es,i2j2)2 += O (T p1p2) . +22 + +(4) +E∥ +T +� +t=1 +R⊤EtW2Ft∥2 +F = E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +riw2,jet,ijFt)2 = T E( +p1 +� +i=1 +p2 +� +j=1 +riw2,jξi,j)2 += T +p1 +� +i1,i2=1 +p2 +� +j1,j2=1 +ri1ri2w2,j1w2,j2E(ξi1,j1ξi2,j2) = O (T p1p2) . +E∥ +T +� +t=1 +R⊤EtW2∥2 +F = E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +riw2,jet,ij)2 = +T +� +s,t=1 +p1 +� +i1,i2=1 +p2 +� +j1,j2=1 +ri1ri2w2,j1w2,j2E(et,i1j1es,i2j2)2 += O (T p1p2) . +(5) Note that +E∥ +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1∥2 +F ≤ ∥W2∥2 +F∥ +T +� +t=1 +W⊤ +1 EtW2Et∥2 +F ≤ ∥W2∥2 +F +p1 +� +i=1 +p2 +� +j=1 +E∥ +T +� +t=1 +W⊤ +1 EtW2et,ij∥2, +while for any i, j, by Assumption 3.2, +E∥ +T +� +t=1 +W ⊤ +1 EtW2et,ij∥2 = E( +T +� +t=1 +p1 +� +i′ +p2 +� +j′ +w1,i′et,i′j′w2,j′et,ij)2 +≲ +T +� +t,s=1 +p1 +� +i1,i2 +p2 +� +j1,j2=1 +|Cov(et,ijet,i1j1, es,ijes,i2j2)| + ( +T +� +t=1 +p1 +� +i′ +p2 +� +j′ +|Eet,i′j′et,ij|)2 += O +� +T p1p2 + T 2� +. +Lemma S2.4. Under Assumption 2.1 (1), Assumption 2.2 and Assumption 3.1-3.4, as min{T, p1, p2} → ∞, +we have +∥ +T +� +s=1 +E⊤ +s ( �R(1) − R �H(1) +r )Fs∥2 +F = Op +� +p2 +1 +p2ν2 +min(H2) + +p1 +ν2 +min(H1)ν4 +min(H2) + +T +p2ν2 +min(H1)ν4 +min(H2) +� +. +Proof. By the proof in Theorem 3.1, �R(1) − R �H(1) +r += II + III + IV, then +T +� +s=1 +E⊤ +s ( �R(1) − R �H(1) +r )Fs = +T +� +s=1 +E⊤ +s (II + III + IV)Fs. +23 + +For the first term, by LemmaS2.3 (2) and (3), we get +∥ +T +� +t=1 +E⊤ +s IIFs∥2 +F = +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s R[( +T +� +t=1 +FtC⊤W2W⊤ +2 E⊤ +t W1)Z−1/2]Fs∥2 +F +≤ +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s RFs∥2 +F∥( +T +� +t=1 +FtC⊤W2W⊤ +2 E⊤ +t W1)Z−1/2∥2 +F +≤ +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s RFs∥2 +F∥ +T +� +t=1 +FtW⊤ +2 E⊤ +t W1∥2 +F ∥Z−1/2∥2 +F ∥C⊤W2∥2 +2 += Op +� +1 +ν2 +min(H1)ν2 +min(H2) +� +. +For the second term, +∥ +T +� +s=1 +E⊤ +s IIIFs∥2 +F = +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s [( +T +� +t=1 +EtW2W⊤ +2 CF⊤ +t R⊤W1)Z−1/2]Fs∥2 +F +≤ +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s ( +T +� +t=1 +EtW2F⊤ +t )Fs∥2 +F∥W⊤ +2 C∥2 +F ∥R⊤W1∥2 +F ∥Z∥2 +F += Op +� +p2 +1 +p2ν2 +min(H2) + +p1 +ν2 +min(H2) +� +, +where +E∥ +T +� +s=1 +E⊤ +s ( +T +� +t=1 +EtW2F⊤ +t )Fs∥2 +F = +p2 +� +j=1 +E( +T +� +s,t=1 +p1 +� +i=1 +p2 +� +j1=1 +w2,j1FsFtes,ijet,ij1)2 = T 2 +p2 +� +j=1 +E( +p1 +� +i=1 +p2 +� +j1=1 +w2,j1ξi,jξi,j1)2 += T 2 +p2 +� +j=1 + +( +p1 +� +i=1 +p2 +� +j1=1 +Eξi,jξi,j1)2 + +p1 +� +i1,i2=1 +p2 +� +j1j2=1 +Cov(ξi1,jξi1,j1, ξi2,jξi2,j2) + + += O +� +T 2p2 +1p2 + T 2p1p2 +2 +� +. +For the third term, by lemma S2.3 (1) and (5), we have +∥ +T +� +s=1 +E⊤ +s IVFs∥2 +F = +1 +T 2p1p2 +2 +∥ +T +� +s=1 +Es[( +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1)Z−1/2]Fs∥2 +F +≤ +1 +T 2p1p2 +2 +∥ +T +� +s=1 +E⊤ +s Fs∥2 +F ∥( +T +� +t=1 +EtW2W⊤ +2 E⊤ +t W1)Z−1/2∥2 +F += Op +� +p1 +ν2 +min(H1)ν4 +min(H2) + +T +p2ν2 +min(H1)ν4 +min(H2) +� +. +As a result, +∥ +T +� +s=1 +E⊤ +s ( �R(1) − R �H(1) +r )Fs∥2 +F = Op +� +p2 +1 +p2ν2 +min(H2) + +p1 +ν2 +min(H1)ν4 +min(H2) + +T +p2ν2 +min(H1)ν4 +min(H2) +� +. +24 + +Lemma S2.5. Under Assumption 2.1 (1), Assumption 2.2 and Assumption 3.1-3.4, take �R(1), �C(1) as the +resultant estimators from the one-step iteration, as min{T, p1, p2} → ∞, then we have +���R⊤( �R(1) − R �H(1) +r ) +��� +F = Op +� +√p1 +√T p2νmin(H1)νmin(H2) + +1 +p2νmin(H1)ν2 +min(H2) +� +, +���C⊤(�C(1) − C �H(1) +c ) +��� +F = Op +� +1 +T νmin(H2) + +√p2 +√T p1νmin(H1)νmin(H2) + +1 +p1νmin(H1)νmin(H2) +� +. +Proof. +��R⊤II +�� +F = +1 +T √p1p2 +�����R⊤R( +T +� +t=1 +FtC⊤W2W⊤ +2 E⊤ +t W1)Z−1/2 +����� +F += +√p1 +T p2 +�����( +T +� +t=1 +FtC⊤W2W⊤ +2 E⊤ +t W1)Z−1/2 +����� +F +≤ +√p1 +T p2 +����� +T +� +t=1 +FtW⊤ +2 E⊤ +t W1 +����� +F +���Z−1/2��� +F +��C⊤W2 +�� +F = Op +� +√p1 +√T p2νmin(H1)νmin(H2) +� +, +��R⊤III +�� +F = +1 +T √p1p2 +�����( +T +� +t=1 +R⊤EtW2W⊤ +2 CF⊤ +t R⊤W1)Z−1/2 +����� +F +≤ +1 +T √p1p2 +����� +T +� +t=1 +R⊤EtW2Ft +����� +F +��W⊤ +2 C +�� +F +��R⊤W1 +�� +F +���Z1/2��� +F += Op +� +√p1 +√T p2νmin(H2) +� +, +��R⊤IV +�� +F = +1 +T √p1p2 +�����( +T +� +t=1 +R⊤EtW2W⊤ +2 E⊤ +t W1)Z−1/2 +����� +F +≤ +1 +T √p1p2 +� +� +� +� +����� +T +� +t=1 +R⊤EtW2 +����� +2 +F +× +����� +T +� +t=1 +W⊤ +2 E⊤ +t W1 +����� +2 +F +���Z−1/2��� +F += Op +� +1 +p2νmin(H1)ν2 +min(H2) +� +. +As R⊤( �R(1) − R �H(1) +r ) = R⊤(II + III + IV), combining the above three items, we get +���R⊤( �R(1) − R �H(1) +r ) +��� +F = Op +� +√p1 +√T p2νmin(H1)νmin(H2) + +1 +p2νmin(H1)ν2 +min(H2) +� +. +As for the second formula, by the proof of Theorem 3.2, we have +�C(1) − C �H(1) +c += √p2 +� +δ(1) +2 (Z(1))−1/2 + δ(1) +3 (Z(1))−1/2 + δ(1) +4 (Z(1))−1/2� +. +25 + +For the first term, +√p2 +���C⊤δ(1) +2 (Z(1))−1/2��� +F = +1 +p1√p2 +�����( +T +� +t=1 +C⊤E⊤ +t �R(1)W⊤ +1 RFtC⊤W2)(Z(1))−1/2 +����� +F +≤ +1 +p1√p2 +����� +T +� +t=1 +C⊤E⊤ +t �R(1)Ft +����� +F +��W⊤ +1 R +�� +2 +��C⊤W2 +�� +2 +���(Z(1) +t )−1/2��� +F += Op +� +1 +T νmin(H2) + +1 +T √p1νmin(H1)ν2 +min(H2) + +1 +√T p2p1νmin(H1)ν2 +min(H2) + +√p2 +√T p1 +� +, +where +����� +T +� +t=1 +C⊤E⊤ +t �R(1)Ft +����� +F +≤ +����� +T +� +t=1 +C⊤E⊤ +t ( �R(1) − R �H(1) +r )Ft +����� +F ++ +����� +T +� +t=1 +C⊤E⊤ +t R �H(1) +r Ft +����� +F +≤ +����� +T +� +t=1 +C⊤E⊤ +t Ft +����� +F +��� �R(1) − R �H(1) +r +��� +F + +����� +T +� +t=1 +C⊤E⊤ +t RFt +����� +F +��� �H(1) +r +��� +F += Op +� +p1 +νmin(H2) + +√p1 +νmin(H1)ν2 +min(H2) + +√ +T +√p2νmin(H1)ν2 +min(H2) + +� +T p1p2 +� +. +For the second term, +√p2 +���C⊤δ(1) +3 (Z(1))−1/2��� +F = +√p2 +p1 +�����( +T +� +t=1 +F⊤ +t R⊤ �R(1)W⊤ +1 EtW2)(Z(1))−1/2 +����� +F +≍ √p2 +�����( +T +� +t=1 +F⊤ +t W⊤ +1 EtW2)(Z(1))−1/2 +����� +F +≤ √p2 +����� +T +� +t=1 +F⊤ +t W⊤ +1 EtW2 +����� +F +���(Z(1))−1/2��� +F += Op +� +√p2 +√T p1νmin(H1)νmin(H2) +� +. +For the third term, +√p2 +���C⊤δ(1) +4 (Z(1))−1/2��� +F = +1 +p1√p2 +�����( +T +� +t=1 +C⊤E⊤ +t �R(1)W⊤ +1 EtW2)(Z(1))−1/2 +����� +F +≤ +1 +p1√p2 +� +� +� +� +����� +T +� +t=1 +C⊤E⊤ +t �R(1) +����� +2 +F +× +����� +T +� +t=1 +W⊤ +1 EtW2 +����� +2 +F +���(Z(1))−1/2��� +F += Op +� +1 +√T p1p2νmin(H1)ν2 +min(H2) + +1 +√T p2p1ν2 +min(H1)ν3 +min(H2) + +1 +p1νmin(H1)νmin(H2) +� +, +26 + +where the last equation holds due to +����� +T +� +t=1 +C⊤E⊤ +t �R(1) +����� +2 +F += +����� +T +� +t=1 +C⊤E⊤ +t ( �R(1) − R �H(1) +r ++ R �H(1) +r ) +����� +2 +F +≤ +����� +T +� +t=1 +C⊤E⊤ +t ( �R(1) − R �H(1) +r ) +����� +2 +F ++ +����� +T +� +t=1 +C⊤E⊤ +t R �H(1) +r +����� +2 +F += Op +� +p2 +1 +ν2 +min(H2) + +p1 +ν2 +min(H1)ν4 +min(H2) + T p1p2 +� +. +Hence, +���C⊤(�C(1) − C �H(1) +c ) +��� +F = Op +� +1 +T νmin(H2) + +√p2 +√T p1νmin(H1)νmin(H2) + +1 +p1νmin(H1)νmin(H2) +� +. +Lemma S2.6. Under Assumption 2.1 (1), Assumption 2.2 and Assumption 3.1-3.4, take �R(s+1), �C(s+1) as +the result of a (s + 1)th step iteration, as min{T, p1, p2} → ∞, then we have +∥R⊤( �R(s+1) − R �H(s+1) +r +)∥F = Op + + +√p1 +√T p2 ++ p1 +� +w(s) +r +√T p2 ++ +� +w(s) +r +p2 ++ +� +p1w(s) +c +√ +T ++ p1 +� +w(s) +r w(s) +c +√ +T ++ +� +w(s) +r w(s) +c + + , +∥C⊤(�C(s+1)−C �H(s+1) +c +)∥F = Op + + +√p2 +√T p1 ++ 1 +p1 ++ +� +w(s) +r +√p1 ++ +� +p2w(s) +r +√ +T ++ p2 +� +w(s) +c +√T p1 ++ p2 +� +w(s) +r w(s) +c +√ +T ++ +� +w(s) +r w(s+1) +r + + . +Proof. By the proof of Theorem 3.4, +�R(s+1) − R �H(s+1) +r += √p1(δ(s+1) +2 ++ δ(s+1) +3 ++ δ(s+1) +4 +)(Z(s+1))−1/2. +By Lemma S2.9 (1) and (2), we have +���√p1R⊤δ(s+1) +2 +(Z(s+1))−1/2��� +F ≍ +����� +√p1 +� T +� +t=1 +Ft �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 +����� +F +≤ √p1 +����� +T +� +t=1 +Ft �C(s)⊤E⊤ +t �R(s) +����� +F +���� +� +Z(s+1)�−1/2���� +F += Op + +p1 +� +w(s) +r w(s) +c +√ +T ++ p1 +� +w(s) +r +√T p2 ++ +� +p1w(s) +c +√ +T ++ +√p1 +√T p2 + + , +where the last equation holds due to +����� +T +� +t=1 +Ft �C(s)⊤E⊤ +t �R(s) +����� +F +≤ +����� +T +� +t=1 +Ft �C(s)⊤E⊤ +t +����� +F +��� �R(s) − R �H(s) +r +��� +F + +����� +T +� +t=1 +Ft �C(s)⊤E⊤ +t R +����� +F +��� �H(ss) +r +��� +F . +27 + +���√p1R⊤δ(s+1) +3 +(Z(s+1))−1/2��� +F ≍ √p1 +����� +� T +� +t=1 +R⊤Et �C(s)F⊤ +t +� � +Z(s+1)�−1/2 +����� +F +≤ √p1 +����� +T +� +t=1 +R⊤Et �C(s)F⊤ +t +����� +F +���� +� +Z(s+1)�−1/2���� +F += Op + + +� +p1w(s) +c +√ +T ++ +√p1 +√T p2 + + . +By the fact that +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t �R(s) +����� +F +≤ +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t ( �R(s) − R �H(s) +r ) +����� +F ++ +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t R �H(s) +r +����� +F +≤ +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t +����� +F +��� �R(s) − R �H(s) +r +��� +F + +����� +T +� +t=1 +R⊤Et �C(s) +����� +2 +F +��� �H(s) +r +��� +F , +and Lemma S2.8, we can get +���√p1R⊤δ(s+1) +4 +(Z(s+1))−1/2��� +F = +����� +1 +√p1p2 +� T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t �R(s) +� � +Z(s+1)�−1/2 +����� +F +≤ +1 +√p1p2 +����� +T +� +t=1 +R⊤Et �C(s) �C(s)⊤E⊤ +t �R(s) +����� +F +���� +� +Z(s+1)�−1/2���� +F += Op + + +� +p1w(s) +r +√T p2 ++ +� +w(s) +r +p2 ++ +� +w(s) +r w(s) +c ++ +� +p1w(s) +r w(s) +c +√ +T ++ +� +w(s) +c +√T p1p2 ++ +1 +√T p1p2p2 + + . +Hence, +∥R⊤( �R(s+1) − R �H(s+1) +r +)∥F = Op + + +√p1 +√T p2 ++ p1 +� +w(s) +r +√T p2 ++ +� +w(s) +r +p2 ++ +� +p1w(s) +c +√ +T ++ p1 +� +w(s) +r w(s) +c +√ +T ++ +� +w(s) +r w(s) +c + + . +And by a similar proof, we can get +∥C⊤(�C(s+1)−C �H(s+1) +c +)∥F = Op + + +√p2 +√T p1 ++ 1 +p1 ++ +� +w(s) +r +√p1 ++ +� +p2w(s) +r +√ +T ++ p2 +� +w(s) +c +√T p1 ++ p2 +� +w(s) +r w(s) +c +√ +T ++ +� +w(s) +r w(s+1) +r + + . +Lemma S2.7. Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed. If Assumptions 1–6 +hold, then we have +(1) +T +� +t=1 +∥Et �C(s)∥2 +F = Op +� +T p1p2 + T p1p2 +2w(s) +c +� +; +(2) +T +� +t=1 +∥E⊤ +t �R(s)∥2 +F = Op +� +T p1p2 + T p2 +1p2w(s) +r +� +. +28 + +Proof. (1) As �T +t=1 ∥EtC∥2 +F = �T +t=1 +�p1 +i=1(�p2 +k=1 et,ikck)2 ≤ c �T +t=1 +�p1 +i=1 +�p2 +k,k′=1 et,iket,ik′ = Op (T p1p2) , +�T +t=1 ∥Et∥2 +F = �T +t=1 +�p1 +i=1 +�p2 +j=1 e2 +t,ij = Op (T p1p2), then +T +� +t=1 +∥Et �C(s)∥2 +F ≤ +T +� +t=1 +∥EtC �H(s) +c ∥2 +F + +T +� +t=1 +∥Et(�C(s) − C �H(s) +c )∥2 +F = Op +� +T p1p2 + T p1p2 +2w(s) +c +� +. +(2) �T +t=1 ∥R⊤Et∥2 +F = �T +t=1 +�p2 +j=1(�p1 +i=1 riet,ij)2 ≤ c �T +t=1 +�p2 +j=1 +�p1 +i1,i2=1 et,i1jet,i2j = Op (T p1p2), +T +� +t=1 +∥E⊤ +t �R(s)∥2 +F ≤ +T +� +t=1 +∥E⊤ +t ( �R(s) − R �H(s) +r )∥2 +F + +T +� +t=1 +∥E⊤ +t R �H(s) +r ∥2 +F = Op +� +T p2 +1p2w(s) +r ++ T p1p2 +� +. +Lemma S2.8. Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed. If Assumption 2.1 +(1), Assumption 2.2 and Assumption 3.1-3.4 hold, then we have +• (1) +∥ +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t R∥2 +F = Op +� +T p2 +1p3 +2 + T 2p1p2 +2 + (T 2p1p4 +2 + T p2 +1p4 +2)w(s)2 +c +� +; +• (2) +∥ +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤EtC∥2 +F = Op +� +T p3 +1p2 +2 + T 2p2 +1p2 + (T p4 +1p2 +2 + T 2p4 +1p2)w(s) +r w(s+1) +r +� +. +Proof. (1) For simplicity, we fix k1 = k2 = m1 = m2 = 1. Note that +∥ +T +� +t=1 +Et �C(s) �C(s)⊤E⊤ +t R∥2 +F ≤ ∥ +T +� +t=1 +EtC �H(s) +c +�H(s)⊤ +c +C⊤E⊤ +t R∥2 +F + ∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c ) �H(s)⊤ +c +C⊤E⊤ +t R∥2 +F ++ ∥ +T +� +t=1 +EtC �H(s) +c (�C(s) − C �H(s) +c )⊤E⊤ +t R∥2 +F + ∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c )(�C(s) − C �H(s) +c )⊤E⊤ +t R∥2 +F. +For the first term, +∥ +T +� +t=1 +EtC �H(s) +c +�H(s)⊤ +c +C⊤E⊤ +t R∥2 +F ≤ ∥ +T +� +t=1 +EtC⊤E⊤ +t R∥2 +F ∥C∥2 +F∥ �H(s) +c ∥4 +F +≤ p2 +p1 +� +i′=1 +p2 +� +j′=1 +( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +ricjet,ijet,i′j′)2 += Op +� +T p2 +1p3 +2 + T 2p1p2 +2 +� +, +where +E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +ricjet,ijet,i′j′)2 ≤ +T +� +t,s=1 +p1 +� +i1,i2=1 +p2 +� +j1,j2=1 +Cov(et,i1j2et,i′j′, es,i2j2es,i′j′) + ( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +Eet,ijet,i′j′)2 += O +� +T p1p2 + T 2� +. +29 + +For the second term, +∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c ) �H(s)⊤ +c +C⊤E⊤ +t R∥2 +F ≤ ∥ +T +� +t=1 +EtC⊤E⊤ +t R∥2 +F ∥�C(s) − C �H(s) +c ∥2 +F ∥ �H(s) +c ∥2 +F += Op +� +(T p2 +1p3 +2 + T 2p1p2 +2)w(s) +c +� +. +For the third term, +∥ +T +� +t=1 +EtC �H(s) +c (�C(s) − C �H(s) +c )⊤E⊤ +t R∥2 +F ≤ ∥ +T +� +t=1 +EtCR⊤Et∥2 +F ∥�C(s) − C �H(s) +c ∥2 +F ∥ �H(s) +c ∥2 +F += +p1 +� +i′=1 +p2 +� +j′=1 +( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +ricjet,i′jet,ij′)2∥�C(s) − C �H(s) +c ∥2 +F ∥ �H(s) +c ∥2 +F += Op +� +(T p2 +1p3 +2 + T 2p1p2 +2)w(s) +c +� +, +where +E( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +ricjet,i′jet,ij′)2 ≤ +T +� +t,s=1 +p1 +� +i1i2=1 +p2 +� +j1j2=1 +Cov(et,i′j1et,i1j′, es,i′j2es,i2j′) + ( +T +� +t=1 +p1 +� +i=1 +p2 +� +j=1 +Eet,i′jet,ij′) += O +� +T p1p2 + T 2� +. +For the last term, +∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c )(�C(s) − C �H(s) +c )⊤E⊤ +t R∥2 +F ≤ ∥�C(s) − C �H(s) +c ∥2 +F ∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c )⊤E⊤ +t R∥2 +F +≤ ∥�C(s) − C �H(s) +c ∥4 +F +p1 +� +i=1 +p2 +� +j=1 +∥ +T +� +t=1 +E⊤ +t Ret,ij∥2 +F +≤ ∥�C(s) − C �H(s) +c ∥4 +F +p1 +� +i=1 +p2 +� +j=1 +p2 +� +k=1 +( +T +� +t=1 +p1 +� +i′=1 +et,i′ket,ij)2 += Op +� +(T 2p1p4 +2 + T p2 +1p4 +2)w(s)2 +c +� +, +where +p1 +� +i=1 +p2 +� +j=1 +p2 +� +k=1 +E( +T +� +t=1 +p1 +� +i′ +et,i′ket,ij)2 ≤ +p1 +� +i=1 +p2 +� +j=1 +p2 +� +k=1 +( +T +� +t=1 +p1 +� +i′=1 +Eet,i′ket,ij)2 ++ +p1 +� +i=1 +p2 +� +j=1 +p2 +� +k=1 +T +� +t,s=1 +p1 +� +i1i2=1 +Cov(et,i1ket,ij, es,i2ket,ij) += Op +� +T 2p1p2 +2 + T p2 +1p2 +2 +� +. +30 + +(1) Note that +∥ +T +� +t=1 +E⊤ +t �R(s+1) �R(s)⊤EtC∥2 +F = ∥ +T +� +t=1 +E⊤ +t ( �R(s+1) − R �H(s+1) +r ++ R �H(s+1) +r +)( �R(s) − R �H(s) +r ++ R �H(s) +r )⊤EtC∥2 +F +≤ ∥ +T +� +t=1 +E⊤ +t R �H(s+1) +r +�H(s)⊤ +r +R⊤EtC∥2 +F + ∥ +T +� +t=1 +E⊤ +t R �H(s+1) +r +( �R(s) − R �H(s) +r )⊤EtC∥2 +F ++ ∥ +T +� +t=1 +E⊤ +t ( �R(s+1) − R �H(s+1) +r +) �H(s)⊤ +r +R⊤EtC∥2 +F ++ ∥ +T +� +t=1 +E⊤ +t ( �R(s+1) − R �H(s+1) +r +)( �R(s) − R �H(s) +r )⊤EtC∥2 +F . +For the first term, +∥ +T +� +t=1 +E⊤ +t R �H(s+1) +r +�H(s)⊤ +r +R⊤EtC∥2 +F ≤ ∥ +T +� +t=1 +E⊤ +t R⊤EtC∥2 +F ∥R∥2 +F∥ �H(s+1) +r +∥2 +F∥ �H(s) +r ∥2 +F = Op +� +T p3 +1p2 +2 + T 2p2 +1p2 +� +. +For the second term, +∥ +T +� +t=1 +E⊤ +t R �H(s+1) +r +( �R(s) − R �H(s) +r )⊤EtC∥2 +F ≤ ∥ +T +� +t=1 +E⊤ +t RC⊤E⊤ +t ∥2 +F ∥ �R(s) − R �H(s) +r ∥2 +F ∥ �H(s+1) +r +∥2 +F += Op +� +(T p3 +1p2 +2 + T 2p2 +1p2)w(s) +r +� +. +For the third term, +∥ +T +� +t=1 +E⊤ +t ( �R(s+1) − R �H(s+1) +r +) �H(s)⊤ +r +R⊤EtC∥2 +F ≤ ∥ +T +� +t=1 +E⊤ +t R⊤EtC∥2 +F∥ �R(s+1) − R �H(s+1) +r +∥2 +F ∥ �H(s) +r ∥2 +F += Op +� +(T p3 +1p2 +2 + T 2p2 +1p2)w(s+1) +r +� +. +For the last term, +∥ +T +� +t=1 +E⊤ +t ( �R(s+1) − R �H(s+1) +r +)( �R(s) − R �H(s) +r )⊤EtC∥2 +F +≤ ∥ +T +� +t=1 +( �R(s) − R �H(s) +r )⊤EtCE⊤ +t ∥2 +F ∥ �R(s+1) − R �H(s+1) +r +∥2 +F +≤ ∥ �R(s) − R �H(s) +r ∥2 +F ∥ �R(s+1) − R �H(s+1) +r +∥2 +F +p1 +� +i′=1 +p2 +� +j′=1 +∥ +T +� +t=1 +EtCet,i′j′∥2 +F +≤ ∥ �R(s) − R �H(s) +r ∥2 +F ∥ �R(s+1) − R �H(s+1) +r +∥2 +F +p1 +� +i′=1 +p2 +� +j′=1 +p1 +� +i=1 +( +T +� +t=1 +p2 +� +j=1 +et,ijet,i′j′)2 += Op +� +(T p4 +1p2 +2 + T 2p4 +1p2)w(s) +r w(s+1) +r +� +, +31 + +where +E( +T +� +t=1 +p2 +� +j=1 +et,ijet,i′j′)2 ≤ +T +� +t,s=1 +p2 +� +j1,j2=1 +Cov(et,ij1et,i′j′, es,ij2es,i′j′) + ( +T +� +t=1 +p2 +� +j=1 +Eet,ijet,i′j′)2 += O +� +T p2 + T 2� +. +Lemma S2.9. Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed. If Assumption 2.1 +(1), Assumption 2.2 and Assumption 3.1-3.4 hold, then we have +(1) ∥ +T +� +t=1 +Et �C(s)F⊤ +t ∥2 +F = Op +� +T p1p2 +2w(s) +c ++ T p1p2 +� +. +(2) ∥ +T +� +t=1 +R⊤Et �C(s)F⊤ +t ∥2 +F = Op +� +T p1p2 +2w(s) +c ++ T p1p2 +� +. +(3) ∥ +T +� +t=1 +E⊤ +t �R(s)Ft∥2 +F = Op +� +T p2 +1p2w(s) +r ++ T p1p2 +� +. +(4) ∥ +T +� +t=1 +C⊤E⊤ +t �R(s)Ft∥2 +F = Op +� +T p2 +1p2w(s) +r ++ T p1p2 +� +Proof. (1) +∥ +T +� +t=1 +Et �C(s)F⊤ +t ∥2 +F ≤ ∥ +T +� +t=1 +Et(�C(s) − C �H(s) +c )F⊤ +t ∥2 +F + ∥ +T +� +t=1 +EtC �H(s) +c F⊤ +t ∥2 +F +≤ ∥ +T +� +t=1 +EtF⊤ +t ∥2 +F ∥�C(s) − C �H(s) +c ∥2 +F + ∥ +T +� +t=1 +EtCF⊤ +t ∥2 +F ∥ �H(s) +c ∥2 +F += Op +� +T p1p2 +2w(s) +c ++ T p1p2 +� +. +(2) +∥ +T +� +t=1 +R⊤Et �C(s)F⊤ +t ∥2 +F ≤ ∥ +T +� +t=1 +R⊤Et(�C(s) − C �H(s) +c )F⊤ +t ∥2 +F + ∥ +T +� +t=1 +R⊤EtC �H(s) +c F⊤ +t ∥2 +F +≤ ∥ +T +� +t=1 +F⊤ +t R⊤Et∥2 +F∥�C(s) − C �H(s) +c ∥2 +F + ∥ +T +� +t=1 +R⊤EtCF⊤ +t ∥2 +F∥ �H(s) +c ∥2 +F += Op +� +T p1p2 +2w(s) +c ++ T p1p2 +� +. +(3) +∥ +T +� +t=1 +E⊤ +t �R(s)Ft∥2 +F ≤ ∥ +T +� +t=1 +E⊤ +t ( �R(s) − R �H(s) +r )Ft∥2 +F + ∥ +T +� +t=1 +E⊤ +t R �H(s) +r Ft∥2 +F +≤ ∥ +T +� +t=1 +E⊤ +t Ft∥2 +F ∥ �R(s) − R �H(s) +r ∥2 +F + ∥ +T +� +t=1 +E⊤ +t RFt∥2 +F∥ �H(s) +r ∥2 +F += Op +� +T p2 +1p2w(s) +r ++ T p1p2 +� +. +32 + +(4) +∥ +T +� +t=1 +C⊤E⊤ +t �R(s)Ft∥2 +F ≤ ∥ +T +� +t=1 +C⊤E⊤ +t ( �R(s) − R �H(s) +r )Ft∥2 +F + ∥ +T +� +t=1 +C⊤E⊤ +t R �H(s) +r Ft∥2 +F +≤ ∥ +T +� +t=1 +C⊤E⊤ +t Ft∥2 +F∥ �R(s) − R �H(s) +r ∥2 +F + ∥ +T +� +t=1 +C⊤E⊤ +t RFt∥2 +F ∥ �H(s) +r ∥2 +F += Op +� +T p2 +1p2w(s) +r ++ T p1p2 +� +. +33 + diff --git a/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/load_file.txt b/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8e952ead0be97e73c1e2eb773715c4112d7ecb7 --- /dev/null +++ b/MtAyT4oBgHgl3EQfgfiG/content/tmp_files/load_file.txt @@ -0,0 +1,2852 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf,len=2851 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='00360v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ME] 1 Jan 2023 Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model by Random Projection Yong He∗, Ran Zhao∗, and Wen-Xin Zhou† January 3, 2023 The matrix factor model has drawn growing attention for its advantage in achieving two-directional dimension reduction simultaneously for matrix-structured observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In this paper, we propose a simple iterative least squares algorithm for matrix factor models, in contrast to the Principal Component Analysis (PCA)-based methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In detail, we first propose to estimate the latent factor matrices by projecting the observations with two deterministic weight matrices, which are chosen to diversify away the idiosyncratic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We show that the inferences on factors are still asymptotically valid even if we overestimate both the row/column factor numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We then estimate the row/column loading matrices by minimizing the squared loss function under certain identifiability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The resultant estimators of the loading matrices are treated as the new weight/projection matrices and thus the above update procedure can be iteratively performed until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theoretically, given the true dimensions of the factor matrices, we derive the convergence rates of the estimators for loading matrices and common components at any s-th step iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thorough numerical simulations are conducted to investigate the finite-sample performance of the proposed methods and two real datasets associated with financial portfolios and multinational macroeconomic indices are used to illustrate practical usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Keyword: Latent low rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Least squares;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Matrix factor model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Random Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 1 Introduction Factor modeling is an extremely popular approach for dimension reduction in large-dimensional time se- ries analysis, which has been successfully applied to large panels of time series for forecasting macroe- conomic variables (Stock and Watson, 2002a), building low-dimensional indicators of the whole economic ∗Institute of Financial Studies, Shandong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E-mail: heyong@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='cn, Zhaoran@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='cn †Department of Mathematical Sciences, University of California, San Diego, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E-mail: wez243@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu 1 activity (Stock and Watson, 2002b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the last two decades, there has been a flourish of literature on large-dimensional factor models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' see, for example, Bai (2003),Onatski (2009), Ahn and Horenstein (2013), Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2013), Trapani (2018), A¨ıt-Sahalia and Xiu (2017), Kong (2017), Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2018), Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019), Barigozzi and Cho (2020), Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021), He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022a) and Fan and Liao (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In economics and finance, observations are usually well structured to be an array/matrix, such as a time list of tables recording several macroeconomic variables across a number of countries or a series of customers’ ratings on a large number of items in an online platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the last few years, the literature has paid increasing attention to factor analysis for matrix time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019) for the first time proposed the following factor model for matrix time series: Xt = RFtC⊤ + Et, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , T, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) where R is the p1×k1 row factor loading matrix exploiting the variations of Xt across the rows, C is the p2×k2 column factor loading matrix reflecting the differences in the columns of Xt, Ft is the k1 ×k2 common factor matrix and Et is the idiosyncratic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019) proposed estimators of the factor loading matrices by an eigen-analysis of the auto-cross-covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Chen and Fan (2021) proposed an α-PCA method by exploiting an eigen-analysis of a weighted average of the mean and the column (row) covariance matrix of the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) proposed a Projection Estimation (PE) method that further improved the estimation efficiency of the factor loading matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021a) established the equivalence between minimizing the squared loss and the PE method by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) and further proposed a robust method by replacing the squared loss with the Huber loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The resultant estimators of factor loading matrices can be simply obtained by an eigen-analysis of weighted sample covariance matrices of the projected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022b) proposed to recover the loading spaces of the matrix elliptical factor model by an eigen- analysis of the generalized row/column matrix Kendall’s tau, which generalizes the multivariate Kendall’s tau to the random matrix setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' However, to our knowledge, all the theoretical studies of the matrix factor model in the literature crucially rely on the assumption that the pair of factor numbers k1 and k2 is consistently estimated, which typically requires that the factors are relatively strong, data have weak serial correlation or the number of observations is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In practical applications, these requirements may fail to hold due to weak signal-to-noise ratio or non-stationarity, making the first top eigenvalues of the row/column covariance matrix less separated from the remaining ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' see also the discussions in Fan and Liao (2022) for vector factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Over-estimating the number of factors would be a promising remedy as discussed in Moon and Weidner (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Westerlund and Urbain (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Barigozzi and Cho (2020) for classical vector factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The impact of over-estimating the pair of factor numbers for matrix factors models remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In this article, we propose a simple iterative least squares algorithm for matrix factor models, in contrast 2 to the Principal Component Analysis (PCA)-based methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In detail, in the first step, we propose to estimate the latent factor matrices by projecting the matrix observations with two deterministic weight matrices, which are chosen to diversify away the idiosyncratic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' This idea is similar in spirit to that in Fan and Liao (2022) and the estimator does not rely on eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the second step, we update the row/column loading matrices by minimizing the squared loss function under the identifiability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then the estimators of the loading matrices are treated as the new weight matrices and we iteratively proceed with the above two steps until a convergence criterion is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The contributions of the current work lie in the following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Firstly, to our knowledge, this is the first work on matrix factor analysis that does not involve any eigen-decomposition of large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The proposed iterative least squares algorithm is quite simple and computationally efficient, with computational complexity O (T p1p2) in contrast to the typical O � T p2 1 + T p2 2 � complexity of the eigen-decomposition based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Secondly, we show that even if both numbers of row and column factors are over-estimated, the inferences on factors are still asymptotically valid, which is new to the literature on matrix factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' At last, given the factor numbers are correctly specified, we establish the convergence rates of the estimators for loading matrices and common components at the s-th iteration for any s ≥ 1 under some strong factor identifiability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Compared to one-step estimation (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2022), the multi-step estimator is proven to be less sensitive to the initial estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In addition, the iterative least squares algorithm also reduces the magnitudes of the idiosyncratic error components in each step, thereby increasing the signal-to-noise ratio and enjoying the same advantage as the projection estimation method by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Section 2, we introduce the factor estimation method via two-directional diversified projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We derive the consistency of the vectorized factor space even when the factor numbers are over-estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Section 3, we propose the iterative least squares estimators for the loading spaces and derive the convergence rates of the estimators at any s-th step iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Section 4, we conduct thorough numerical studies to illustrate the advantages of the proposed methods over the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Section 5, we analyze a financial dataset and a multinational macroeconomic indices dataset to illustrate the empirical usefulness of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We discuss possible future research directions and conclude the paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The proofs of the main theorems and additional details are collected in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We end this section by introducing some notations that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For any vector µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , µp)⊤ ∈ Rp, let ∥µ∥2 = (�p i=1 µ2 i )1/2, ∥µ∥∞ = maxi |µi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For a real number a, denote [a] as the largest integer smaller than or equal to a, let sgn(a) = 1 if a ≥ 0 and sgn(a) = −1 if a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let I(·) be the indicator function and diag(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , ap) be a p × p diagonal matrix, whose diagonal entries are a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For a matrix A, let Aij (or Ai,j) be the (i, j)-th entry of A, A⊤ the transpose of A, tr(A) the trace of A, rank(A) the rank of A, A+ the Moore-Penrose generalized inverse of A and diag(A) a vector composed of 3 the diagonal elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Denote PA as the projection matrix PA = A(A⊤A)−1A⊤ and span(A) as the space spanned by the columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Denote λj(A) as the j-th largest eigenvalue of a nonnegative definitive matrix A, and let ∥A∥ be the spectral norm of matrix A and ∥A∥F be the Frobenius norm of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For two series of random variables, Xn and Yn, Xn ≍ Yn means Xn = Op(Yn) and Yn = Op(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For two random variables (vectors) X and Y , Var(X) denotes the variance (covariance matrix) of X, X d= Y means the distributions of X and Y are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The constants c, C1, C2 in different lines can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 2 Factor Estimation via Two-directional Diversified Projections In this section, we introduce a simple two-directional diversified projection method to estimate the factor score matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also investigate the theoretical properties of the estimators under cases of finite samples and overestimation of the factor numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 The estimation of factors In this section, we propose a simple two-directional diversified projection method to estimate the factor score matrices, which does not involve eigen-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Wi = (wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='mi) be a given exogenous (or deterministic) pi × mi matrix i = 1, 2, where wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j, the j-th column of the matrix Wi, is a vector of “diversified weights” in the sense that its strength should be approximately equally distributed across most of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We call W1 and W2 the “left projection matrix” and “right projection matrix”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We propose to estimate Ft simply by �Ft = 1 p1p2 W⊤ 1 XtW2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) By the matrix factor model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1), we have �Ft = 1 p1p2 W⊤ 1 RFtC⊤W2 + 1 p1p2 W⊤ 1 EtW2 := H1FtH⊤ 2 + Et, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2) where H1 = W⊤ 1 R/p1, H2 = W⊤ 2 C/p2 and Et = W⊤ 1 EtW2/(p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thus �Ft estimates Ft up to two affine transformation with Et as the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The assumption that W1, W2 should be diversified guarantees that as min(p1, p2) → ∞, Et is diversified away (converging to zero in probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We call the new factor matrix estimator “bi-diversified factors”, which reduces the dimensions of Xt from p1 × p2 to m1 × m2 by two-directional diversified projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Due to the clean expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2), the mathematics for theoretical analysis is much simpler than the most benchmark estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Intuitively, �Ft would lead to valid inferences in factor-augmented models so long as mi ≥ ki, i = 1, 2, in the same spirit as Fan and Liao (2022) for vector factor model, and we leave this to future work as the matrix factor-augmented model is still in its 4 infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Theoretical analysis for the estimators of Factors In this section, we investigate the theoretical properties of the estimators of factors under finite sample cases and the overestimation of the factor numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We assume that the predetermined constants mi, i = 1, 2 do not grow with pi, which are named as “the working (pseudo) numbers of row (column) factors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Since in practice we do not know the true number of factors, we often take slightly large numbers mi such that mi ≥ ki are likely to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Wi be either deterministic or random but independent of the σ-algebra generated by {Et : t ∈ [T ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We further assume that the projection matrices Wi, i = 1, 2 satisfy the following: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' There are positive constants c1 and c2, such that (almost surely if W1, W2 are random) as p1, p2 → ∞, (1) max i≤p1 |w1,ij| < c1, max i≤p2 |w2,ij| < c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) the m1 × m1 matrix W⊤ 1 W1/p1 and m2 × m2 matrix W⊤ 2 W2/p2 satisfy λmin(W⊤ 1 W1/p1) ≥ c1, λmin(W⊤ 2 W2/p2) ≥ c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) W1, W2 are independent of Et, t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Vectorizing the matrix �Ft in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2), we have Vec(�Ft) = (H2 ⊗ H1)Vec(Ft) + 1 p1p2 (W2 ⊗ W1)⊤Vec(Et) := HVec(Ft) + 1 p1p2 W⊤Vec(Et), where H = H2 ⊗ H1 and W = W2 ⊗ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Therefore, Vec(�Ft) can be treated as an estimate of Vec(Ft) up to a transformation matrix, where H ∈ Rm1m2×k1k2, with estimation error equal to W⊤Vec(Et)/(p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' When each element of Et is cross-sectional weakly dependent, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 guarantees the cross-sectional central limit theorem of W⊤Vec(Et)/(p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For example, assume each element of Et is cross-sectional independent, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, as p1p2 → ∞, we get 1 √p1p2 W⊤Vec(Et) d→ N(0, V), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) where V = limp1p2→∞ W⊤Var(Vec(Et))W/(p1p2), assuming it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) shows that for each t ≤ T , √p1p2(Vec(�Ft) − HVec(Ft)) is asymptotically normal regardless of whether T goes to infinity, m1 = k1, m2 = k2 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' It only requires p1p2 → ∞, which is particularly useful for analyzing short matrix time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In addition, the factor components should not be diversified away, which entails the following conditions on transformation matrix Hi, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let νmin(Hi), νmax(Hi) denote the minimum and maximum nonzero singular values of Hi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 5 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose m1 ≥ k1, m2 ≥ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Almost surely (1) rank(H1) = k1, rank(H2) = k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) there exist constants c1 and c2 such that ν2 min(Hi) ≫ 1 pi , νmax(Hi) ≤ c1νmin(Hi), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (1) requires W1 to have at least k1 columns that are not orthogonal to R and W2 to have at least k2 columns that are not orthogonal to C so that R and C are not diversified away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' It also ensures that the space spanned by Vec(�Ft) is asymptotically equal to the space spanned by Vec(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' This is the key assumption imposed, but not stringent in the context of over-estimating factors (Barigozzi and Cho, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Fan and Liao, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (2) determines the rate of convergence in recovering the space spanned by the factors and ensures that the weight matrix and the loading matrix are not orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assume νmin(W⊤ 1 R) = pα1 1 , νmin(W⊤ 2 C) = pα2 2 , then Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (2) entails that α1, α2 ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 are direct generalizations of Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 in Fan and Liao (2022) to the matrix factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the matrix factor model, once we vectorize the observations, the model reduces to a vector factor model X = F(C ⊗ R)⊤ + E, where F = (Vec(F1), Vec(F2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , Vec(FT ))⊤, X = (Vec(X1), Vec(X2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , Vec(XT ))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E = (Vec(E1), Vec(E2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , Vec(ET ))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2), we have �F = FH⊤ + E, where �F = (Vec(�F1), Vec(�F2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , Vec(�FT ))⊤, E = (Vec(E1), Vec(E2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , Vec(ET ))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' To derive the theo- retical properties of the factor estimators, we further impose the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' There are constants c, C such that (1) c < λmin(�T t=1 Vec(Ft)Vec(Ft)⊤/T ) ≤ λmax(�T t=1 Vec(Ft)Vec(Ft)⊤/T ) < C, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) (2i) 1/T �T s=1 �T T =1 E∥Ft∥F ∥Fs∥F tr(E(EtE⊤ s |F)) < c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2ii) ∥E(EtE⊤ t )∥2 ≤ c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) λmin � E � E⊤E/T �� ≥ c0, E∥E � Vec(Et)Vec(Et)⊤|F � ∥2 < c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (4) for any t ∈ [T ], i1, i1 ∈ [p1], j1, j2 ∈ [p2], �T s=1 �p1 u1,v1=1 �p2 u2,v2=1 |Cov(et,i1i2et,j1j2, es,u1u2es,v1v2)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 requires weak dependence among idiosyncratic errors, which is common in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' see, for example, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021b), Chen and Fan (2021) and Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The following theorem establishes the convergence of �Ft to the true Ft up to transformation matrices, regardless of whether T → ∞ or overestimating the factor numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 6 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (2) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (2i) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As min{p1, p2} → ∞ and T is finite or T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then for any bounded m1 ≥ k1, m2 ≥ k2, for any t ∈ [T ], ��M⊤ 1 �FtM2 − Ft �� 2 = Op � 1 √p1p2 ν−1 min(H1)ν−1 min(H2) � , where M1 = (H1H⊤ 1 )+H1 ∈ Rm1×k1, M2 = (H2H⊤ 2 )+H2 ∈ Rm2×k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The following theorem shows that the linear space spanned by �F equals to the linear space spanned by F asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For any bounded m1 ≥ k1, m2 ≥ k2, we have as min{p1, p2} → ∞ that ∥P�FPF − PF∥2 = Op � 1 √p1p2 ν−1 min(H1)ν−1 min(H2) � , ∥P�FM − PF∥2 = Op � 1 √p1p2 ν−1 min(H1)ν−1 min(H2) � , where M = (HH⊤)+H and H = H2 ⊗ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 establishes that when m1 ≥ k1, m2 ≥ k2, the linear space spanned by �F is asymptotically the same as the linear space spanned by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 also shows that a particular subspace of span(�F) is consistent for span(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In particular, when m1 = k1, m2 = k2, we have P�FM = P�F as M is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' It then degenerates to the usual space consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Consistent estimation of the number of factors k1, k2 typically requires strong conditions, which are difficult to fulfill in finite samples case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' One advantage of the proposed method for estimating the factor matrices is that it is still robust against overestimating the number of factors in many statistical inference problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the choices of weight matrices, one can select W1 and W2 following the same strategies by Fan and Liao (2022), such as the Hadamard Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, {�Ft, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , T } would lead to valid inferences in factor-augmented models so long as mi ≥ ki, i = 1, 2 and we leave this to our future work as the matrix factor-augmented model is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 3 Iterative Least Squares Estimators for Loading Spaces In this section, we introduce the iterative least squares estimators for the column/row loading spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also derive the convergence rates of the estimators for loading matrices at the s-th iteration (for any s ≥ 1) provided that the pair of factor numbers are correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In case that the pair of factor numbers are unknown, many methods have been proposed in the literature to estimate them consistently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' see for example the α-PCA-ER in Chen and Fan (2021) and the Iter-ER in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 The Estimation of Loading Spaces In Section 2, we introduce the way to estimate the factor matrices with two diversified projection matrices W1, W2 and denote the estimators as {�Ft, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' , T }, which are of dimension m1 × m2 with m1 ≥ k1, m2 ≥ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Given {�Ft}, it is straightforward to estimate the row factor loading matrix R by minimizing the squared Frobenius loss under the identifiability condition: min R L1(R) = min R 1 T T � t=1 ∥Xt − R�FtW⊤ 2 ∥2 F , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t 1 p1 R⊤R = Im1, m1 ≥ k1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) where W2 is the column projection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The objective function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) can be simplified as L1(R) = 1 T T � t=1 � tr � X⊤ t Xt � − 2tr � X⊤ t R�FtW⊤ 2 � + p1tr � W2�F⊤ t �FtW⊤ 2 �� , and the Lagrangian function is min R,Θ L1(R, Θ) = min R,Θ � L1(R) + tr � Θ � 1 p1 R⊤R − Im1 ��� , where the Lagrangian multipliers Θ is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Taking ∂L1(R, Θ)/∂R = 0 and ∂L1(R, Θ)/∂Θ = 0, we obtain ∂L1(R, Θ) ∂R = 1 T T � t=1 � − 2XtW2�F⊤ t + 2 p1 RΘ � = 0, ∂L1(R, Θ) ∂Θ = 1 p1 R⊤R − Im1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2) Further, we can derive the explicit expression for �Θ and �R satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2), that is, �Θ = √p1 T �� T � t=1 �FtW⊤ 2 X⊤ t � � T � t=1 XtW2�F⊤ t ��1/2 , �R = √p1 � T � t=1 XtW2�F⊤ t � �� T � t=1 �FtW⊤ 2 X⊤ t � � T � t=1 XtW2�F⊤ t ��−1/2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) 8 and we set �R as the one-step estimator of the row loading matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similarly, by minimizing the following loss function under the identifiability condition: min C L2(C) = min C 1 T T � t=1 ��Xt − W1�FtC⊤��2 F , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t 1 p2 C⊤C = Im2, m2 ≥ k2, with the Lagrangian function min C,Λ L2(C, Λ) = min C,Λ � 1 T T � t=1 ��Xt − W1�FtC⊤��2 F + tr � Λ � 1 p2 C⊤C − Im2 ��� , we get the following estimator of the column factor loading matrix �C = √p2 � T � t=1 X⊤ t W1�Ft � �� T � t=1 �F⊤ t W⊤ 1 Xt � � T � t=1 X⊤ t W1�Ft ��−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) To be consistent with the following iterative algorithm, we also denote �Ft in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2) as �F(1) t and �R in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) as �R(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Once we get the estimator �R(1), we can obtain �C(1) by replacing W1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) with �R(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', �C(1) = √p2 � T � t=1 X⊤ t �R(1)�Ft � �� T � t=1 �F⊤ t �R(1)⊤Xt � � T � t=1 X⊤ t �R(1)�Ft ��−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5) Given �C(1), we update the estimator of Ft as �F(2) t = �R(1)⊤Xt �C(1)/(p1p2), thereby updating the estimators of R and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In detail, update the estimator of R as �R(2) by replacing �Ft, W2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) with �F(2) t , �C(1), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' update the estimator of C as �C(2) by replacing �Ft, W1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) with �F(2) t , �R(2), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we repeat the above steps until a convergence criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' At the (s + 1)-th iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' the estimators �F(s+1) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �R(s+1) and �C(s+1) have the following expressions: �F(s+1) t = 1 p1p2 �R(s)⊤Xt �C(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �R(s+1) = √p1 � T � t=1 Xt �C(s)�F(s+1)⊤ t � �� T � t=1 �F(s+1) t �C(s)⊤X⊤ t � � T � t=1 Xt �C(s)�F(s+1)⊤ t ��−1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �C(s+1) = √p2 � T � t=1 X⊤ t �R(s+1)�F(s+1) t � �� T � t=1 �F(s+1)⊤ t �R(s+1)⊤Xt � � T � t=1 X⊤ t �R(s+1)�F(s+1) t ��−1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where �R(s) and �C(s) are the estimators from the s-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The Random Projection-based Iterative Least Squares (RPILS) procedure for the matrix factor model is summarized in Algorithm 1 below and the 9 theoretical analysis is presented in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the convergence criterion, denote the common component matrix at the (s+1)-th step as �S(s+1) = �R(s)�F(s+1) t �C(s)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In our simulation studies, the iterative procedure is terminated either when a pre-specified maximum iteration number (maxiter = 100) is reached or when ∥�S(s+1) − �S(s)∥F ≤ ǫ, where ǫ is a small constant (10−6) given in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Algorithm 1 Random Projection based Iterative Least Squares (RPILS) procedure for matrix factor model Input: Data matrices {Xt}, t ≤ T , the pair of pseudo row and column factor numbers m1(m1 ≥ k1) and m2(m2 ≥ k2), the diversified projection matrices W1, W2 Output: Factor loading matrices �R ∈ Rp1×m1, �C ∈ Rp2×m2 and factor matrix �Ft ∈ Rm1×m2, t ≤ T 1: obtain the initial estimator �F(1) t by �F(1) t = W⊤ 1 XtW2/(p1p2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 2: given �F(1) t and W2, get an estimator �R(1) by Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 3: given �F(1) t and �R(1), get an estimator �C(1) by replacing W1 in the Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) with �R(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 4: update �F(2) t by �F(2) t = �R(1)⊤Xt �C(1)/(p1p2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 5: update �R(2) by replacing �Ft, W2 in the Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) with �F(2) t , �C(1), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 6: update �C(2) by replacing �Ft, W1 in the Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) with �F(2) t , �R(2), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 7: repeat steps 4-6 until convergence and output the estimators from the last step denoted as �R, �C and {�Ft, t ≤ T }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Convergence rates In this section, we establish the convergence rates of the estimators of loading matrices at s-th iteration for any s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' To this end, we first impose some additional conditions that are common in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The factor matrix satisfies E(Ft) = 0, E∥Ft∥4 ≤ c < ∞ for some constant c > 0, and 1 T T � t=1 FtF⊤ t p→ Σ1 and 1 T T � t=1 F⊤ t Ft p→ Σ2, where Σi, i = 1, 2 is a ki × ki positive definite matrix with bounded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' There exist positive constants c, c0 < ∞ such that (1) E(et,ij) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) for any t ∈ [T ], i ∈ [p1], j ∈ [p2], �T s=1 �p1 l=1 �p2 h=1 |E(et,ijes,lh)| ≤ c and �p1 l=1 �p2 h=1 |E(et,ljet,ih)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) for any T ∈ [T ], i1, i1 ∈ [p1], j1, j2 ∈ [p2], �T s=1 �p1 u1,v1=1 �p2 u2,v2=1 |Cov(et,i1i2et,j1j2, es,u1u2es,v1v2)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' There exist positive constants c1, c2 such that ∥R∥max ≤ c1, ∥C∥max ≤ c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' There exists a constant c > 0 such that (1) for any deterministic vectors v and w satisfying ∥v∥ = 1 and ∥w∥ = 1, E �� 1 √ T �T t=1(Ftv⊤Etw) ��2 ≤ c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) for any i ∈ [p1], j ∈ [p2], ∥ �p1 i′=1 �p2 j′=1 E(ξi,j ⊗ 10 ξi′,j′)∥max ≤ c, where ξi,j = Vec(T −1/2 �T t=1 Ftet,ij);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) for any i1, i2 ∈ [p1], j1, j2 ∈ [p2], ∥ �p1 i′ 1,i′ 1=1 �p2 j′ 1,j′ 2=1 Cov(ξi1,j1 ⊗ ξi2,j2, ξi′ 1,j′ 1 ⊗ ξi′ 2,j′ 2)∥max ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 are standard in the literature on matrix factor models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' see for example Chen and Fan (2021), Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) and He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ensures the (cross-sectional and time series) summability of the idiosyncratic terms Et, which allows for (weak) dependence in both space and time domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 requires that the common factors are pervasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 allows the common factors Ft and errors Et to be weakly correlated, which is satisfied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', when {Ft} and {Et} are two mutually independent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' One may refer to the detailed discussions on these assumptions by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following theorem, we first establish the convergence rates of the one-step estimator �R ( �R(1)) and �C defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose that m1 = k1, m2 = k2 and the factor numbers k1, k2 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Further assume that ν2 min(H2) ≫ max(1/T, 1/p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, as T, p1, p2 go to infinity, there exists an asymptotic orthogonal matrix �H(1) r , such that 1 p1 ∥ �R(1) − R �H(1) r ∥2 F = Op � 1 T p2ν2 min(H2) + 1 T p1p2ν2 min(H1)ν4 min(H2) + 1 p2 1p2 2ν2 min(H1)ν4 min(H2) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6) where H1 = W⊤ 1 R/p2, H2 = W⊤ 2 C/p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Symmetrically, assume ν2 min(H1) ≫ max(1/T, 1/p1) hold, then there exists an asymptotic orthogonal matrix �Hc, such that 1 p2 ∥�C − C �Hc∥2 F = Op � 1 T p1ν2 min(H1) + 1 T p1p2ν4 min(H1)ν2 min(H2) + 1 p2 1p2 2ν4 min(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The condition ν2 min(H2) ≫ max(1/T, 1/p2) guarantees that the matrices �Hr are asymptotic orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' This condition in essence requires that the space spanned by the columns of the initial projection matrix W2 does not deviate far from that spanned by the columns of C, and fails to hold if the two spaces are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' This is conceivable that the iterative algorithm would never converge to the true space if we start from its orthogonal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 shows that the closer the initial projection directions (space spanned by the columns of W2) are to the true loading directions (space spanned by the columns of C), the faster the estimated loading matrix �R(1) converges to the true loading matrix R up to an orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In particular, if νmin(H1) = νmin(H2) = Op(1) (as long as we start from the consistent α-PCA estimators in Chen and Fan (2021) as projection directions), then we have 1 p1 ∥ �R(1) − R �H(1) r ∥2 F = Op � 1 T p2 + 1 p2 1p2 2 � , 1 p2 ∥�C − C �Hc∥2 F = Op � 1 T p1 + 1 p2 1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following theorem, we establish the convergence rate of the one-step estimator �C(1) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 11 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the same conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, there exists an asymptotic orthogonal matrix �H(1) c , such that w(1) c = 1 p2 ∥�C(1)−C �H(1) c ∥2 F = Op � 1 T p1 + 1 p2 1p2 2ν2 min(H1)ν2 min(H2) + 1 T 2p2 2ν2 min(H2) + 1 T p1p2ν2 min(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In addition, w(1) r = 1 p1 ∥ �R(1) − R �H(1) r ∥2 F is the rate derived in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assume that νmin(H1) = νmin(H2) = Op(1), then the derived convergence rate of �C(1) is 1 p2 ∥�C(1) − C �H(1) c ∥2 F = Op � 1 T p1 + 1 p2 1p2 2 + 1 T 2p2 2 � , which is the same as that derived in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As long as we get the estimators of row/column loading matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', �R(1) and �C(1), the update of the factor matrix can be obtained by �F(2) t = �R(1)⊤Xt �C(1)/(p1p2) and the corresponding common-components matrix is then updated by �S(2) = �R(1)�F(2) t �C(1)⊤ = (�S(2) t,ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The following theorem provides the convergence rates of the estimated factors and common components after one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the same conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' as min{T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p2} → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' for any t ∈ [T ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' i ∈ [p1] and j ∈ [p2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have �����F(2) t − ( �H(1) r )−1Ft � ( �H(1) c )−1�⊤���� F = Op � 1 √ Tp1 + 1 √ Tp2νmin(H2) + 1 √p1p2 + 1 p1p2νmin(H1)ν2 min(H2) + γf � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where γf = 1 √T p1p2νmin(H1)νmin(H2) + 1 T p1√p2νmin(H1)ν2 min(H2) + 1 T p1p2ν2 min(H1)ν3 min(H2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' and for the common components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have |�S(2) t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij−St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij| = Op � 1 √T p1 + 1 √p1p2 + 1 √T p2νmin(H2) + 1 √T p1p2νmin(H1)ν2 min(H2) + 1 p1p2νmin(H1)ν2 min(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij is the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' j)-th entry of S = RFtC⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The derived convergence rates in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 are the same as those derived in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5 of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) when νmin(H1) = νmin(H2) = Op (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following theorem, we establish the recurrence formula of the convergence rate for the estimators �R(s+1) and �C(s+1) given any integer s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the same conditions stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' there exist asymptotic orthogonal matrices 12 �H(s+1) r and �H(s+1) c such that for any integer s ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have w(s+1) r = 1 p1 ∥ �R(s+1) − R �H(s+1) r ∥2 F = Op � 1 T p2 + 1 p2 1p2 2 + γ(s+1) r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' w(s+1) c = 1 p2 ∥�C(s+1) − C �H(s+1) c ∥2 F = Op � 1 T p1 + 1 p2 1p2 2 + γ(s+1) c � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where γ(s+1) r = 1 p2 2 w(s) r + 1 T w(s) c + w(s) r w(s) c p2 + w(s) r w(s)2 c + 1 p2 1 w(s)2 c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' γ(s+1) c = 1 T p2 w(s) r + 1 T w(s+1) r + 1 p2 1 w(s) c + 1 p1 w(s) r w(s) c + 1 T w(s) r w(s) c + w(s) r w(s+1) r w(s) c + 1 p2 2 w(s) r w(s+1) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following theorem, we also establish the recurrence formula of the convergence rate for the esti- mators �F(s+1) and �S(s+1) given any integer s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the same conditions stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' as min{T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p2} → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' for any t ∈ [T ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' i ∈ [p1] and j ∈ [p2] and any integer s > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have �����F(s+1) t − ( �H(s) r )−1Ft � ( �H(s) c )−1�⊤���� F = Op \uf8eb \uf8ed � w(s) r p2 + � w(s−1) r T p2 + � w(s) c p1 + � w(s−1) c T p1 + 1 √p1p2 + γ(s+1) f \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where γ(s+1) f = � w(s) r w(s) c + � w(s−1) r w(s−1) c T + � w(s−1) r w(s−1) c p1 + � w(s−1) r √p1p2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' and for the common components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have |�S(s+1) t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij − St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij| = Op \uf8eb \uf8ed � w(s) r + � w(s−1) r T p2 + � w(s) c + � w(s−1) c T p1 + 1 √p1p2 + γ(s+1) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where γ(s+1) = � w(s−1) r w(s−1) c T + � w(s−1) r w(s−1) c p1 + � w(s−1) r √p1p2 and St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij is the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' j)-th entry of S = RFtC⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Theoretical analysis for the estimators of loading matrices above relies on the correct specification of factor numbers (note that we suppose m1 = k1, m2 = k2 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) and the strong factor conditions R⊤R/p1 = Ik1, C⊤C/p2 = Ik2, which means that the row and column factors are pervasive along both dimensions and is an extension of the pervasive assumption in Stock and Watson (2002a) to the matrix regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 13 4 Simulation Study In this section, we investigate the empirical performance of the Random Projection-based Iterative Least Squares (RPILS) procedure in terms of estimating the loading and factor spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We first introduce the data generation mechanism of the synthetic dataset, which is similar to He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We set k1 = 3, k2 = 3, draw the entries of R and C independently from uniform distribution U(−1, 1), and let Ft = φFt−1 + � 1 − φ2ǫt, Et = ψEt−1 + � 1 − ψ2Ut, where Vec(ǫt) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='d ∼ N(0, Ik1×k2), Ut i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='d ∼ MN(0, UE, VE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Vec(Ut) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='d ∼ N(0, VE ⊗UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The parameters φ and ψ control the temporal correlations, and UE and VE are matrices with ones on the diagonal, and the off-diagonal entries are 1/p1 and 1/p2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We compare the performances of our Random Projection based Iterative Least Squares (RPILS) method with the α-PCA method (α = 0) by Chen and Fan (2021) and the PE method by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) in terms of estimating the loading and factor spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the RPILS method, the initial weight matrices W1, W2 are randomly chosen and all their entries from a standard normal distribution and the column dimensions m1, m2 are set as the true dimensions of factor matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e, m1 = k1 = 3, m2 = k2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' To show the impact of the initial weight matrices, we also compare with the One-Step Estimators (OSE), either with completely random standard normal elements as entries of the initial weight matrices or with α-PCA estimators as initial weight matrices, denoted as OSE1 and OSE2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We point out that in unreported simulations, we have tried to use the Walsh-Hadamard matrices (Fan and Liao, 2022) as the initial weight matrices for the RPILS method, and find that the performances are almost the same as using initial weight matrices with standard normal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We consider the following two scenarios of parameter settings: Scenario A: p1 = 20, T = p2 ∈ {20, 50, 100, 150, 200}, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Scenario B: p2 = 20, T = p1 ∈ {20, 50, 100, 150, 200}, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' To measure the performances of various methods in terms of estimating loading/factor spaces, we adopt a metric between linear spaces which was also utilized in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For two column- wise orthogonal matrices (Q1)p×q1 and (Q2)p×q2, we define D(Q1, Q2) = � 1 − 1 max (q1, q2)Tr � Q1Q⊤ 1 Q2Q⊤ 2 ��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the definition of D(Q1, Q2), we can easily see that 0 ≤ D(Q1, Q2) ≤ 1, which measures the distance between the column spaces spanned by Q1 and Q2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', span(Q1) and span(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In particular, span(Q1) and span(Q2) are the same when D(Q1, Q2) = 0, while span(Q1) and span(Q2) are orthogonal when D(Q1, Q2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The Gram-Schmidt orthogonalization can be used to make Q1 and Q2 column-orthogonal 14 Table 1: Averaged estimation errors (standard errors in parentheses) in terms of D( �R, R), D(�C, C) and D(Vec(Ft), Vec(�Ft)) for Scenarios A and B under Matrix Normal distribu- tion over 500 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Evaluation T p1 p2 OSE1 OSE2 RPILS α-PCA PE Setting A: p1 = 20, p2 = T D( �R, R) 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6178(0.' metadata={'source': 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+page_content='0051) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0518(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0047) 15 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Table 1 reported the averaged estimation errors (standard errors in parentheses) in terms of D( �R, R), D(�C, C) and D(Vec(Ft), Vec(�Ft)) for Scenarios A and B over 500 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' All methods benefit from large dimensions in terms of estimating loading spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By comparing the results for OSE1 and OSE2, we conclude that the better the initial projection directions, the faster the loading/factor spaces converge to the corresponding true ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for RPILS, the results indicate that even if we start from a random guess of the projection directions, we can finally get satisfactory estimators via the iterative procedure in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In addition, the RPILS performs comparably with the PE method and better than the α-PCA method, which is consistent with our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In other words, the RPILS method also reduces the magnitudes of the idiosyncratic error components, thereby increasing the signal-to-noise ratio and enjoying the same advantage as PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' However, compared with the eigen-decomposition-based PE method, the RPILS method is computationally simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 5 Real Data Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 Fama-French 10 × 10 portfolios In this section, we study a financial portfolio dataset studied in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019) and Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The dataset is composed of monthly returns of 100 portfolios, well structured into a 10 × 10 matrix at each time point, with rows corresponding to 10 levels of market capital size (denoted as S1-S10) and columns corresponding to 10 levels of book-to-equity ratio (denoted as BE1-BE10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The dataset collects monthly returns from January 1964 to December 2019 covering a total of 672 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The details are available at the website http://mba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='tuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu/pages/faculty/ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='french/data_library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Follow- ing the same preprocessing as in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019), we adjusted the return series by first subtracting the corresponding monthly market excess returns and then standardizing each of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We imputed the missing values by the factor-model-based method introduced in Xiong and Pelger (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' All augmented Dickey-Fuller tests reject the null hypothesis, which indicates the stationarity of all the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the factor numbers, we take (k1, k2) = (2, 2) as in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Table 2 shows the estimated row and column loading matrices after varimax rotation and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' From the table, we can see the proposed RPILS method performs similarly to PE, α-PCA and ACCE methods in terms of the estimated loading matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' From the perspective of Size, small Size portfolios load heavily on the first factor while large Size portfolios load on the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' From the perspective of Book-to-Equity, small BE portfolios load heavily on the second factor while large BE portfolios load mainly on the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also use a rolling-validation scheme as in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2019) to further compare the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For each year t from 1996 to 2019, we repeatedly use n (bandwidth) years before t to fit the 16 Table 2: Loading matrices for Fama-French data set after varimax rotation and scaling by 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Factor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='S1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Table 3: Rolling validation for the Fama-French portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The sample size of the training set is 12n and k1 = k2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ¯ MSE, ¯ρ, ¯v are the mean pricing error, mean unexplained proportion of total variances and mean variation of the estimated loading space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' MSE ¯ρ ¯v n k RPILS PE α-PCA ACCE RPILS PE α-PCA ACCE RPILS PE α-PCA ACCE 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8001 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2986 matrix-variate factor model and estimate the loading matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The estimated loadings are then used to estimate the factors and corresponding residuals of the 12 months in the current year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In detail, let Yi t and 17 �Yi t be the observed and estimated price matrix of month i in year t, denote ¯Yt as the mean price matrix, and define MSEt = 1 12 × 10 × 10 12 � i=1 ∥ �Yi t − Yi t∥2 F, ρt = �12 i=1 ∥ �Yi t − Yi t∥2 F �12 i=1 ∥Yi t − ¯Yt∥2 F , as the mean squared pricing error and unexplained proportion of total variances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The variation of loading space is measured by vt = D(�Ct ⊗ �Rt, �Ct−1 ⊗ �Rt−1), during the rolling-validation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Table 3 reports the results of the means of MSE, ρ, v by the proposed RPILS method and the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For different bandwidth n and the number of factors k1, k2, our RPILS method is comparable to the PE method and better than the other methods in terms of the averaged MSE, ρ, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Multinational macroeconomic indices In this section, we analyze a multinational macroeconomic index dataset collected from Organization for Eco- nomic Co-operation and Development (OECD), which contains 10 macroeconomic indices across 8 countries over 130 quarters from 1988-Q1 to 2020-Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The 8 countries are the United States, the United Kingdom, Canada, France, Germany, Norway, Australia and New Zealand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The indices are from 4 major groups, namely consumer price, interest rate, production, and international trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the preprocessing proce- dure of the dataset, we refer to Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the factor numbers, we take the advice (k1, k2) = (3, 4) by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022) for better illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The estimated loading matrices are shown in Table 4 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The proposed RPILS method behaves almost the same as the PE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As concluded in Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022), the countries excluding Germany naturally divide into 3 groups, Oceania, North American and European.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' On the other hand, the macroeconomic indices divide into 4 groups, consumer price, interest rate, production and international trade, which coincide with economic interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also adopt a rolling prediction procedure to further investigate the practical use of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' First, we consider the change of inflation (second-order difference of the log level of the total consumer price index–CPI:Tot) of a selected country at time t, denoted as yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let xt be the vector of all the other 9 indices of the selected country at time t, and Zt be the 8×10 panel at time t, with rows corresponding to the countries and column corresponding to all macroeconomic indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We predict yt+1 by the following Auto-Regression (AR) model (Model 1) and Factor-Augmented-Auto-Regression (FAAR) models (Models 2–4), similar to the Diffusion Index forecasting by Stock and Watson (2002b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Model 1 yt+1 = a + byt + ǫt+1, Model 2 yt+1 = a + byt + β⊤f1t + ǫt+1, where f1t’s are estimated from the vector factor model with 18 Table 4: Row loading matrices by different methods for multinational macroeconomic index dataset, varimax rotated and multiplied by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Factor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='AUS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='NZL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='USA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='CAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='NOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='DEU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='FRA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='GBR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='RPILS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='observations {xt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Model 3 yt+1 = a + byt + β⊤f2t + ǫt+1, where f2t’s are estimated from the vector factor model with observations {Vec(Zt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Model 4 yt+1 = a + byt + β⊤Vec(Ft) + ǫt+1, where Ft’s are estimated from the matrix factor model with observations {Zt}, by the RPILS, PE, ACCE, and α-PCA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The models for comparison here are exactly the same with Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2022), we explain these models here again for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' First, Model 1 is a simple auto-regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Model 2 adds common 19 index factors of the selected country into the auto-regression model in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Model 3 and Model 4, both index and country factors are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The difference is that Model 4 considers the more parsimonious matrix factor structure while Model 3 vectorizes the matrix time series and considers the vector factor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' To avoid possible over-fitting in prediction, we also use the LASSO (Tibshirani, 1996) to select factors and estimate the coefficients for Models 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Table 6: MAPEs for inflation and the growth rate of GDP (both at an annual rate) for different countries with different methods, (k1, k2) = (3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Model AUS NZL USA CAN NOR DEU FRA GBR MAPEs for inflation rates Model 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5880 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8866 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8346 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4605 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9359 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8751 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5776 Model 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6258 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9086 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5835 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2349 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2607 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9574 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8322 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5019 Model 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5989 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0507 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4709 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9653 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3865 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7279 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2411 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1980 Model 4 (RPILS) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5607 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9319 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6829 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2736 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1377 1.' 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8273 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4442 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3822 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3348 Model 4 (α-PCA) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5981 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9167 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8535 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3285 Model 4 (RPILS) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8904 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0531 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5087 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7084 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7467 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3315 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7672 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3198 Model 4 (PE) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8731 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9268 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5819 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5505 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9012 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4485 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8079 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3523 Model 4 (ACCE) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9118 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0370 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5067 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6869 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6739 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5838 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7446 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4251 Model 4 (α-PCA) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8709 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='0364 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6078 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6544 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8955 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5226 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8701 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3843 For each quarter t from 2008-Q1 to 2020-Q2, we use the 80 neighboring observations before t to train the models and predict yt+1 (denoted as �yt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As yt was standardized in preprocessing, we transformed the predicted yt+1 to match the change of inflation rate by multiplying the standard deviation and adding back the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For simplicity of notation, we still denote the transformed predictor as �yt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The inflation It+1 is then predicted by integrating �yt+1 and It, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', �It+1 = �yt+1 + It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Model 2 and Model 3, the factor numbers before model selection are set as k2 and k1 × k2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also focus on the case that (k1, k2) = (3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The top panel of Table 6 shows the mean absolute prediction errors (MAPEs) 20 for the annualized inflation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the largest Oceania country, Australia, Model 4 with RPILS has the best prediction performance in terms of MAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For Norway and New Zealand, Model 1 performs the best, indicating that the index and country factors act as noises in Model 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the USA, Canada, France, Great Britain and Germany, both index and country factors are useful for improving prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that the results also show that the matrix factor structure can further improve the prediction for two American countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We also consider the rolling prediction of the GDP growth rate (first-order difference of the log level of GDP) for all countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The results shown in the bottom panel of Table 6 demonstrate that for the strong manufacturing American and European countries, USA, Germany, France and Great Britain, the simple AR model suffices to predict the GDP growth rates well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the other countries, the country and the index factors contribute to improving the prediction performance of the GDP growth rates, while for Canada and Norway, the advantage of the matrix factor structure is more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 6 Discussion We propose a simple iterative least squares algorithm for the matrix factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the first step, we estimate the latent factor matrices by projecting the observations with two deterministic weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We show that the inferences on factors are still asymptotically valid under some regularity conditions, even if both row and column factor numbers are overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the second step, we estimate the row/column loading matrices by minimizing the squared Frobenius loss function under some identifiability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The resultant estimators of the loading matrices are further treated as the new weight/projection matrices and we iteratively perform the above two steps until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Given the true dimensions of the factor matrices, we establish the convergence rates of the estimators for loading matrices and common components at the s-th iteration for any s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As a future direction, our methodology could be generalized to tensor-valued time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Intuitively, if one substitutes the squared loss in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) with the Huber loss, it would lead to a more robust estimator, which is of independent interest because real-world financial returns and macroeconomic indexes often exhibit heavy tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Since a significant amount of additional work is still needed, we leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' References Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Trapani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' A randomised sequential procedure to determine the number of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of the American Statistical Association 113, 1341–1349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Factor models for matrix-valued high-dimensional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of Econometrics 208, 231–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Westerlund, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Urbain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Cross-sectional averages versus principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of Econo- metrics 185, 372–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Xiong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Pelger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Large dimensional latent factor modeling with missing observations and appli- cations to causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of Econometrics, in press .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Projected estimation for large-dimensional matrix factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of Econometrics 229, 201–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Robust factor number specification for large-dimensional elliptical factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Journal of Multivariate analysis 174, 104543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 23 Supplementary Materials for “Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model by Random Projection” Yong He ∗, Ran Zhao∗, Wen-Xin Zhou†, This document provides the detailed proofs of the main theorems and additional lemmas and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' S1 Proofs of the main theorems S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the fact that H⊤ 1 (H1H⊤ 1 )+H1 = Ik1, H⊤ 2 (H2H⊤ 2 )+H2 = Ik2, then �Ft = H1FtH⊤ 2 + Et im- plies M⊤ 1 �FtM2 − Ft = M⊤ 1 EtM2 with M1 = (H1H⊤ 1 )+H1, M2 = (H2H⊤ 2 )+H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As ∥(H1H⊤ 1 )+H1∥2 = Op(ν−1 min(H1)) and (H2H⊤ 2 )+H2 = Op(ν−1 min(H2)), by Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (1), we have ∥M⊤ 1 �FtM2 − Ft∥2 = Op � 1 √p1p2 ν−1 min(H1)ν−1 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Proposition S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, λmin( 1 T M⊤�F⊤�FM) ≥ λmin( 1 T �F⊤�F)λmin(M⊤M) ≥ c(p1p2)−1λmin(D−2 H ) with large probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the SVD of H⊤, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=', H⊤ = UH(DH, 0)E⊤ H, we conclude that P�FM is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As �F = FH⊤ + E, then we have �FM − F = E(HH⊤)+H with M = (HH⊤)+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Further by the fact that ∥(HH⊤)+H∥2 = Op � ν−1 min � and Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (2), (3), we have 1 √ T ∥�FM − F∥2 = 1 √ T ∥E(HH⊤)+H∥2 ≤ 1 √ T ∥E∥2∥(HH⊤)+H∥2 = Op � 1 √p1p2 ν−1 min � , 1 T ∥F⊤(�FM − F)∥2 ≤ 1 T ∥F⊤E∥2∥(HH⊤)+H∥2 = Op � 1 √T p1p2 ν−1 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Further by ��� 1 T M⊤�F⊤�FM − 1 T F⊤F ��� 2 = ��� 1 T (�FM − F)⊤(�FM − F) + 1 T F⊤(�FM − F) + 1 T (�FM − F)⊤F ��� 2 ≤ 1 T ∥�FM − F∥2 2 + 2∥F⊤(�FM − F)∥2 = Op � 1 √T p1p2 ν−1 min + 1 p1p2 ν−2 min � , we have ���( 1 T M⊤�F⊤�FM)−1��� 2 = Op (1) and ��� � 1 T M⊤�F⊤�FM �−1 − � 1 T F⊤F �−1��� 2 = Op � 1 √T p1p2 ν−1 min + 1 p1p2 ν−2 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∗Institute of Financial Studies, Shandong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E-mail:heyong@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='cn, Zhaoran@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='cn †Department of Mathematical Sciences, University of California, San Diego, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E-mail:wez243@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='edu 1 As a result, P�FM − PF = 1 √ T �FM � ( 1 T M⊤�F⊤�FM)−1 − ( 1 T F⊤F)−1 � ( 1 √ T �FM)⊤ + 1 √ T �FM( 1 T F⊤F)−1 1 √ T (�FM − F)⊤ + 1 √ T (�FM − F)( 1 T F⊤F)−1 1 √ T F⊤, ∥P�FM − PF∥2 = Op � 1 √p1p2 ν−1 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Finally, noting that P�FP�FM = P�FM, we have ∥P�FPF − PF∥2 ≤ ∥P�F(PF − P�FM)∥2 + ∥P�FM − PF∥2 = Op � 1 √p1p2 ν−1 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Without loss of generality, we assume in the following that m1 = k1 = 1, m2 = k2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 1 T T � t=1 XtW2�F⊤ t = 1 T p1p2 T � t=1 XtW2W⊤ 2 X⊤ t W1 = 1 T p1p2 T � t=1 (RFtC⊤ + Et)W2W⊤ 2 (RFtC⊤ + Et)⊤W1 = 1 T p1p2 T � t=1 RFtC⊤W2W⊤ 2 CF⊤ t R⊤W1 + 1 T p1p2 T � t=1 RFtC⊤W2W⊤ 2 E⊤ t W1 + 1 T p1p2 T � t=1 EtW2W⊤ 2 CF⊤ t R⊤W1 + 1 T p1p2 T � t=1 EtW2W⊤ 2 E⊤ t W1 := δ1 + δ2 + δ3 + δ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following, we analyze δ1, δ2, δ3, δ4 term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the first term δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have ∥δ1∥F = ∥ 1 T p1p2 T � t=1 RFtC⊤W2W⊤ 2 CF⊤ t R⊤W1∥F = ∥p2 T T � t=1 RFtH⊤ 2 H2F⊤ t H1∥F ≤ p2∥R∥F∥ 1 T T � t=1 FtF⊤ t ∥F ∥∥H2∥2 2∥H1∥2 = Op �√p1p2νmax(H1)ν2 max(H2) � = Op �√p1p2νmin(H1)ν2 min(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the first term δ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have ∥δ2∥F = ∥ 1 T p1p2 T � t=1 RFtC⊤W2W⊤ 2 E⊤ t W1∥F ≤ ∥ 1 T p1p2 T � t=1 RFtW⊤ 2 E⊤ t W1∥F∥C⊤W2∥2 ≤ 1 T p1 ∥R∥F∥ T � t=1 FtW⊤ 2 E⊤ t W1∥F ∥H2∥2 = Op ��p2 T νmin(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where the last equation is derived according to Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 2 By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (3), we have E∥ �T t=1 FtW⊤ 2 E⊤ t ∥2 F = O(T p1p2), thus, ∥δ3∥F = ∥ 1 T p1p2 T � t=1 EtW2W⊤ 2 CF⊤ t R⊤W1∥F ≤ 1 T p1p2 ∥ T � t=1 EtW2F⊤ t ∥F ∥W⊤ 2 C∥2∥R⊤W1∥2 = Op ��p1p2 T νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (4), ∥δ4∥F = ∥ 1 T p1p2 T � t=1 EtW2W⊤ 2 E⊤ t W1∥F = Op ��p2 T + 1 √p1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Z = 1 T 2 ��T t=1 �FtW⊤ 2 X⊤ t � ��T t=1 XtW2�F⊤ t � , then Z = 1 p2 1p2 2T 2 ( T � t=1 W⊤ 1 XtW2W⊤ 2 X⊤ t )( T � t=1 XtW2W⊤ 2 X⊤ t W1) = (δ1 + δ2 + δ3 + δ4)⊤(δ1 + δ2 + δ3 + δ4) = 4 � i=1 4 � j=1 δ⊤ i δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We can prove that when ν2 min(H2) ≫ max( 1 T , 1 p2 ), the leading term of Z is δ⊤ 1 δ1, which is of the order Op � p1p2 2ν2 min(H1)ν4 min(H2) � , thus ∥Z−1/2∥F = Op � 1 √p1p2νmin(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �R(1) = √p1 � T � t=1 XtW2�F⊤ t � �� T � t=1 �FtW⊤ 2 X⊤ t � � T � t=1 XtW2�F⊤ t ��−1/2 = √p1 � 1 T T � t=1 XtW2�F⊤ t � Z−1/2 = R � 1 √p1p2 � 1 T T � t=1 FtC⊤W2W⊤ 2 CF⊤ t R⊤W1 � Z−1/2 � + 1 √p1p2 � 1 T T � t=1 RFtC⊤W2W⊤ 2 E⊤ t W1 � Z−1/2 + 1 √p1p2 � 1 T T � t=1 EtW2W⊤ 2 CF⊤ t R⊤W1 � Z−1/2 + 1 √p1p2 � 1 T T � t=1 EtW2W⊤ 2 E⊤ t W1 � Z−1/2 := I + II + III + IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let �H(1) r = 1 √p1p2T ��T t=1 FtC⊤W2W⊤ 2 CF⊤ t R⊤W1 � Z−1/2 then, �R(1) − R �H(1) r = II + III + IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1) 3 As ��� 1 T T � t=1 FtC⊤W2W⊤ 2 CF⊤ t R⊤W1 ��� 2 ≤ ��� 1 T T � t=1 FtC⊤W2W⊤ 2 CF⊤ t ��� 2∥R⊤W1∥2 = p1p2 2∥ 1 T T � t=1 FtH⊤ 2 H2F⊤ t ∥2∥H1∥2 ≤ p1p2 2∥ 1 T T � t=1 F⊤ t Ft∥2∥H2∥2 2∥H1∥2 = Op � p1p2 2νmin(H1)ν2 min(H2) � , thus we have the following results: ∥ �H(1) r ∥F = ∥ 1 √p1p2 ( 1 T T � t=1 FtC⊤W2W⊤ 2 CF⊤ t R⊤W1)Z−1/2∥F ≤ 1 √p1p2 ∥ 1 T T � t=1 FtC⊤W2W⊤ 2 CF⊤ t R⊤W1∥2∥Z−1/2∥F = Op (1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥II∥F = ∥ 1 √p1p2 � 1 T T � t=1 RFtC⊤W2W⊤ 2 E⊤ t W1 � Z−1/2∥F = Op � 1 √T p2 ν−1 min(H1)ν−1 min(H2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥III∥F = ∥ 1 √p1p2 � 1 T T � t=1 EtW2W⊤ 2 CF⊤ t R⊤W1 � Z−1/2∥F = Op �� p1 T p2 ν−1 min(H2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥IV∥F = ∥ 1 √p1p2 � 1 T T � t=1 EtW2W⊤ 2 E⊤ t W1 � Z−1/2∥F = Op � 1 √T p2 ν−1 min(H1)ν−2 min(H2) + 1 √p1p2 ν−1 min(H1)ν−2 min(H2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Finally, we can get 1 p1 ∥ �R(1) − R �H(1) r ∥2 F = Op � 1 T p2ν2 min(H2) + 1 T p1p2ν2 min(H1)ν4 min(H2) + 1 p2 1p2 2ν2 min(H1)ν4 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' It remains to show that �H(1)⊤ r �H(1) r p→ Ik1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under the condition that ν2 min(H2) ≫ max( 1 T , 1 p2 ), we can obtain 1 p1 ∥ �R(1) − R �H(1) r ∥2 F = op (1) , ∥ 1 p1 R⊤( �R(1) − R �H(1) r )∥F ≤ (∥R∥2 F p1 ∥ �R(1) − R �H(1) r ∥2 F p1 )1/2 = op(1), ∥ 1 p1 �R(1)⊤( �R(1) − R �H(1) r )∥F = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that p−1 1 �R(1)⊤ �R(1) = Ik1, while p−1 1 R⊤R = Ik1, then Ik1 = 1 p1 �R(1)⊤R �H(1) r + op(1) = �H(1)⊤ r �H(1) r + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 4 In the following we show the row-wise consistency of �R(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By equation (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1), we have � R(1) i· − �H(1)⊤ r Ri· = 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 EtW2W⊤ 2 CF⊤ t Ri· � + 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 RFtC⊤W2W⊤ 2 et,i· � + 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 EtW2W⊤ 2 et,i· � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (3), we have ����� 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 EtW2W⊤ 2 CF⊤ t Ri· ������ 2 ≤ 1 T √p1p2 ���Z−1/2��� F ����� T � t=1 W⊤ 1 EtW2F⊤ t ����� F ��W⊤ 2 C �� 2 = Op � 1 √T p1p2νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similar to the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (3), we can get ����T t=1 FtW⊤ 2 et,i· ��� 2 F = Op (T p2), then ����� 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 RFtC⊤W2W⊤ 2 et,i· ������ 2 ≤ 1 T √p1p2 ���Z−1/2��� F ��W⊤ 1 R �� 2 ��C⊤W2 �� 2 ����� T � t=1 FtW⊤ 2 et,i· ����� F = Op � 1 √T p2νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similar to the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (5), we can also get ��� �T t=1 W⊤ 1 EtW2W⊤ 2 et,i· ��� 2 F = Op � T p1p2 2 + T 2p2 � , ����� 1 T √p1p2 Z−1/2 � T � t=1 W⊤ 1 EtW2W⊤ 2 et,i· ������ 2 ≤ 1 T √p1p2 ���Z−1/2��� F ∥W2∥F ����� T � t=1 W⊤ 1 EtW2W⊤ 2 et,i· ����� F = Op � 1 √T p1p2νmin(H1)ν2 min(H2) + 1 p1p2νmin(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Symmetrically, the convergence rate of the �C can be similarly derived by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4) and we omit the proof for saving space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 5 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' First we decompose �T t=1 X⊤ t �R(1)�Ft into four terms: T � t=1 X⊤ t �R(1)�Ft = 1 p1p2 T � t=1 X⊤ t �R(1)W⊤ 1 XtW2 = 1 p1p2 T � t=1 (RFtC⊤ + Et)⊤ �R(1)W⊤ 1 (RFtC⊤ + Et)W2 = 1 p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 + 1 p1p2 T � t=1 E⊤ t �R(1)W⊤ 1 RFtC⊤W2 + 1 p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 EtW2 + 1 p1p2 T � t=1 E⊤ t �R(1)W⊤ 1 EtW2 = δ(1) 1 + δ(1) 2 + δ(1) 3 + δ(1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As ���� 1 T δ(1) 1 ���� F = ����� 1 T p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 ����� F ≍ ����� 1 T p2 T � t=1 CF⊤ t W⊤ 1 RFtC⊤W2 ����� F = p1∥C∥F ����� 1 T T � t=1 F⊤ t Ft ����� F ∥H1∥2 ∥H2∥2 = Op (p1 √p2νmin(H1)νmin(H2)) , thus, we have ∥δ(1) 1 ∥F = Op � T p1√p2νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we get ����� T � s=1 E⊤ s ( �R(1) − R �H(1) r )Fs ����� 2 F = Op � p2 1 p2ν2 min(H2) + p1 ν2 min(H1)ν4 min(H2) + T p2ν2 min(H1)ν4 min(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥δ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ∥F = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(1)W⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 RFtC⊤W2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ( �R(1) − R �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ R �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r )W⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 RFtC⊤W2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='νmin(H2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (3), we also have E∥ �T t=1 E⊤ t W1Ft∥2 F = O (T p1p2), thus 6 ���δ(1) 3 ��� F = ����� 1 p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 EtW2 ����� F ≍ 1 p2 ����� T � t=1 CF⊤ t W⊤ 1 EtW2 ����� F ≤ 1 p2 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ( �R(1) − R �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r )W⊤ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r W⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 EtW2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t W⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 EtW2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='��� �R(1) − R �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t W⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 EtW2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥R∥F ∥ �H(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ∥F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= Op ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T p1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Z(1) = (�T t=1 �F⊤ t �R(1)⊤Xt)(�T t=1 X⊤ t �R(1)�Ft), then Z(1) = �4 i=1 �4 j=1 δ(1)⊤ i δ(1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We can prove the leading term in Z(1) is δ(1)⊤ 1 δ(1) 1 , hence ∥(Z(1))−1/2∥F = Op � 1 T p1√p2νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �C(1) = √p2 � T � t=1 X⊤ t �R(1)�Ft � �� T � t=1 �F⊤ t �R(1)⊤Xt � � T � t=1 X⊤ t �R(1)�Ft ��−1/2 = √p2 � 1 p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 � � Z(1)�−1/2 + √p2 � 1 p1p2 T � t=1 E⊤ t �R(1)W⊤ 1 RFtC⊤W2 � � Z(1)�−1/2 + √p2 � 1 p1p2 T � t=1 CF⊤ t R⊤ �R(1)W⊤ 1 EtW2 � � Z(1)�−1/2 + √p2 � 1 p1p2 T � t=1 E⊤ t �R(1)W⊤ 1 EtW2 � � Z(1)�−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' = √p2 � δ(1) 1 (Z(1))−1/2 + δ(1) 2 (Z(1))−1/2 + δ(1) 3 (Z(1))−1/2 + δ(1) 4 (Z(1))−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let �H(1) c = � 1 p1√p2 �T t=1 F⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 � � Z(1)�−1/2, then �C(1) − C �H(1) c = √p2δ(1) 2 (Z(1))−1/2 + √p2δ(1) 3 (Z(1))−1/2 + √p2δ(1) 4 (Z(1))−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 7 ∥ �H(1) c ∥F = ����� � 1 p1√p2 T � t=1 F⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 � � Z(1)�−1/2 ����� F ≤ ����� 1 p1√p2 T � t=1 F⊤ t R⊤ �R(1)W⊤ 1 RFtC⊤W2 ����� F ���� � Z(1)�−1/2���� F = Op (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥δ(1) 2 (Z(1))−1/2∥F = Op � 1 √T p1 + 1 T p2 ν−1 min(H2) + 1 T √p1p2 ν−1 min(H1)ν−2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥δ(1) 3 (Z(1))−1/2∥F = Op � 1 √T p1p2 ν−1 min(H1)ν−1 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥δ(1) 4 (Z(1))−1/2∥F = Op � 1 √T p1p2 ν−1 min(H1)ν−1 min(H2) + 1 p1p2 ν−1 min(H1)ν−1 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As a result 1 p2 ∥�C(1) − C �H(1) c ∥2 F = Op � 1 T p1 + 1 p2 1p2 2ν2 min(H1)ν2 min(H2) + 1 T 2p2 2ν2 min(H2) + 1 T p1p2ν2 min(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ��� 1 p2 C⊤(�C(1) − C �H(1) c ) ��� F ≤ (∥C∥2 F p2 ∥�C(1) − C �H(1) c ∥2 F p2 )1/2 = op (1) , ��� 1 p2 �C(1)⊤(�C(1) − C �H(1) c ) ��� F = op (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that p−1 2 C⊤C = Ik2, p−1 2 �C(1)⊤ �C(1) = Ik2, then Ik2 = 1 p2 �C(1)⊤C �H(1) c + op (1) = �H(1)⊤ c �H(1) c + op (1) , �H(1)⊤ c �H(1) c p→ Ik2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In the following, we shows the row-wise consistency of �C(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For j ≤ p2, � C(1) j· − �H(1)⊤ c Cj· = 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 CF⊤ t R⊤W1 �R(1)⊤et,·j � + 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤RFtCj· � + 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤et,·j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Firstly, similar to the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we can obtain ����� T � t=1 F⊤ t ( �R(1) − R �H(1) r )⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� 2 F = Op � p2 1 p2 2ν2 min(H2) + p1 p2ν2 min(H1)ν4 min(H2) + T p2 2ν2 min(H1)ν4 min(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 8 ����� 1 p1√p2 T � t=1 W⊤ 2 CF⊤ t R⊤W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� 2 ≤ 1 p1√p2 ����� T � t=1 F⊤ t �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ∥W⊤ 2 C∥2∥R⊤W1∥2 ≤ 1 p1√p2 ����� T � t=1 F⊤ t ( �R(1) − Rt �H(1) r )⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ∥W⊤ 2 C∥2∥R⊤W1∥2 + 1 p1√p2 ����� T � t=1 F⊤ t R⊤ t et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ∥W⊤ 2 C∥2∥R⊤W1∥2∥ �H(1) c ∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 CF⊤ t R⊤W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ������ 2 ≤ 1 p1√p2 ����� T � t=1 W⊤ 2 CF⊤ t R⊤W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ���(Z(1))−1/2��� F = Op � 1 T p2νmin(H2) + 1 T √p1p2νmin(H1)ν2 min(H2) + 1 √T p1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Secondly, ����� 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤RFtCj· ������ 2 ≍ ����� 1 √p2 (Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1Ft ������ 2 = Op � 1 √T p1p2νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thirdly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F = ����� T � t=1 W⊤ 2 E⊤ t W1( �R(1) − R �H(1) r + R �H(1) r )⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ≤ ����� T � t=1 W⊤ 2 E⊤ t W1e⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ��� �R(1) − R �H(1) r ��� F + ����� T � t=1 W⊤ 2 E⊤ t W1e⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ����� F ∥R∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� 1 p1√p2 (Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ������ F ≤ 1 p1√p2 �����(Z(1))−1/2 � T � t=1 W⊤ 2 E⊤ t W1 �R(1)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='·j ������ F = Op � 1 p1p2νmin(H1)νmin(H2) + 1 √T p1p2νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' The conclusion is obtained by summarizing the above three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 9 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By definition, �F(2) t = 1 p1p2 �R(1)⊤Xt �C(1) = 1 p1p2 �R(1)⊤RFtC⊤ �C(1) + 1 p1p2 �R(1)⊤Et �C(1) = 1 p1p2 �R(1)⊤ � R − �R(1)( �H(1) r )−1 + �R(1)( �H(1) r )−1� Ft � C − �C(1)( �H(1) c )−1 + �C(1)( �H(1) c )−1�⊤ �C(1) + 1 p1p2 � �R(1) − R �H(1) r + R �H(1) r �⊤ Et � �C(1) − C �H(1) c + C �H(1) c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that �R(1)⊤ �R(1) = p1Ik1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �C(1)⊤ �C(1) = p2Ik2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' then �F(2) t − ( �H(1) r )−1Ft � ( �H(1) c )−1�⊤ = 1 p1p2 �R(1)⊤ � R − �R(1)( �H(1) r )−1� Ft � C − �C( �H(1) c )−1�⊤ �C(1) + 1 p1 �R(1)⊤ � R − �R(1)( �H(1) r )−1� Ft � ( �H(1) c )−1�⊤ + 1 p2 ( �H(1) r )−1Ft � C − �C( �H(1) c )−1�⊤ �C(1) + 1 p1p2 � �R(1) − R �H(1) r �⊤ Et � �C(1) − C �H(1) c � + 1 p1p2 � �R(1) − R �H(1) r �⊤ EtC �H(1) c + 1 p1p2 �H(1)⊤ r R⊤Et � �C(1) − C �H(1) c � + 1 p1p2 �H(1)⊤ r R⊤EtC �H(1) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5, we prove that ���R⊤( �R(1) − R �H(1) r ) ��� F = Op � √p1 √T p2νmin(H1)νmin(H2) + 1 p2νmin(H1)ν2 min(H2) � , ���C⊤(�C(1) − C �H(1) c ) ��� F = Op � 1 T νmin(H2) + √p2 √T p1νmin(H1)νmin(H2) + 1 p1νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Therefore, by Cauchy-Schwartz inequality, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, and the bounds for ∥R⊤Et∥2 F , ∥EtC∥2 F and ∥R⊤EtC∥2 F , we have �����F(2) t − ( �H(1) r )−1Ft � ( �H(1) c )−1�⊤���� F = Op � 1 √ Tp1 + 1 √ Tp2νmin(H2) + 1 √p1p2 + 1 p1p2νmin(H1)ν2 min(H2) + γf � , where γf = 1 √T p1p2νmin(H1)νmin(H2) + 1 T p1√p2νmin(H1)ν2 min(H2) + 1 T p1p2ν2 min(H1)ν3 min(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Next, for any t, i, j �S(2) t,ij − St,ij = � R(1)⊤ i· �F(2) t � C(1) j· − R⊤ i· FtCj· = ( � R(1) i· − �H(1)⊤ r Ri· + �H(1)⊤ r Ri·)⊤�F(2) t ( � C(1) j· − �H(1)⊤ c Cj· + �H(1)⊤ c Cj·) − R⊤ i· FtCj· = ( � R(1) i· − �H(1)⊤ r Ri·)⊤�F(2) t ( � C(1) j· − �H(1)⊤ c Cj·) + R⊤ i· �H(1) r �F(2) t ( � C(1) j· − �H(1)⊤ c Cj·) + ( � R(1) i· − �H(1)⊤ r Ri·)⊤�F(2) t �H(1)⊤ c Cj· + R⊤ i· ( �H(1) r �F(2) t �H(1)⊤ c − Ft)Cj·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 10 Then, by Cauchy-Schwartz inequality, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and the consistency of the estimators for the factor matrix, we have |�S(2) t,ij−St,ij| = Op � 1 √T p1 + 1 √p1p2 + 1 √T p2νmin(H2) + 1 √T p1p2νmin(H1)ν2 min(H2) + 1 p1p2νmin(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Recall first that �R(s+1) = √p1 � T � t=1 Xt �C(s)�F(s+1)⊤ t � �� T � t=1 �F(s+1) t �C(s)⊤X⊤ t � � T � t=1 Xt �C(s)�F(s+1)⊤ t ��−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' T � t=1 Xt �C(s)�F(s+1)⊤ t = 1 p1p2 T � t=1 Xt �C(s) �C(s)⊤X⊤ t �R(s) = 1 p1p2 T � t=1 (RFtC⊤ + Et)�C(s) �C(s)⊤(RFtC⊤ + Et)⊤ �R(s) = 1 p1p2 T � t=1 RFtC⊤ �C(s) �C(s)⊤CF⊤ t R⊤ �R(s) + 1 p1p2 T � t=1 RFtC⊤ �C(s) �C(s)⊤E⊤ t �R(s) + 1 p1p2 T � t=1 Et �C(s) �C(s)⊤CF⊤ t R⊤ �R(s) + 1 p1p2 T � t=1 Et �C(s) �C(s)⊤E⊤ t �R(s) = δ(s+1) 1 + δ(s+1) 2 + δ(s+1) 3 + δ(s+1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As ∥ 1 T δ(s+1) 1 ∥2 F = ∥ 1 T p1p2 T � t=1 RFtC⊤ �C(s) �C(s)⊤CF⊤ t R⊤ �R(s)∥2 F ≍ p2 2∥ 1 T T � t=1 RFtF⊤ t ∥2 F ≤ p2 2∥R∥2 F∥ 1 T T � t=1 FtF⊤ t ∥2 F = Op � p1p2 2 � , thus, ∥δ(s+1) 1 ∥2 F = Op � T 2p1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥δ(s+1) 2 ∥2 F = ∥ 1 p1p2 T � t=1 RFtC⊤ �C(s) �C(s)⊤E⊤ t �R(s)∥2 F ≍ 1 p2 1 ∥ T � t=1 RFt �C(s)⊤E⊤ t �R(s)∥2 F = 1 p2 1 ∥ T � t=1 RFt �C(s)⊤E⊤ t ( �R(s) − R �H(s) r )∥2 F + 1 p2 1 ∥ T � t=1 RFt �C(s)⊤E⊤ t R �H(s) r ∥2 F ≤ 1 p2 1 ∥R∥2 F∥ T � t=1 Ft �C(s)⊤E⊤ t ∥2 F ∥ �R(s) − R �H(s) r ∥2 F + 1 p2 1 ∥R∥2 F∥ T � t=1 Ft �C(s)⊤E⊤ t R∥2 F∥ �H(s) r ∥2 F = Op � T p1p2 2w(s) r w(s) c + T p1p2w(s) r + T p2 2w(s) c + T p2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 11 the last equality is due to Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥δ(s+1) 3 ∥2 F = ∥ 1 p1p2 T � t=1 Et �C(s) �C(s)⊤CF⊤ t R⊤ �R(s)∥2 F ≍ ∥ T � t=1 Et �C(s)F⊤ t ∥2 F = Op � T p1p2 2w(s) c + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥δ(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F = ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ( �R(s) − R �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r )∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F + ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥Et �C(s)∥4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ∥ �R(s) − R �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ∥ �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= Op ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T p2 + T 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p1w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p1p2w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='w(s)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2w(s)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' the last equality is according to Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7 and Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Z(s+1) = ��T t=1 �F(s+1) t �C(s)⊤X⊤ t � ��T t=1 Xt �C(s)�F(s+1)⊤ t � , Z(s+1) = 1 p2 1p2 2 � T � t=1 �R(s)⊤Xt �C(s) �C(s)⊤X⊤ t � � T � t=1 Xt �C(s) �C(s)⊤X⊤ t �R(s) � = 4 � i=1 4 � j=1 δ(s+1)⊤ i δ(s+1) j , then we can prove ��(Z(s+1))−1/2��2 F = Op � 1 T 2p1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�R(s+1) = √p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Xt �C(s)�F(s+1)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� �� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�F(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�C(s)⊤X⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Xt �C(s)�F(s+1)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='��−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= √p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Xt �C(s)�F(s+1)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Z(s+1)�1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='FtC⊤ �C(s) �C(s)⊤CF⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R⊤ �R(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Z(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='RFtC⊤ �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Z(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤CF⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R⊤ �R(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Z(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='√p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Et �C(s) �C(s)⊤E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Z(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 12 Denote �H(s+1) r = 1 √p1p2 ��T t=1 FtC⊤ �C(s) �C(s)⊤CF⊤ t R⊤ �R(s)� � Z(s+1)�−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have �R(s+1) − R �H(s+1) r = 1 √p1p2 � T � t=1 RFtC⊤ �C(s) �C(s)⊤E⊤ t �R(s) � � Z(s+1)�−1/2 + 1 √p1p2 � T � t=1 Et �C(s) �C(s)⊤CF⊤ t R⊤ �R(s) � � Z(s+1)�−1/2 + 1 √p1p2 � T � t=1 Et �C(s) �C(s)⊤E⊤ t �R(s) � � Z(s+1)�−1/2 = √p1(δ(s+1) 2 + δ(s+1) 3 + δ(s+1) 4 )(Z(s+1))−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 1 p1 ∥ �R(s+1) − R �H(s+1) r ∥2 F = Op � 1 T p2 + 1 p2 1p2 2 + 1 p2 2 w(s) r + 1 T w(s) c + w(s) r w(s) c p2 + w(s) r w(s)2 c + 1 p2 1 w(s)2 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the row-consistency of �R(s+1), we have � R(s+1) i· − �H(s+1)⊤ r Ri· = 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤Et �C(s) �C(s)⊤CF⊤ t Ri· � + 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤RFtC⊤ �C(s) �C(s)⊤et,i· � + 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤Et �C(s) �C(s)⊤et,i· � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' First, by Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9 (1) and (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� T � t=1 �R(s)⊤Et �C(s) �C(s)⊤CF⊤ t Ri· ����� 2 F ≍ p2 2 ����� T � t=1 �R(s)⊤Et �C(s)F⊤ t ����� 2 F ≤ p2 2 ����� T � t=1 Et �C(s)F⊤ t ����� 2 F ����R(s) − R �H(s) r ��� 2 F + p2 2 ����� T � t=1 R⊤Et �C(s)F⊤ t ����� 2 F = Op � T p2 1p4 2w(s) r w(s) c + T p2 1p3 2w(s) r + T p1p4 2w(s) c + T p1p3 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤Et �C(s) �C(s)⊤CF⊤ t Ri· ������ 2 F ≤ 1 p1p2 2 ���(Z(s+1))−1/2��� 2 F ����� T � t=1 �R(s)⊤Et �C(s) �C(s)⊤CF⊤ t Ri· ����� 2 F = Op � w(s) r w(s) c T + w(s) r T p2 + w(s) c T p1 + 1 T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 13 Second, similar to the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9 (1), we have ����� T � t=1 Ft �C(s)⊤et,i· ����� 2 F = Op � T p2 2w(s) c + T p2 � , ����� 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤RFtC⊤ �C(s) �C(s)⊤et,i· ������ 2 F ≲ p1 ���� � Z(s+1)�−1/2���� 2 F ����� T � t=1 Ft �C(s)⊤et,i· ����� 2 F = Op � w(s) c T + 1 T p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Third, similar to the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7 (1) and Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8 (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have ����� T � t=1 �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F = Op � T p2 2w(s) c + T p2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� T � t=1 R⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F = Op � T p1p3 2 + T 2p2 2 + (T 2p4 2 + T p1p4 2)w(s)2 c � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ����� T � t=1 �R(s)⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F ≤ ����� T � t=1 ( �R(s) − R �H(s) r )⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F + ����� T � t=1 R⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F ≤ ��� �R(s) − R �H(s) r ��� 2 F ����� T � t=1 Et �C(s) ����� 2 F ����� T � t=1 �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F + ����� T � t=1 R⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F = Op � T p1p3 2 + T 2p2 2 + T 2p2 1p2 2w(s) r + T p1p4 2w(s)2 c + T 2p4 2w(s)2 c + T 2p2 1p4 2w(s)2 c ws r + T 2p2 1p3 2w(s) r w(s) c � ����� 1 √p1p2 � Z(s+1)�−1/2 � T � t=1 �R(s)⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ������ 2 F ≤ 1 p1p2 2 ���� � Z(s+1)�−1/2���� 2 F ����� T � t=1 �R(s)⊤Et �C(s) �C(s)⊤et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i· ����� 2 F = Op � 1 T p1p2 + 1 p2 1p2 2 + w(s) r p2 2 + w(s)2 c T p1 + w(s)2 c p2 1 + w(s) r w(s)2 c + w(s) r w(s) c p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As a result ∥ � R(s+1) i· − �H(s+1)⊤ r Ri·∥2 2 = Op � 1 T p2 + 1 p2 1p2 2 + 1 p2 2 w(s) r + 1 T w(s) c + w(s) r w(s) c p2 + w(s) r w(s)2 c + 1 p2 1 w(s)2 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the �C(s+1), �C(s+1) = √p2 � T � t=1 X⊤ t �R(s+1)�F(s+1) t � �� T � t=1 �F(s+1)⊤ t �R(s+1)⊤Xt � � T � t=1 X⊤ t �R(s+1)�F(s+1) t ��−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 14 T � t=1 X⊤ t �R(s+1)�F(s+1) t = 1 p1p2 T � t=1 X⊤ t �R(s+1) �R(s)⊤Xt �C(s) = 1 p1p2 T � t=1 (RFtC⊤ + Et)⊤ �R(s+1) �R(s)⊤(RFtC⊤ + Et)�C(s) = 1 p1p2 T � t=1 CF⊤ t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s) + 1 p1p2 T � t=1 E⊤ t �R(s+1) �R(s)⊤RFtC⊤ �C(s) + 1 p1p2 T � t=1 CF⊤ t R⊤ �R(s+1) �R(s)⊤Et �C(s) + 1 p1p2 T � t=1 E⊤ t �R(s+1) �R(s)⊤Et �C(s) = ∆(s+1) 1 + ∆(s+1) 2 + ∆(s+1) 3 + ∆(s+1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As ∥ 1 T ∆(s+1) 1 ∥F = ∥ 1 T p1p2 T � t=1 CF⊤ t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s)∥2 F ≍ p2 1∥ 1 T T � t=1 CF⊤ t Ft∥2 F = Op � p2 1p2 � , then ∥∆(s+1) 1 ∥2 F = Op � T 2p2 1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9, ∥∆(s+1) 2 ∥2 F = ∥ 1 p1p2 T � t=1 E⊤ t �R(s+1) �R(s)⊤RFtC⊤ �C(s)∥2 F ≍ ∥ T � t=1 E⊤ t �R(s+1)Ft∥F = Op � T p2 1p2w(s+1) r + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥∆(s+1) 3 ∥2 F = ∥ 1 p1p2 T � t=1 CF⊤ t R⊤ �R(s+1) �R(s)⊤Et �C(s)∥2 F ≍ 1 p2 2 ∥ T � t=1 CF⊤ t �R(s)⊤Et �C(s)∥2 F ≤ 1 p2 2 ∥ T � t=1 CF⊤ t �R(s)⊤Et(�C(s) − C �H(s) c )∥2 F + 1 p2 2 ∥ T � t=1 CF⊤ t �R(s)⊤EtC �H(s) c ∥2 F ≤ 1 p2 2 ∥C∥2 F∥ T � t=1 F⊤ t �R(s)⊤Et∥2 F ∥�C(s) − C �H(s) c ∥2 F + 1 p2 2 ∥C∥2 F∥ T � t=1 F⊤ t �R(s)⊤EtC∥2 F∥ �H(s) c ∥2 F = Op � T p2 1p2w(s) r w(s) c + T p1p2w(s) c + T p2 1w(s) r + T p1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' According to Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7 and Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥∆(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F = ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤Et �C(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤Et(�C(s) − C �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c )∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤EtC �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ∥�C(s) − C �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤EtC∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ∥ �H(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= Op ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T p1 + T 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p2w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p1p2w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='w(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r w(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let Y(s+1) = ��T t=1 �F(s+1)⊤ t �R(s+1)⊤Xt � ��T t=1 X⊤ t �R(s+1)�F(s+1) t � , then Y(s+1) = �4 i=1 �4 j=1 ∆(s+1) i ∆(s+1) j , 15 we can prove ���� � Y(s+1)�−1/2���� = Op � 1 T 2p2 1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�C(s+1) = √p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='X⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1)�F(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� �� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�F(s+1)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�R(s+1)⊤Xt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='X⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1)�F(s+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='��−1/2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Y(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1√p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='F⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Y(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1√p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤RFtC⊤ �C(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Y(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1√p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='CF⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t R⊤ �R(s+1) �R(s)⊤Et �C(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Y(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='p1√p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t �R(s+1) �R(s)⊤Et �C(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Y(s+1)�−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Denote �H(s+1) c = 1 p1√p2 ��T t=1 F⊤ t R⊤ �R(s+1) �R(s)⊤RFtC⊤ �C(s)� � Y(s+1)�−1/2, then �C(s+1) − C �H(s+1) c = 1 p1√p2 � T � t=1 E⊤ t �R(s+1) �R(s)⊤RFtC⊤ �C(s) � � Y(s+1)�−1/2 + 1 p1√p2 � T � t=1 CF⊤ t R⊤ �R(s+1) �R(s)⊤Et �C(s) � � Y(s+1)�−1/2 + 1 p1√p2 � T � t=1 E⊤ t �R(s+1) �R(s)⊤Et �C(s) � � Y(s+1)�−1/2 = √p2(∆(s+1) 2 + ∆(s+1) 3 + ∆(s+1) 4 )(Y(s+1))−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' thus 1 p2 ∥�C(s+1) − C �H(s+1) c ∥2 F = Op � 1 T p1 + 1 p2 1p2 2 + γ(s+1) c � , where γ(s+1) c = 1 T p2 w(s) r + 1 T w(s+1) r + 1 p2 1 w(s) c + 1 p1 w(s) r w(s) c + 1 T w(s) r w(s) c + w(s) r w(s+1) r w(s) c + 1 p2 2 w(s) r w(s+1) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similar to the proof the row-consistency of �R(s+1), for any j ∈ [p2], we can obtain ∥ � C(s+1) j· − �H(s+1)⊤ c Cj·∥2 2 = Op � 1 T p1 + 1 p2 1p2 2 + γ(s+1) c � , where γ(s+1) c = 1 T p2 w(s) r + 1 T w(s+1) r + 1 p2 1 w(s) c + 1 p1 w(s) r w(s) c + 1 T w(s) r w(s) c + w(s) r w(s+1) r w(s) c + 1 p2 2 w(s) r w(s+1) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 16 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By definition, �F(s+1) t = 1 p1p2 �R(s)⊤Xt �C(s) = 1 p1p2 �R(s)⊤RFtC⊤ �C(s) + 1 p1p2 �R(s)⊤Et �C(s) = 1 p1p2 �R(s)⊤ � R − �R(s)( �H(s) r )−1 + �R(s)( �H(s) r )−1� Ft � C − �C(s)( �H(s) c )−1 + �C(s)( �H(s) c )−1�⊤ �C(s) + 1 p1p2 � �R(s) − R �H(s) r + R �H(s) r �⊤ Et � �C(s) − C �H(s) c + C �H(s) c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that �R(s)⊤ �R(s) = p1Ik1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �C(s)⊤ �C(s) = p2Ik2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' then �F(s+1) t − ( �H(s) r )−1Ft � ( �H(s) c )−1�⊤ = 1 p1p2 �R(s)⊤ � R − �R(s)( �H(s) r )−1� Ft � C − �C( �H(s) c )−1�⊤ �C(s) + 1 p1 �R(s)⊤ � R − �R(s)( �H(s) r )−1� Ft � ( �H(s) c )−1�⊤ + 1 p2 ( �H(s) r )−1Ft � C − �C( �H(s) c )−1�⊤ �C(s) + 1 p1p2 � �R(s) − R �H(s) r �⊤ Et � �C(s) − C �H(s) c � + 1 p1p2 � �R(s) − R �H(s) r �⊤ EtC �H(s) c + 1 p1p2 �H(s)⊤ r R⊤Et � �C(s) − C �H(s) c � + 1 p1p2 �H(s)⊤ r R⊤EtC �H(s) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Therefore, by Cauchy-Schwartz inequality, Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, we have �����F(s+1) t − ( �H(s) r )−1Ft � ( �H(s) c )−1�⊤���� F = Op \uf8eb \uf8ed � w(s) r √p2 + � w(s−1) r √T p2 + � w(s) c √p1 + � w(s−1) c √T p1 + 1 √p1p2 + γ(s+1) f \uf8f6 \uf8f8 , where γ(s+1) f = � w(s) r w(s) c + � w(s−1) r w(s−1) c √ T + � w(s−1) r w(s−1) c p1 + � w(s−1) r √p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Next, for any t, i, j �S(s+1) t,ij − St,ij = � R(s)⊤ i· �F(s+1) t � C(s) j· − R⊤ i· FtCj· = ( � R(s) i· − �H(s)⊤ r Ri· + �H(s)⊤ r Ri·)⊤�F(s+1) t ( � C(s) j· − �H(s)⊤ c Cj· + �H(s)⊤ c Cj·) − R⊤ i· FtCj· = ( � R(s) i· − �H(s)⊤ r Ri·)⊤�F(s+1) t ( � C(s) j· − �H(s)⊤ c Cj·) + R⊤ i· �H(s) r �F(s+1) t ( � C(s) j· − �H(s)⊤ c Cj·) + ( � R(s) i· − �H(s)⊤ r Ri·)⊤�F(s+1) t �H(s)⊤ c Cj· + R⊤ i· ( �H(s) r �F(s+1) t �H(s)⊤ c − Ft)Cj·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then, by Cauchy-Schwartz inequality, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and the consistency of the estimators for the factor matrices, we have |�S(s+1) t,ij − St,ij| = Op \uf8eb \uf8ed � w(s) r + � w(s−1) r √T p2 + � w(s) c + � w(s−1) c √T p1 + 1 √p1p2 + γ(s+1) \uf8f6 \uf8f8 , 17 where γ(s+1) = � w(s−1) r w(s−1) c √ T + � w(s−1) r w(s−1) c p1 + � w(s−1) r √p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' S2 Propositions and Lemmas In this section, we first give some propositions and lemmas which are essential for the proofs of main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proposition S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose m1 ≥ k1, m2 ≥ k2 and T, p1, p2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Under Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3, we have λmin( 1 T �F⊤�F) ≥ c(p1p2)−1 with probability approaching one for some c > 0, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' It is easy to get that �F = FH⊤ + E, where E = EW/(p1p2), W = W2 ⊗ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let ∆ := 1 T HF⊤E + 1 T E⊤FH⊤ + 1 T EE⊤E + 1 T (E⊤E − EE⊤E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then we have 1 T �F⊤�F = 1 T HF⊤FH⊤ + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the assumption λmin � 1 T EE⊤E � = λmin � 1 T T � t=1 EVec(Et)Vec(Et)⊤ � ≥ c0, and the property of the Kronecker product, we can get λmin � 1 T EE⊤E � ≥ λmin � 1 T EE⊤E � λmin � 1 p2 1p2 2 W⊤W � ≥ c0(p1p2)−1 for some c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' In addition, in the following Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (4), we show that ∥ 1 T (E⊤E − EE⊤E)∥2 = Op � 1 p1p2 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Therefore, the inequality ∥ 1 T (E⊤E − EE⊤E)∥2 ≤ 1 2λmin( 1 T EE⊤E) hold with large probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' We now continue the argument conditioning on this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let v be the unit vector such that v⊤ 1 T �F⊤�Fv = λmin( 1 T �F⊤�F), as v⊤ 1 T �F⊤�Fv = v⊤ 1 T HF⊤FH⊤v + v⊤∆v, then we have λmin( 1 T �F⊤�F) ≥ 1 T v⊤HF⊤FH⊤v + 2 T v⊤HF⊤Ev + c0 2p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' If v⊤H = 0, then λmin( 1 T �F⊤�F) ≥ c0 2p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' If v⊤H ̸= 0, we have 1 T v⊤HF⊤FH⊤v > 0 with large probability because 1 T F⊤F is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Let α2 v = 1 T v⊤HF⊤FH⊤v, X = ( α2 v T p1p2 )−1/22v⊤ 1 T HF⊤Ev, 2v⊤ 1 T HF⊤Ev = X � α2 v T p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 18 Then λmin( 1 T �F⊤�F) ≥ α2 v + X � α2 v T p1p2 + c0 2p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' First, we prove that X = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3, it holds that λmin( 1 T F⊤F) > c > 0, then we have α2 v ≥ λmin( 1 T F⊤F)v⊤HH⊤v > c∥v⊤H∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (3) in the following, we have ∥ 1 T F⊤E∥2 2 = Op � 1 T p1p2 � and as a result, |X|2 ≤ 4T p1p2α−2 v ∥v⊤H∥2 2∥ 1 T F⊤E∥2 2 ≤ Op(1)α−2 v ∥v⊤H∥2 2 ≤ Op(1)c−1∥v⊤H∥−2 2 ∥v⊤H∥2 2 = Op (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then we consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Case one: α2 v ≤ 4|X| � α2 v T p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Then |αv| ≤ 4|X| 1 √T p1p2 and λmin( 1 T �F⊤�F) ≥ c0 2p1p2 − |X||αv| 1 √T p1p2 ≥ c0 2p1p2 − 4|X|2 1 T p1p2 ≥ c0 4p1p2 , where the last inequality holds with probability approaching to 1 by the fact that X = Op(1) and T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Case two: α2 v > 4|X| � α2 v T p1p2 , then λmin( 1 T �F⊤�F) ≥ α2 v − |X| � α2 v T p1p2 + c0 2p1p2 ≥ 3 4α2 v + c0 2p1p2 ≥ c0 2p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' From what has been discussed above, we conclude that with probability approaching one, we have λmin( 1 T �F⊤�F) > c0/(p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For any m1 ≥ 1 and m2 ≥ 1, (note that m1, m2 can be either smaller, equal to or larger than k1, k2), (1) ∥E(E⊤ t Et)∥2 = O � 1 p1p2 � and ∥Et∥2 = Op � 1 √p1p2 � , t ∈ [T ], (2) ∥E∥2 = Op �� T p1p2 � , (3) E∥ 1 T F⊤E∥2 2 ≤ O � 1 T p1p2 � , (4) ∥ 1 T (E⊤E − EE⊤E)∥2 ≤ Op � 1 p1p2 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (1) By Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3, we have ∥E(EtE⊤ t )∥2 ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 19 Thus, ∥E(E⊤ t W1W⊤ 1 Et)∥2 ≤ tr(EE⊤ t W1W⊤ 1 Et) ≤ E(tr(E⊤ t W1W⊤ 1 Et)) = tr(W⊤ 1 E(EtE⊤ t )W1) ≤ m1∥W1∥2 2∥E(EtE⊤ t )∥2 ≤ cm1p1, where the penultimate inequality is derived as follows: let vi be the i-th eigenvector of W⊤ 1 E(EtE⊤ t )W1, then tr(W⊤ 1 E(EtE⊤ t )W1) = m1 � i=1 v⊤ i W⊤ 1 E(EtE⊤ t )W1vi ≤ ∥E(EtE⊤ t )∥2 m1 � i=1 ∥W1vi∥2 2 ≤ m1∥E(EtE⊤ t )∥2∥W1∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥E(E⊤ t Et)∥2 = 1 p2 1p2 2 ∥E(W⊤ 2 E⊤ t W1W⊤ 1 EtW2)∥2 = 1 p2 1p2 2 ∥W⊤ 2 E(E⊤ t W1W⊤ 1 Et)W2∥2 ≤ 1 p2 1p2 2 ∥W2∥2 2∥E(E⊤ t W1W⊤ 1 Et)∥2, which imply ∥E(E⊤ t Et)∥2 ≤ c p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thus, we have E∥E2 t∥2 ≤ tr(EE⊤ t Et) ≤ m2∥EE⊤ t Et∥2 ≤ cm1m2 p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) It holds that E = 1 p1p2 EW, W = W2 ⊗ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3, ��� 1 T E(E⊤E) ��� 2 = ��� 1 T T � t=1 EVec(Et)Vec(Et)⊤��� 2 = ∥EVec(Et)Vec(Et)⊤∥2 ≤ E∥E � Vec(Et)Vec(Et)⊤|F � ∥2 ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thus E ��� 1 p1p2 EW ��� 2 2 ≤ 1 p2 1p2 2 trE(W⊤E⊤EW) ≤ 1 p2 1p2 2 ∥W∥2 F∥EE⊤E∥2 ≤ cT p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) By the Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have 1 T T � s=1 T � T =1 E(∥Ft∥F ∥Fs∥F )tr(E(EtE⊤ s |F)) < c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='then we can obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E∥ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T F⊤E∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E∥F⊤EW∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='W⊤Vec(Et)Vec(Ft)⊤∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='W⊤Vec(Et)Vec(Ft)⊤∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� �����F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='tr( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Ft)Vec(Et)⊤WW⊤Vec(Es)Vec(Fs)⊤) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� �����F ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Ft)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Et)⊤WW⊤Vec(Es) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='���F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Fs)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Ft)Vec(Fs)⊤E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='Vec(Et)⊤WW⊤Vec(Es) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='���F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijE � Vec(Et)⊤WW⊤Vec(Es) ���F �� = 1 T 2p2 1p2 2 k1 � i=1 k2 � j=1 T � t=1 T � s=1 E � ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijtr � W⊤E � Vec(Es)Vec(Et)⊤���F � W �� ≤ 1 T 2p2 1p2 2 k1 � i=1 k2 � j=1 T � t=1 T � s=1 E � |ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij|∥W∥2 F∥E � Vec(Es)Vec(Et)⊤���F �� ≤ c T 2p1p2 T � t=1 T � s=1 E∥Ft∥F ∥∥Fs∥F E � Vec(Es)Vec(Et)⊤���F � ∥ ≤ C T p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Thus, E∥ 1 T F⊤E∥2 2 ≤ O � 1 T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (4) By the Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have E∥ 1 T (E⊤E − EE⊤E)∥2 2 ≤ E∥ 1 T (E⊤E − EE⊤E)∥2 F ≤ m1m2 � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='q=1 E \uf8eb \uf8ed 1 T p2 1p2 2 T � t=1 p1p2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j=1 wikwjq(etietj − Eetietj) \uf8f6 \uf8f8 2 = m1m2 � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='q=1 E \uf8eb \uf8ed 1 T p2 1p2 2 T � t=1 p1p2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j=1 wikwjq (Vec(Et)iVec(Et)j − EVec(Et)iVec(Et)j) \uf8f6 \uf8f8 2 ≤ 1 T p2 1p2 2 m1m2 � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='q=1 1 T p2 1p2 2 T � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 p1p2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='v=1 ��wikwjqwukwvq ����Cov (Vec(Et)iVec(Et)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Vec(Es)uVec(Es)v) �� ≤ c T p2 1p2 2 1 T p2 1p2 2 T � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 p1 � i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='v1=1 p2 � i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='v2=1 |Cov(et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i1i2et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='u1u2es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='v1v2)| ≤ c T p2 1p2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' as min{T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' p2} → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have 21 (1) �T t=1 E∥E⊤ t R∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' �T t=1 E∥EtC∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 E⊤ t Ft∥2 F = O (T p1p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) E∥ �T t=1 FtC⊤E⊤ t ∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 F⊤ t R⊤Et∥2 F = O (T p1p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 FtC⊤E⊤ t R∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 F⊤ t R⊤EtC∥2 F = O (T p1p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) E∥ �T t=1 FtW⊤ 2 E⊤ t W1∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 E⊤ t W1Ft∥2 F = O (T p1p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 FtW⊤ 2 E⊤ t ∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 W⊤ 1 EtW2∥2 F = O (T p1p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (4) E∥ �T t=1 R⊤EtW2F⊤ t ∥2 F = O (T p1p2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ �T t=1 R⊤EtW2∥2 F = O (T p1p2) (5) E∥ �T t=1 EtW2W⊤ 2 E⊤ t W1∥2 F = O � T p2 1p3 2 + T 2p1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Assume m1 = k1 = 1, m2 = k2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (1) By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, we can get E∥E⊤ t R∥2 F = p2 � j=1 E( p1 � i=1 et,ijri)2 = p2 � j=1 p1 � i1,i2=1 E(ri1ri2et,i1jet,i2j) = O (p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Similarly, �T t=1 E∥EtC∥2 F∥2 F = O (T p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4(2), E∥ T � t=1 E⊤ t Ft∥2 F = p1 � i=1 p2 � j=1 E( T � t=1 Ftet,ij)2 = T p1 � i=1 p2 � j=1 E(ξi,j)2 = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) The results hold directly by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) By Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 (2), we have E∥ T � t=1 FtW⊤ 2 E⊤ t W1∥2 F = E( T � t=1 p1 � i=1 p2 � j=1 Ftw1,iet,ijw2,j)2 = T E( p1 � i=1 p2 � j=1 ξi,jw1,iw2,j)2 = T p1 � i1,i2=1 p2 � j1,j2=1 E(ξi1,j1ξi2,j2)w1,i1w1,i2w2,j1w2,j2 = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ T � t=1 E⊤ t W1Ft∥2 F = p2 � j=1 E( T � t=1 p1 � i=1 et,ijw1,jFt)2 = T p2 � j=1 E( p1 � i=1 ξi,jw1,j)2 = T p2 � j=1 p1 � i,i′=1 E(ξi,jξi′,j)w1,iw1,i′ = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Analogously, E∥ �T t=1 FtW⊤ 2 E⊤ t ∥2 F = O(T p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 (2), we can also obtain E∥ T � t=1 W⊤ 1 EtW2∥2 F = E( T � t=1 p1 � i=1 p2 � j=1 w1,iw2,jet,ij)2 = T � s,t=1 p1 � i1,i2=1 p2 � j1,j2=1 w1,i1w1,i2w2,j1w2,j2E(et,i1j1es,i2j2)2 = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 22 (4) E∥ T � t=1 R⊤EtW2Ft∥2 F = E( T � t=1 p1 � i=1 p2 � j=1 riw2,jet,ijFt)2 = T E( p1 � i=1 p2 � j=1 riw2,jξi,j)2 = T p1 � i1,i2=1 p2 � j1,j2=1 ri1ri2w2,j1w2,j2E(ξi1,j1ξi2,j2) = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' E∥ T � t=1 R⊤EtW2∥2 F = E( T � t=1 p1 � i=1 p2 � j=1 riw2,jet,ij)2 = T � s,t=1 p1 � i1,i2=1 p2 � j1,j2=1 ri1ri2w2,j1w2,j2E(et,i1j1es,i2j2)2 = O (T p1p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (5) Note that E∥ T � t=1 EtW2W⊤ 2 E⊤ t W1∥2 F ≤ ∥W2∥2 F∥ T � t=1 W⊤ 1 EtW2Et∥2 F ≤ ∥W2∥2 F p1 � i=1 p2 � j=1 E∥ T � t=1 W⊤ 1 EtW2et,ij∥2, while for any i, j, by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, E∥ T � t=1 W ⊤ 1 EtW2et,ij∥2 = E( T � t=1 p1 � i′ p2 � j′ w1,i′et,i′j′w2,j′et,ij)2 ≲ T � t,s=1 p1 � i1,i2 p2 � j1,j2=1 |Cov(et,ijet,i1j1, es,ijes,i2j2)| + ( T � t=1 p1 � i′ p2 � j′ |Eet,i′j′et,ij|)2 = O � T p1p2 + T 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, as min{T, p1, p2} → ∞, we have ∥ T � s=1 E⊤ s ( �R(1) − R �H(1) r )Fs∥2 F = Op � p2 1 p2ν2 min(H2) + p1 ν2 min(H1)ν4 min(H2) + T p2ν2 min(H1)ν4 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the proof in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1, �R(1) − R �H(1) r = II + III + IV, then T � s=1 E⊤ s ( �R(1) − R �H(1) r )Fs = T � s=1 E⊤ s (II + III + IV)Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 23 For the first term, by LemmaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (2) and (3), we get ∥ T � t=1 E⊤ s IIFs∥2 F = 1 T 2p1p2 2 ∥ T � s=1 E⊤ s R[( T � t=1 FtC⊤W2W⊤ 2 E⊤ t W1)Z−1/2]Fs∥2 F ≤ 1 T 2p1p2 2 ∥ T � s=1 E⊤ s RFs∥2 F∥( T � t=1 FtC⊤W2W⊤ 2 E⊤ t W1)Z−1/2∥2 F ≤ 1 T 2p1p2 2 ∥ T � s=1 E⊤ s RFs∥2 F∥ T � t=1 FtW⊤ 2 E⊤ t W1∥2 F ∥Z−1/2∥2 F ∥C⊤W2∥2 2 = Op � 1 ν2 min(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥ T � s=1 E⊤ s IIIFs∥2 F = 1 T 2p1p2 2 ∥ T � s=1 E⊤ s [( T � t=1 EtW2W⊤ 2 CF⊤ t R⊤W1)Z−1/2]Fs∥2 F ≤ 1 T 2p1p2 2 ∥ T � s=1 E⊤ s ( T � t=1 EtW2F⊤ t )Fs∥2 F∥W⊤ 2 C∥2 F ∥R⊤W1∥2 F ∥Z∥2 F = Op � p2 1 p2ν2 min(H2) + p1 ν2 min(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where E∥ T � s=1 E⊤ s ( T � t=1 EtW2F⊤ t )Fs∥2 F = p2 � j=1 E( T � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='t=1 p1 � i=1 p2 � j1=1 w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1FsFtes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij1)2 = T 2 p2 � j=1 E( p1 � i=1 p2 � j1=1 w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='jξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1)2 = T 2 p2 � j=1 \uf8ee \uf8f0( p1 � i=1 p2 � j1=1 Eξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='jξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1)2 + p1 � i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i2=1 p2 � j1j2=1 Cov(ξi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='jξi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ξi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='jξi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j2) \uf8f9 \uf8fb = O � T 2p2 1p2 + T 2p1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the third term, by lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='3 (1) and (5), we have ∥ T � s=1 E⊤ s IVFs∥2 F = 1 T 2p1p2 2 ∥ T � s=1 Es[( T � t=1 EtW2W⊤ 2 E⊤ t W1)Z−1/2]Fs∥2 F ≤ 1 T 2p1p2 2 ∥ T � s=1 E⊤ s Fs∥2 F ∥( T � t=1 EtW2W⊤ 2 E⊤ t W1)Z−1/2∥2 F = Op � p1 ν2 min(H1)ν4 min(H2) + T p2ν2 min(H1)ν4 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As a result, ∥ T � s=1 E⊤ s ( �R(1) − R �H(1) r )Fs∥2 F = Op � p2 1 p2ν2 min(H2) + p1 ν2 min(H1)ν4 min(H2) + T p2ν2 min(H1)ν4 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 24 Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, take �R(1), �C(1) as the resultant estimators from the one-step iteration, as min{T, p1, p2} → ∞, then we have ���R⊤( �R(1) − R �H(1) r ) ��� F = Op � √p1 √T p2νmin(H1)νmin(H2) + 1 p2νmin(H1)ν2 min(H2) � , ���C⊤(�C(1) − C �H(1) c ) ��� F = Op � 1 T νmin(H2) + √p2 √T p1νmin(H1)νmin(H2) + 1 p1νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ��R⊤II �� F = 1 T √p1p2 �����R⊤R( T � t=1 FtC⊤W2W⊤ 2 E⊤ t W1)Z−1/2 ����� F = √p1 T p2 �����( T � t=1 FtC⊤W2W⊤ 2 E⊤ t W1)Z−1/2 ����� F ≤ √p1 T p2 ����� T � t=1 FtW⊤ 2 E⊤ t W1 ����� F ���Z−1/2��� F ��C⊤W2 �� F = Op � √p1 √T p2νmin(H1)νmin(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ��R⊤III �� F = 1 T √p1p2 �����( T � t=1 R⊤EtW2W⊤ 2 CF⊤ t R⊤W1)Z−1/2 ����� F ≤ 1 T √p1p2 ����� T � t=1 R⊤EtW2Ft ����� F ��W⊤ 2 C �� F ��R⊤W1 �� F ���Z1/2��� F = Op � √p1 √T p2νmin(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ��R⊤IV �� F = 1 T √p1p2 �����( T � t=1 R⊤EtW2W⊤ 2 E⊤ t W1)Z−1/2 ����� F ≤ 1 T √p1p2 � � � � ����� T � t=1 R⊤EtW2 ����� 2 F × ����� T � t=1 W⊤ 2 E⊤ t W1 ����� 2 F ���Z−1/2��� F = Op � 1 p2νmin(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As R⊤( �R(1) − R �H(1) r ) = R⊤(II + III + IV), combining the above three items, we get ���R⊤( �R(1) − R �H(1) r ) ��� F = Op � √p1 √T p2νmin(H1)νmin(H2) + 1 p2νmin(H1)ν2 min(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' As for the second formula, by the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2, we have �C(1) − C �H(1) c = √p2 � δ(1) 2 (Z(1))−1/2 + δ(1) 3 (Z(1))−1/2 + δ(1) 4 (Z(1))−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 25 For the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' √p2 ���C⊤δ(1) 2 (Z(1))−1/2��� F = 1 p1√p2 �����( T � t=1 C⊤E⊤ t �R(1)W⊤ 1 RFtC⊤W2)(Z(1))−1/2 ����� F ≤ 1 p1√p2 ����� T � t=1 C⊤E⊤ t �R(1)Ft ����� F ��W⊤ 1 R �� 2 ��C⊤W2 �� 2 ���(Z(1) t )−1/2��� F = Op � 1 T νmin(H2) + 1 T √p1νmin(H1)ν2 min(H2) + 1 √T p2p1νmin(H1)ν2 min(H2) + √p2 √T p1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where ����� T � t=1 C⊤E⊤ t �R(1)Ft ����� F ≤ ����� T � t=1 C⊤E⊤ t ( �R(1) − R �H(1) r )Ft ����� F + ����� T � t=1 C⊤E⊤ t R �H(1) r Ft ����� F ≤ ����� T � t=1 C⊤E⊤ t Ft ����� F ��� �R(1) − R �H(1) r ��� F + ����� T � t=1 C⊤E⊤ t RFt ����� F ��� �H(1) r ��� F = Op � p1 νmin(H2) + √p1 νmin(H1)ν2 min(H2) + √ T √p2νmin(H1)ν2 min(H2) + � T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the second term, √p2 ���C⊤δ(1) 3 (Z(1))−1/2��� F = √p2 p1 �����( T � t=1 F⊤ t R⊤ �R(1)W⊤ 1 EtW2)(Z(1))−1/2 ����� F ≍ √p2 �����( T � t=1 F⊤ t W⊤ 1 EtW2)(Z(1))−1/2 ����� F ≤ √p2 ����� T � t=1 F⊤ t W⊤ 1 EtW2 ����� F ���(Z(1))−1/2��� F = Op � √p2 √T p1νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the third term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' √p2 ���C⊤δ(1) 4 (Z(1))−1/2��� F = 1 p1√p2 �����( T � t=1 C⊤E⊤ t �R(1)W⊤ 1 EtW2)(Z(1))−1/2 ����� F ≤ 1 p1√p2 � � � � ����� T � t=1 C⊤E⊤ t �R(1) ����� 2 F × ����� T � t=1 W⊤ 1 EtW2 ����� 2 F ���(Z(1))−1/2��� F = Op � 1 √T p1p2νmin(H1)ν2 min(H2) + 1 √T p2p1ν2 min(H1)ν3 min(H2) + 1 p1νmin(H1)νmin(H2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 26 where the last equation holds due to ����� T � t=1 C⊤E⊤ t �R(1) ����� 2 F = ����� T � t=1 C⊤E⊤ t ( �R(1) − R �H(1) r + R �H(1) r ) ����� 2 F ≤ ����� T � t=1 C⊤E⊤ t ( �R(1) − R �H(1) r ) ����� 2 F + ����� T � t=1 C⊤E⊤ t R �H(1) r ����� 2 F = Op � p2 1 ν2 min(H2) + p1 ν2 min(H1)ν4 min(H2) + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Hence, ���C⊤(�C(1) − C �H(1) c ) ��� F = Op � 1 T νmin(H2) + √p2 √T p1νmin(H1)νmin(H2) + 1 p1νmin(H1)νmin(H2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, take �R(s+1), �C(s+1) as the result of a (s + 1)th step iteration, as min{T, p1, p2} → ∞, then we have ∥R⊤( �R(s+1) − R �H(s+1) r )∥F = Op \uf8eb \uf8ed √p1 √T p2 + p1 � w(s) r √T p2 + � w(s) r p2 + � p1w(s) c √ T + p1 � w(s) r w(s) c √ T + � w(s) r w(s) c \uf8f6 \uf8f8 , ∥C⊤(�C(s+1)−C �H(s+1) c )∥F = Op \uf8eb \uf8ed √p2 √T p1 + 1 p1 + � w(s) r √p1 + � p2w(s) r √ T + p2 � w(s) c √T p1 + p2 � w(s) r w(s) c √ T + � w(s) r w(s+1) r \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4, �R(s+1) − R �H(s+1) r = √p1(δ(s+1) 2 + δ(s+1) 3 + δ(s+1) 4 )(Z(s+1))−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9 (1) and (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' we have ���√p1R⊤δ(s+1) 2 (Z(s+1))−1/2��� F ≍ ����� √p1 � T � t=1 Ft �C(s)⊤E⊤ t �R(s) � � Z(s+1)�−1/2 ����� F ≤ √p1 ����� T � t=1 Ft �C(s)⊤E⊤ t �R(s) ����� F ���� � Z(s+1)�−1/2���� F = Op \uf8eb \uf8edp1 � w(s) r w(s) c √ T + p1 � w(s) r √T p2 + � p1w(s) c √ T + √p1 √T p2 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where the last equation holds due to ����� T � t=1 Ft �C(s)⊤E⊤ t �R(s) ����� F ≤ ����� T � t=1 Ft �C(s)⊤E⊤ t ����� F ��� �R(s) − R �H(s) r ��� F + ����� T � t=1 Ft �C(s)⊤E⊤ t R ����� F ��� �H(ss) r ��� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 27 ���√p1R⊤δ(s+1) 3 (Z(s+1))−1/2��� F ≍ √p1 ����� � T � t=1 R⊤Et �C(s)F⊤ t � � Z(s+1)�−1/2 ����� F ≤ √p1 ����� T � t=1 R⊤Et �C(s)F⊤ t ����� F ���� � Z(s+1)�−1/2���� F = Op \uf8eb \uf8ed � p1w(s) c √ T + √p1 √T p2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' By the fact that ����� T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t �R(s) ����� F ≤ ����� T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t ( �R(s) − R �H(s) r ) ����� F + ����� T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t R �H(s) r ����� F ≤ ����� T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t ����� F ��� �R(s) − R �H(s) r ��� F + ����� T � t=1 R⊤Et �C(s) ����� 2 F ��� �H(s) r ��� F , and Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8, we can get ���√p1R⊤δ(s+1) 4 (Z(s+1))−1/2��� F = ����� 1 √p1p2 � T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t �R(s) � � Z(s+1)�−1/2 ����� F ≤ 1 √p1p2 ����� T � t=1 R⊤Et �C(s) �C(s)⊤E⊤ t �R(s) ����� F ���� � Z(s+1)�−1/2���� F = Op \uf8eb \uf8ed � p1w(s) r √T p2 + � w(s) r p2 + � w(s) r w(s) c + � p1w(s) r w(s) c √ T + � w(s) c √T p1p2 + 1 √T p1p2p2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Hence, ∥R⊤( �R(s+1) − R �H(s+1) r )∥F = Op \uf8eb \uf8ed √p1 √T p2 + p1 � w(s) r √T p2 + � w(s) r p2 + � p1w(s) c √ T + p1 � w(s) r w(s) c √ T + � w(s) r w(s) c \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' And by a similar proof, we can get ∥C⊤(�C(s+1)−C �H(s+1) c )∥F = Op \uf8eb \uf8ed √p2 √T p1 + 1 p1 + � w(s) r √p1 + � p2w(s) r √ T + p2 � w(s) c √T p1 + p2 � w(s) r w(s) c √ T + � w(s) r w(s+1) r \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' If Assumptions 1–6 hold, then we have (1) T � t=1 ∥Et �C(s)∥2 F = Op � T p1p2 + T p1p2 2w(s) c � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) T � t=1 ∥E⊤ t �R(s)∥2 F = Op � T p1p2 + T p2 1p2w(s) r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 28 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (1) As �T t=1 ∥EtC∥2 F = �T t=1 �p1 i=1(�p2 k=1 et,ikck)2 ≤ c �T t=1 �p1 i=1 �p2 k,k′=1 et,iket,ik′ = Op (T p1p2) , �T t=1 ∥Et∥2 F = �T t=1 �p1 i=1 �p2 j=1 e2 t,ij = Op (T p1p2), then T � t=1 ∥Et �C(s)∥2 F ≤ T � t=1 ∥EtC �H(s) c ∥2 F + T � t=1 ∥Et(�C(s) − C �H(s) c )∥2 F = Op � T p1p2 + T p1p2 2w(s) c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) �T t=1 ∥R⊤Et∥2 F = �T t=1 �p2 j=1(�p1 i=1 riet,ij)2 ≤ c �T t=1 �p2 j=1 �p1 i1,i2=1 et,i1jet,i2j = Op (T p1p2), T � t=1 ∥E⊤ t �R(s)∥2 F ≤ T � t=1 ∥E⊤ t ( �R(s) − R �H(s) r )∥2 F + T � t=1 ∥E⊤ t R �H(s) r ∥2 F = Op � T p2 1p2w(s) r + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' If Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 hold, then we have (1) ∥ T � t=1 Et �C(s) �C(s)⊤E⊤ t R∥2 F = Op � T p2 1p3 2 + T 2p1p2 2 + (T 2p1p4 2 + T p2 1p4 2)w(s)2 c � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) ∥ T � t=1 E⊤ t �R(s+1) �R(s)⊤EtC∥2 F = Op � T p3 1p2 2 + T 2p2 1p2 + (T p4 1p2 2 + T 2p4 1p2)w(s) r w(s+1) r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (1) For simplicity, we fix k1 = k2 = m1 = m2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Note that ∥ T � t=1 Et �C(s) �C(s)⊤E⊤ t R∥2 F ≤ ∥ T � t=1 EtC �H(s) c �H(s)⊤ c C⊤E⊤ t R∥2 F + ∥ T � t=1 Et(�C(s) − C �H(s) c ) �H(s)⊤ c C⊤E⊤ t R∥2 F + ∥ T � t=1 EtC �H(s) c (�C(s) − C �H(s) c )⊤E⊤ t R∥2 F + ∥ T � t=1 Et(�C(s) − C �H(s) c )(�C(s) − C �H(s) c )⊤E⊤ t R∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the first term, ∥ T � t=1 EtC �H(s) c �H(s)⊤ c C⊤E⊤ t R∥2 F ≤ ∥ T � t=1 EtC⊤E⊤ t R∥2 F ∥C∥2 F∥ �H(s) c ∥4 F ≤ p2 p1 � i′=1 p2 � j′=1 ( T � t=1 p1 � i=1 p2 � j=1 ricjet,ijet,i′j′)2 = Op � T p2 1p3 2 + T 2p1p2 2 � , where E( T � t=1 p1 � i=1 p2 � j=1 ricjet,ijet,i′j′)2 ≤ T � t,s=1 p1 � i1,i2=1 p2 � j1,j2=1 Cov(et,i1j2et,i′j′, es,i2j2es,i′j′) + ( T � t=1 p1 � i=1 p2 � j=1 Eet,ijet,i′j′)2 = O � T p1p2 + T 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 29 For the second term, ∥ T � t=1 Et(�C(s) − C �H(s) c ) �H(s)⊤ c C⊤E⊤ t R∥2 F ≤ ∥ T � t=1 EtC⊤E⊤ t R∥2 F ∥�C(s) − C �H(s) c ∥2 F ∥ �H(s) c ∥2 F = Op � (T p2 1p3 2 + T 2p1p2 2)w(s) c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the third term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥ T � t=1 EtC �H(s) c (�C(s) − C �H(s) c )⊤E⊤ t R∥2 F ≤ ∥ T � t=1 EtCR⊤Et∥2 F ∥�C(s) − C �H(s) c ∥2 F ∥ �H(s) c ∥2 F = p1 � i′=1 p2 � j′=1 ( T � t=1 p1 � i=1 p2 � j=1 ricjet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′jet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij′)2∥�C(s) − C �H(s) c ∥2 F ∥ �H(s) c ∥2 F = Op � (T p2 1p3 2 + T 2p1p2 2)w(s) c � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where E( T � t=1 p1 � i=1 p2 � j=1 ricjet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′jet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij′)2 ≤ T � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 p1 � i1i2=1 p2 � j1j2=1 Cov(et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j1et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i1j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j2es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i2j′) + ( T � t=1 p1 � i=1 p2 � j=1 Eet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′jet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij′) = O � T p1p2 + T 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the last term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥ T � t=1 Et(�C(s) − C �H(s) c )(�C(s) − C �H(s) c )⊤E⊤ t R∥2 F ≤ ∥�C(s) − C �H(s) c ∥2 F ∥ T � t=1 Et(�C(s) − C �H(s) c )⊤E⊤ t R∥2 F ≤ ∥�C(s) − C �H(s) c ∥4 F p1 � i=1 p2 � j=1 ∥ T � t=1 E⊤ t Ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij∥2 F ≤ ∥�C(s) − C �H(s) c ∥4 F p1 � i=1 p2 � j=1 p2 � k=1 ( T � t=1 p1 � i′=1 et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′ket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij)2 = Op � (T 2p1p4 2 + T p2 1p4 2)w(s)2 c � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' where p1 � i=1 p2 � j=1 p2 � k=1 E( T � t=1 p1 � i′ et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′ket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij)2 ≤ p1 � i=1 p2 � j=1 p2 � k=1 ( T � t=1 p1 � i′=1 Eet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′ket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij)2 + p1 � i=1 p2 � j=1 p2 � k=1 T � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 p1 � i1i2=1 Cov(et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i1ket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i2ket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij) = Op � T 2p1p2 2 + T p2 1p2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 30 (1) Note that ∥ T � t=1 E⊤ t �R(s+1) �R(s)⊤EtC∥2 F = ∥ T � t=1 E⊤ t ( �R(s+1) − R �H(s+1) r + R �H(s+1) r )( �R(s) − R �H(s) r + R �H(s) r )⊤EtC∥2 F ≤ ∥ T � t=1 E⊤ t R �H(s+1) r �H(s)⊤ r R⊤EtC∥2 F + ∥ T � t=1 E⊤ t R �H(s+1) r ( �R(s) − R �H(s) r )⊤EtC∥2 F + ∥ T � t=1 E⊤ t ( �R(s+1) − R �H(s+1) r ) �H(s)⊤ r R⊤EtC∥2 F + ∥ T � t=1 E⊤ t ( �R(s+1) − R �H(s+1) r )( �R(s) − R �H(s) r )⊤EtC∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the first term, ∥ T � t=1 E⊤ t R �H(s+1) r �H(s)⊤ r R⊤EtC∥2 F ≤ ∥ T � t=1 E⊤ t R⊤EtC∥2 F ∥R∥2 F∥ �H(s+1) r ∥2 F∥ �H(s) r ∥2 F = Op � T p3 1p2 2 + T 2p2 1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the second term, ∥ T � t=1 E⊤ t R �H(s+1) r ( �R(s) − R �H(s) r )⊤EtC∥2 F ≤ ∥ T � t=1 E⊤ t RC⊤E⊤ t ∥2 F ∥ �R(s) − R �H(s) r ∥2 F ∥ �H(s+1) r ∥2 F = Op � (T p3 1p2 2 + T 2p2 1p2)w(s) r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the third term, ∥ T � t=1 E⊤ t ( �R(s+1) − R �H(s+1) r ) �H(s)⊤ r R⊤EtC∥2 F ≤ ∥ T � t=1 E⊤ t R⊤EtC∥2 F∥ �R(s+1) − R �H(s+1) r ∥2 F ∥ �H(s) r ∥2 F = Op � (T p3 1p2 2 + T 2p2 1p2)w(s+1) r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' For the last term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' ∥ T � t=1 E⊤ t ( �R(s+1) − R �H(s+1) r )( �R(s) − R �H(s) r )⊤EtC∥2 F ≤ ∥ T � t=1 ( �R(s) − R �H(s) r )⊤EtCE⊤ t ∥2 F ∥ �R(s+1) − R �H(s+1) r ∥2 F ≤ ∥ �R(s) − R �H(s) r ∥2 F ∥ �R(s+1) − R �H(s+1) r ∥2 F p1 � i′=1 p2 � j′=1 ∥ T � t=1 EtCet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′∥2 F ≤ ∥ �R(s) − R �H(s) r ∥2 F ∥ �R(s+1) − R �H(s+1) r ∥2 F p1 � i′=1 p2 � j′=1 p1 � i=1 ( T � t=1 p2 � j=1 et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′)2 = Op � (T p4 1p2 2 + T 2p4 1p2)w(s) r w(s+1) r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 31 where E( T � t=1 p2 � j=1 et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′)2 ≤ T � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='s=1 p2 � j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='j2=1 Cov(et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij1et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ij2es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′) + ( T � t=1 p2 � j=1 Eet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='ijet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='i′j′)2 = O � T p2 + T 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' Suppose that T, p1, p2 tend to infinity, and m1 = k1, m2 = k2 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' If Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1 (1), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content='4 hold, then we have (1) ∥ T � t=1 Et �C(s)F⊤ t ∥2 F = Op � T p1p2 2w(s) c + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) ∥ T � t=1 R⊤Et �C(s)F⊤ t ∥2 F = Op � T p1p2 2w(s) c + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) ∥ T � t=1 E⊤ t �R(s)Ft∥2 F = Op � T p2 1p2w(s) r + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (4) ∥ T � t=1 C⊤E⊤ t �R(s)Ft∥2 F = Op � T p2 1p2w(s) r + T p1p2 � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (1) ∥ T � t=1 Et �C(s)F⊤ t ∥2 F ≤ ∥ T � t=1 Et(�C(s) − C �H(s) c )F⊤ t ∥2 F + ∥ T � t=1 EtC �H(s) c F⊤ t ∥2 F ≤ ∥ T � t=1 EtF⊤ t ∥2 F ∥�C(s) − C �H(s) c ∥2 F + ∥ T � t=1 EtCF⊤ t ∥2 F ∥ �H(s) c ∥2 F = Op � T p1p2 2w(s) c + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (2) ∥ T � t=1 R⊤Et �C(s)F⊤ t ∥2 F ≤ ∥ T � t=1 R⊤Et(�C(s) − C �H(s) c )F⊤ t ∥2 F + ∥ T � t=1 R⊤EtC �H(s) c F⊤ t ∥2 F ≤ ∥ T � t=1 F⊤ t R⊤Et∥2 F∥�C(s) − C �H(s) c ∥2 F + ∥ T � t=1 R⊤EtCF⊤ t ∥2 F∥ �H(s) c ∥2 F = Op � T p1p2 2w(s) c + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' (3) ∥ T � t=1 E⊤ t �R(s)Ft∥2 F ≤ ∥ T � t=1 E⊤ t ( �R(s) − R �H(s) r )Ft∥2 F + ∥ T � t=1 E⊤ t R �H(s) r Ft∥2 F ≤ ∥ T � t=1 E⊤ t Ft∥2 F ∥ �R(s) − R �H(s) r ∥2 F + ∥ T � t=1 E⊤ t RFt∥2 F∥ �H(s) r ∥2 F = Op � T p2 1p2w(s) r + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 32 (4) ∥ T � t=1 C⊤E⊤ t �R(s)Ft∥2 F ≤ ∥ T � t=1 C⊤E⊤ t ( �R(s) − R �H(s) r )Ft∥2 F + ∥ T � t=1 C⊤E⊤ t R �H(s) r Ft∥2 F ≤ ∥ T � t=1 C⊤E⊤ t Ft∥2 F∥ �R(s) − R �H(s) r ∥2 F + ∥ T � t=1 C⊤E⊤ t RFt∥2 F ∥ �H(s) r ∥2 F = Op � T p2 1p2w(s) r + T p1p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQfgfiG/content/2301.00360v1.pdf'} diff --git a/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/2301.02236v1.pdf.txt b/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/2301.02236v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a06d42941394c271809d96cd40b2bbcda2228f5 --- /dev/null +++ b/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/2301.02236v1.pdf.txt @@ -0,0 +1,1860 @@ +arXiv:2301.02236v1 [math.AP] 5 Jan 2023 +A MINIMIZATION PROBLEM WITH FREE BOUNDARY FOR +p-LAPLACIAN WEAKLY COUPLED SYSTEM +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Abstract. In this paper we consider a weakly coupled p-Laplacian system of a +Bernoulli type free boundary problem, through minimization of a corresponding +functional. We prove various properties of any local minimizer and the corre- +sponding free boundary. +1. Introduction +1.1. Problem setting. For Ω ⊂ Rn (n ≥ 2), we consider the problem of minimizing +the functional +(1) +J(u) = +� +Ω +m +� +i=1 +|∇ui|p + Qpχ{|u|>0}dx, +1 < p < ∞, +in the class of vectorial functions +K := {u = (u1, . . . , um) ∈ W1,p(Ω; Rm) : u = g on ∂Ω and ui ≥ 0 for i = 1, · · · , m}, +with a given boundary data g ∈ W1,p(Ω; Rm). Here, Q is a H¨older function satisfying +0 < Qmin ≤ Q ≤ Qmax < ∞, +for some constants Qmin and Qmax. +We are interested in regularity properties of minimizers u, as well as the free +boundary Γ = ∂{|u| > 0} ∩ Ω. Any local minimizer satisfies the p-Laplace equation +div (|∇ui|p−2∇ui) = 0, +in {ui > 0}, +also denoted by ∆pui = 0. In fact, ∆pui is a nonnegative Radon measure with +support on free boundary, Γ. The problem is to find a reasonable representation of +this measure and put it into some pde-context for further analysis. +This problem is referred to as Bernoulli-type free boundary problem, and is well +studied in the literature, for the scalar case and for p = 2, starting with seminal work +of H.W. Alt and L.A. Caffarelli [2], and also for any 1 < p < ∞ in [8]. There are very +few results for Bernoulli-type problems that involve systems (see [7, 11, 20]). In [7], +the authors study the minimum problem (1) for p = 2 and show the smoothness +of the regular part of free boundary as well as some partial result for Hausdorff +dimension of singular part. Indeed, they apply a reduction method to reduce the +problem to its scalar counterpart and the same result for the scalar case can be +extended to the vectorial problem. Also, a vectorial Bernoulli problem with no +Date: January 6, 2023. +2020 Mathematics Subject Classification. 35R35. +Key words and phrases. p-Laplacian, minimizers, free boundary regularity, system. +This project was carried out during the program Geometric aspects of nonlinear PDE at Institute +Mittag Leffler, Stockholm, Sweden. H. Shahgholian was supported by Swedish Research Council. +1 + +2 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +sign assumption on the components is studied in [20]. In [11], the same result has +been obtained by the viscosity approach and improvement of flatness. +In this paper, we deal with a (weakly coupled) cooperative system for p-laplacian +version of Bernoulli-type problem, following similar procedure as that in [7]. +Remark 1.1. It should be remarked that our approaches in this paper, with some extra +efforts, can be adapted to variable exponent case, as well as variable coefficient one. Similar +types of results are then expected. +1.2. Notation. For clarity of exposition we shall introduce some notations and +definitions, which are used frequently in this text. +Throughout this paper, Rn will be equipped with the Euclidean inner product +x · y and the induced norm |x|, Br(x0) will denote the open n-dimensional ball with +center x0, radius r and its boundary with ∂Br(x0). In addition, Br = Br(0) and +∂Br = ∂Br(0). For the target space, Rm, we use several norms +|u|p = ((u1)p + · · · + (um)p)1/p, +|u(x)|∞ = max +1≤j≤m |uj(x)|. +For convenience, we denote the Euclidean norm without the index, |u| = |u|2. We +also use the Euclidean norm in the definition of L∞-norm, that is +∥u∥L∞ := sup +x +|u(x)|. +1.3. Plan of the paper. The paper is organized as follows: In Section 2, we study the +existence of minimizer (Theorem 2.1) and show that minimizers are p-subharmonic +(Lemma 2.3). Section 3 is devoted to the regularity property of solutions, includ- +ing the H¨older regularity (Lemma 3.2) and the Lipschitz regularity (Theorem 3.5). +Section 4 consists of the proof of nondegeneracy property (Lemma 4.1). Also, in +Theorem 4.2 an estimate for the density of the free boundary is obtained which is +enough to prove that the free boundary has zero Lebesgue measure. The vector- +valued measure ∆pu (Theorem 5.1) and (n−1)-Hausdorff dimension of free bound- +ary (Theorem 5.2) are discussed in Section 5. The main result in Section 6 is the +flatness of regular part of the free boundary (Theorem 6.5). We prove a partial +result for the regularity of free boundary in Section 7 (Theorem 7.5), along with +C1,α-regularity of the free boundary when p is sufficiently close to 2 (Theorem +7.6). In Appendix A we deal with NTA properties of the free boundary. Also, in +Appendix B we present an auxiliary lemma to study the asymptotic behaviour of +p-harmonic functions. +2. Existence of a minimizer +Theorem 2.1. If J(g) < ∞, then there exists an absolute minimizer of J over class K. +Proof. Obviously the functional is nonnegative, and hence it takes an infimum +value. Let uk be a minimizing sequence +inf +v∈K J(v) = lim +k→∞ J(uk). + +3 +Then uk − g is bounded in W1,p +0 (Ω; Rm) and up to a subsequence we can assume +that +uk ⇀ u, +weakly in W1,p(Ω; Rm), +uk → u, +a.e. in Ω, +for some u ∈ K. The latter convergence implies +� +Ω +χ{|uk|>0}dx → +� +Ω +χ{|u|>0}dx, +and the weakly lower semicontinuity of the norm implies that +� +Ω +m +� +i=1 +|∇ui|pdx ≤ lim inf +k→∞ +� +Ω +m +� +i=1 +|∇ui +k|pdx. +These together show that J(u) ≤ limk→∞ J(uk) and hence u ∈ K is an absolute +minimizer. +□ +Remark 2.2. We say u ∈ K is a local minimizer of J, if J(u) ≤ J(v) for any v ∈ K with +∥∇u − ∇v∥Lp(Ω) + ∥χ{|u|>0} − χ{|v|>0}∥L1(Ω) ≤ ε, +for some ε > 0. Although, all results in this paper are proved for local minimizers, for the +sake of convenience we argue with absolute minimizers. +Lemma 2.3. If u is a (local) minimizer, then ui is p-subharmonic for all i = 1, . . ., m, i.e., +� +Ω +|∇ui|p−2∇ui · ∇ϕ dx ≤ 0, +for all ϕ ∈ C∞ +0 (Ω), ϕ ≥ 0. +Moreover, in each component of {|u| > 0} for i = 1, . . ., m, either ui is identically vanishing +or it is positive. Hence, +(2) +∆pui = 0, +in {|u| > 0}, for all i = 1, . . ., m, +and consequently (by the maximum principle) +∥u∥L∞(Ω) ≤ ∥g∥L∞(Ω). +Proof. Let +ϕ ∈ C∞ +0 (Ω), ϕ ≥ 0, +t > 0, and define vj = uj for j � i and vi = (ui − tϕ)+. Then v ∈ K and we can choose +v as a competitor, so +0 ≤ 1 +t (J(v) − J(u)) =1 +t +� +Ω +|∇vi|p − |∇ui|p + Qp �χ{|v|>0} − χ{|u|>0} +� dx +≤1 +t +� +Ω +|∇(ui − tϕ)|p − |∇ui|pdx +→ +� +Ω +−p|∇ui|p−2∇ui · ∇ϕ dx. +The second statement in the lemma relies on the strong minimum principle for +p-harmonic functions. In fact, let x ∈ {|u| > 0} but ui(x) = 0 for some i. Choose an +index j such that uj > 0 inside Br(x) where r is small enough. Now denote by ˜ui the +p-harmonic extension of ui inside Br(x) and consider the competitor ˜u by defining +˜uk = uk for k � i. Since {|u| > 0} = {| ˜u| > 0}, we get J(˜u) < J(u) unless ˜ui = ui and ui is +p-harmonic in Br(x). But this violates the strong minimum principle if ui(x) = 0. +□ + +4 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +3. Regularity of local minimizers +Lemma 3.1. Let u be a (local) minimizer of J, and vi the harmonic replacement (majorant) +for ui in B ⊂ Ω (for B a small ball). Then there is a universal constant C = C(n, p) such +that +� +B +|∇(ui − vi)|p dx ≤ CQp +max|{|u| = 0} ∩ B|, +when 2 ≤ p, +� +B +|∇(ui − vi)|p dx ≤ C(Qmax)p2/2|{|u| = 0} ∩ B|p/2 +�� +B +|∇ui|p dx +�1−p/2 +, when 1 < p ≤ 2. +Proof. Let vj = uj for all j � i, and extend vi by ui in Ω \ B. If B is small enough +(when u is absolute minimizer, we do not need this assumption), then we have +J(u) ≤ J(v) and consequently +� +B +|∇ui|p − |∇vi|p dx ≤ Qp +max |{|u| = 0} ∩ B| . +Set now ws(x) = sui(x) + (1 − s)vi(x) for 0 ≤ s ≤ 1. Then +� +B +|∇ui|p − |∇vi|p dx = +� +B +|∇w1|p − |∇w0|p dx +=p +� 1 +0 +ds +� +B +|∇ws|p−2∇ws · ∇(ui − vi) dx +=p +� 1 +0 +ds +� +B +� +|∇ws|p−2∇ws − |∇vi|p−2∇vi� +· ∇(ui − vi) dx +=p +� 1 +0 +ds +s +� +B +� +|∇ws|p−2∇ws − |∇vi|p−2∇vi� +· ∇(ws − vi) dx, +where for the third equality we have used ∆pvi = 0. Next using +� +|b|p−2b − |a|p−2a +� +· (b − a) ≥ γ + +|b − a|2(|b| + |a|)p−2, +1 < p ≤ 2, +|b − a|p, +2 ≤ p, +we obtain for p ≥ 2 +� +B +|∇ui|p − |∇vi|p dx ≥ γp +� 1 +0 +ds +s +� +B +|∇(ws − vi)|p dx = γp +� 1 +0 +sp−1ds +� +B +|∇(ui − vi)|p dx, +which shows the desired estimate for p ≥ 2. +In case 1 < p ≤ 2, we have +� +B +|∇ui|p − |∇vi|p dx ≥ γp +� 1 +0 +ds +s +� +B +|∇(ws − vi)|2 � +|∇ws| + |∇vi| +�p−2 dx +≥ γp +� 1 +0 +sds +� +B +|∇(ui − vi)|2 � +s|∇ui| + (2 − s)|∇vi| +�p−2 dx, +≥ C +� +B +|∇(ui − vi)|2 � +|∇ui| + |∇vi| +�p−2 dx. +On the other hand, using the H¨older inequality, we have +� +B +|∇(ui − vi)|p dx ≤ +�� +B +|∇(ui − vi)|2 � +|∇ui| + |∇vi| +�p−2 dx +�p/2 �� +B +� +|∇ui| + |∇vi| +�p�1−p/2 +. + +5 +We conclude the proof by applying +� +B |∇vi|p ≤ +� +B |∇ui|p (since vi is p-harmonic). +□ +Lemma 3.2. (H¨older regularity) Let u be a (local) minimizer of J in B1. Then for some +α = α(n, p) +∥u∥Cα(B3/4) ≤ C(n, p, ∥u∥L∞(B1)). +Proof. Let M = ∥u∥L∞(B1) and Br = Br(y) for y ∈ B3/4 and r < 1/8. Since ui is a +p-subsolution, a Caccioppoli type inequality (see [16], Lemma 3.27) implies that +� +Br +|∇ui|p dx ≤ C +rp +� +B2r +(ui)p dx ≤ CMprn−p. +On the other hand, if vi is the p-harmonic replacement of ui inside Br, we have the +gradient estimate (see [17]) +sup +Br/2 +|∇vi| ≤ +� +C +rn +� +Br +|∇vi|p dx +�1/p +≤ CM +r . +Now, let us take some ρ < r/2 which will be specified below and apply Lemma 3.1 +in Br(y) +∥∇ui∥Lp(Bρ) ≤∥∇(ui − vi)∥Lp(Bρ) + ∥∇vi∥Lp(Bρ) +≤∥∇(ui − vi)∥Lp(Br) + Cρn/p∥∇vi∥L∞(Br/2) +≤C + +rn/p + Mρn/pr−1 +for 2 ≤ p, +M1−p/2rn/p−1+p/2 + Mρn/pr−1 +for 1 < p ≤ 2. +Thus for r = ρ1−α, if we take α = α(n, p) sufficiently small, we obtain +∥∇ui∥Lp(Bρ) ≤ C(M, n, p, Qmax)ρn/p−1+α. +By virtue of Morrey’s theorem (see [19]) we conclude the proof of the lemma. +□ +The next lemma is essential to prove the Lipschitz regularity of the minimizers. +Lemma 3.3. Let u = (u1, . . . , um) be a bounded minimizer in B1 and ui(0) = 0 for some +1 ≤ i ≤ m. Then there exists a constant C = C(n, p, Qmax) > 0 such that +∥ui∥L∞(B1/4) ≤ C. +We need to remark that the constant C is independent of the boundary values +of u on ∂Ω. In other words when going away from a free boundary, but staying +uniformly inside the domain Ω, the minimizer cannot grow too large, regardless +of the boundary values. In other words, for large enough boundary values, the +origin cannot be a free boundary point. +Proof. For the sake of convenience consider i = 1. Towards a contradiction, assume +that there is a sequence of bounded solutions uk in B1 such that +∥u1 +k∥L∞(B1/4) > k. +Set +dk(x) := dist(x, {u1 +k = 0}) +in B1, +and define +Ok := {x ∈ B1 : dk(x) ≤ (1 − |x|)/3}. + +6 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Obviously, B1/4 ⊂ Ok. We have also +mk := sup +Ok +(1 − |x|)u1 +k(x) ≥ 3 +4 max +B1/4 u1 +k > 3 +4k, +Since u1 +k is bounded (for fixed k), we get (1 − |x|)u1 +k(x) → 0 as |x| → 1, and therefore +mk is attained at some point xk ∈ Ok. So, +u1 +k(xk) = +mk +1 − |xk| ≥ mk > 3 +4k. +Now let yk ∈ ∂{u1 +k > 0} ∩ B1 be such that |yk − xk| = dk(xk) =: δk, which satisfies +δk ≤ (1 − |xk|)/3 due to xk ∈ Ok. This implies that +B2δk(yk) ⊂ B1 +and +Bδk/2(yk) ⊂ Ok. +Indeed, if z ∈ B2δk(yk), +|z| ≤ |z − yk| + |yk − xk| + |xk| ≤ 2δk + δk + |xk| ≤ 1, +and if z ∈ Bδk/2(yk), +1 − |z| ≥ 1 − |xk| − |xk − yk| − |yk − z| ≥ 1 − |xk| − δk − δk/2 ≥ 3δk/2 ≥ 3|z − yk| ≥ 3dk(z). +Also, we have 1 − |z| ≥ (1 − |xk|)/2 for any z ∈ Bδk/2(yk). Then +1 − |xk| +2 +max +Bδk/2(yk) u1 +k ≤ +max +z∈Bδk/2(yk)(1 − |z|)u1 +k(z) ≤ max +z∈Ok (1 − |z|)u1 +k(z) = (1 − |xk|)u1 +k(xk) +or +max +Bδk/2(yk) u1 +k ≤ 2u1 +k(xk). +Since Bδk(xk) ⊂ {u1 +k > 0}, then u1 +k is p-harmonic inside Bδk(xk), i.e. ∆pu1 +k = 0. By the +Harnack inequality for p-harmonic functions, there is a constant c = c(n, p) such +that +min +B4δk/5(xk) u1 +k ≥ cu1 +k(xk). +In particular, +max +Bδk/4(yk) u1 +k ≥ cu1 +k(xk). +We define the sequence +wk(x) := uk(yk + (δk/2)x) +u1 +k(xk) +, +whose first component satisfies +(3) +max +B1 w1 +k ≤ 2, +max +B1/2 w1 +k ≥ c > 0, +w1 +k(0) = 0. +Moreover, wk is a minimizer of +Jk(w) = +� +B1 +m +� +i=1 +|∇wi|p + Qp +kχ{|w|>0}dx, +where Qk(x) = δkQ(yk+(δk/2)x) +2u1 +k(xk) +→ 0. Now consider v1 +k to be p-harmonic replacement of +u1 +k in B3/4 and apply Lemma 3.1 +(4) +� +B3/4 +|∇(w1 +k − v1 +k)|p dx ≤ C(max Qk)p → 0, + +7 +when 2 ≤ p. Similar statement holds for 1 < p ≤ 2, we just need to note that +∥∇w1 +k∥Lp is uniformly bounded (w1 +k is p-subsolution and uniformly bounded in B1). +Furthermore, w1 +k and v1 +k are uniformly Cα in B5/8 and we can extract a subsequence +(still denoted by w1 +k and v1 +k) such that w1 +k → w0 and v1 +k → v0 uniformly in B5/8. +Observe that ∆pv0 = 0 in B5/8 and (4) implies that w0 = v0 + c. Hence, w0 is also +p-harmonic and by the strong maximum principle, w0 ≡ 0 in B5/8, since w0 ≥ 0 and +w0(0) = 0. On the other hand, (3) necessitates +max +B1/2 w0 ≥ c > 0, +which is a contradiction. +□ +A direct consequence of the above lemma is the following estimate. +Lemma 3.4. Let u be a (local) minimizer in Ω. If dist(x0, {ui = 0}) < 1 +5dist(x0, ∂Ω) then +ui(x0) ≤ 4Cdist(x0, {ui = 0}), +where C is the constant defined in Lemma 3.3. +Proof. Choose y0 ∈ {ui = 0} such that dist(x0, {ui = 0}) = |x0 − y0| = d0. Now apply +Lemma 3.3 to +v(x) := u(y0 + 4d0x) +4d0 +to get +ui(x0) ≤ 4Cd0. +□ +With the above two results we will obtain uniform Lipschitz regularity for +minimizers. +Theorem 3.5. Let u be a (local) minimizer in Ω, then u is Lipschitz. Moreover, for +every K ⋐ Ω such that K ∩ ∂{ui > 0} � ∅ for some 1 ≤ i ≤ m, there is a constant +C = C(n, p, Qmax, dist(K, ∂Ω), Ω) > 0 such that +∥∇ui∥L∞(K) ≤ C. +Once again we remark that the constant C does not depend on the boundary +values of the minimizer, as long as we stay uniformly inside the domain. +Proof. Step 1: We show that ui is bounded in K with a universal constant C depend- +ing on the following ingredients n, p, Qmax, dist(K, ∂Ω), Ω. Let r0 = 1 +5dist(K, ∂Ω) and +for any arbitrary point x ∈ K there is a sequence of points x = x0, . . ., xk ∈ K with +(we can assume K is connected, otherwise replace it with a bigger one which is +connected) +xj ∈ Br0/2(xj−1), +for j = 1, . . ., k, +Br0(xj) ⊂ {ui > 0} for j = 0, . . ., k − 1 and Br0(xk) ∩ {ui = 0} � ∅. Note that k, the +number of points, only depends on Ω and dist(K, ∂Ω). From Lemma 3.4, we get +ui(xk) ≤ 4Cr0. +Since ui is p-harmonic in Br0(xj), j = 0, . . ., k − 1, by virtue of Harnack’s inequality, +there is a constant c such that +ui(xj+1) ≥ cui(xj). + +8 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Thus +ui(x) ≤ 4c−kCr0. +Step 2: Here we find a control on ∇ui at points close to {ui = 0}. If d = dist(y, {ui = +0}) < +1 +11dist(y, ∂Ω), every points x0 ∈ Bd(y) satisfy condition Lemma 3.4. Then +ui(x0) ≤ 4Cdist(x0, {ui = 0}) ≤ 8Cd. +Let us define +v(x) := ui(y + dx) +d +which is a p-harmonic in B1 and ∥v∥L∞(B1) ≤ 8C. By p-Laplacian estimate for gradient, +we obtain +|∇v(0)| ≤ ˜C(n, p, Qmax), +that is |∇ui(y)| ≤ C(n, p, Qmax). +Step 3: Let r1 = +1 +11dist(K, ∂Ω). If dist(x, {ui = 0}) ≤ r1, by the result of Step 2 we +have already |∇ui(x)| ≤ C. If dist(x, {ui = 0}) > r1, then ui is p-harmonic inside Br1(x) +and ∥ui∥L∞(Br0) is universally bounded by the result of Step 1. Thus |∇ui(x)| will be +universally bounded. +□ +A straightforward corollary to this theorem, that can be useful later, is the +following +Corollary 3.6. Let u be a (local) minimizer for our functional. For every K ⋐ Ω there +exists constant C = C(n, p, Qmax, dist(K, ∂Ω), Ω) such that +1 +r − +� +∂Br +ui dx > C implies ui > 0 in Br. +Proof. If Br ⊂ K contains a free boundary point, then by Theorem 3.5, ui ≤ Cr on +∂Br. +□ +4. Nondegeneracy +Lemma 4.1. For any 0 < κ < 1 there exists a constant c = c(κ, n, m, p, Qmin) > 0 such +that for every minimizer u and for any (small) ball Br ⊂ Ω +∥u∥L∞(Br) < cr implies u = 0 in Bκr. +Proof. Without loss of generality, we may assume r = 1. Let +M = ∥u∥L∞(B √κ). +Let φ(x) = φκ(|x|) be the solution of +∆pφ = 0, in B √κ \ Bκ, +φ = 0 on ∂Bκ, +φ = 1 on ∂B √κ +and extend φ = 0 in Bκ. Set v = M √κφ and wi = min(ui, v) for all i = 1, . . ., m. Since +v ≥ ui on ∂B √κ, so wi = ui on ∂B √κ. Therefore J(u) ≤ J(w), or equivalently +� +B √κ +m +� +i=1 +|∇ui|p + Qpχ{|u|>0} dx ≤ +� +B √κ\Bκ +m +� +i=1 +|∇wi|p + Qpχ{|w|>0} dx. + +9 +Since {|w| > 0} ⊂ {|u| > 0}, we get +� +Bκ +m +� +i=1 +|∇ui|p + Qpχ{|u|>0} dx ≤ +� +B √κ\Bκ +m +� +i=1 +� +|∇wi|p − |∇ui|p� +dx +≤ p +� +B √κ\Bκ +m +� +i=1 +|∇wi|p−2∇wi · ∇(wi − ui) dx += −p +� +∂Bκ +m +� +i=1 +|∇wi|p−2(wi − ui)(∇wi · ν) dHn−1 += p +� +∂Bκ +m +� +i=1 +|∇v|p−2ui(∇v · ν) dHn−1. +Since |∇v| ≤ C(p, κ, n)M on ∂Bκ, we find out that +(5) +� +Bκ +m +� +i=1 +|∇ui|p + Qpχ{|u|>0} dx ≤ CMp−1 +m +� +i=1 +� +∂Bκ +ui dHn−1. +On the other hand, +� +∂Bκ +ui dHn−1 ≤ C(n, κ) +� +Bκ +ui + |∇ui| dx += C(n, κ) +� +Bκ +� +ui + |∇ui| +� +χ{ui>0} dx +≤ C(n, κ, p, Qmin) +� +Bκ +MQpχ{ui>0} + |∇ui|p + Qpχ{ui>0} dx +≤ C(n, κ, p, Qmin)(1 + M) +� +Bκ +|∇ui|p + Qpχ{ui>0} dx. +Comparing with (5), we we arrive at +� +Bκ +m +� +i=1 +|∇ui|p + Qpχ{|u|>0} dx ≤ CMp−1(1 + M) +� +Bκ +m +� +i=1 +|∇ui|p + Qpχ{|u|>0} dx. +Therefore, if M is small enough, we obtain that u = 0 in Bκ. +□ +An immediate consequence of the above lemma is the following. For any K ⋐ Ω +there are positive constants c0, C0 such that if Br(x) ⊂ K ∩ {|u| > 0} touches ∂{|u| > 0} +then +(6) +c0r ≤ |u(x)| ≤ C0r. +Theorem 4.2. For K ⋐ Ω there exists constant 0 < c = c(n, m, p, K, Ω) < 1 such that for +any (local) minimizer u and for any (small) ball Br(x) ⊂ K with x ∈ ∂{|u| > 0}, +(7) +c < Ln(Br(x) ∩ {|u| > 0}) +Ln(Br(x)) +< 1 − c. +Proof. By Lemma 4.1, there exists y ∈ Br/2 such that |u(y)| ≥ cr > 0. Using Lipschitz +continuity we get +− +� +∂Bκr(y) +|u| ≥ cr +2 , + +10 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +provided κ is small enough. Hence +1 +κr− +� +∂Bκr(y) +|u| ≥ c +2κ, +and also for at least one component ui +1 +κr− +� +∂Bκr(y) +ui ≥ +c +2κm, +which by Corollary 3.6 implies |u| > 0 in Bκr(y). This gives the lower estimate in +(7). +To prove the estimate from above we assume, for simplicity, r = 1 and suppose +(towards a contradiction) that there is a sequence of minimizers uk in B1(0) such +that 0 ∈ ∂{|uk| > 0} and +Ln({|uk| = 0}) =: εk → 0. +Let vi +k be a p-harmonic function in B1/2 with boundary data vi +k = ui +k on ∂B1/2. From +Lemma 3.1, we obtain that +(8) +� +B1/2 +|∇(vi +k − ui +k)|p dx ≤ C(εk) → 0. +Since ui +k and vi +k are both uniformly Lipschitz in B1/4, we may assume that ui +k → ui +0 +and vi +k → vi +0 uniformly in B1/4. Observe that ∆pvi +0 = 0 and (8) implies that ui +0 = vi +0+c. +Thus ∆pui +0 = 0 in B1/4 and from the strong minimum principle (since ui +0(0) = 0) +it follows ui +0 ≡ 0 in B1/4, since ui +0 ≥ 0 and ui +0(0) = 0. On the other hand form +nondegeneracy property, Lemma 4.1, we know +∥uk∥L∞(B1/2) ≥ c > 0, +which implies a similar inequality for u0, and hence a contradiction. +□ +Remark 4.3. Theorem 4.2, along with the Lebesgue density theorem implies that the free +boundary has zero Lebesgue measure +Ln(∂{|u| > 0}) = 0. +5. The vector-valued measure ∆pu +Let 0 ≤ ζ ∈ C∞ +0 (Ω) be a test function, and define the measure λi by +� +ζ dλi = − +� +|∇ui|p−2∇ui · ∇ζ dx, +which in virtue of Lemma 2.3 is a bounded non-negative measure, i.e. a Radon +measure. Obviously λi is the formal way of expressing ∆pui in Ω. +Since each ui is p-subharmonic in Ω and ui ≥ 0 we have that λi is a positive +Radon measure. Because ui is also p-harmonic in {ui > 0} we have that the support +of λi is in Ω ∩ ∂{ui > 0} ⊆ Ω ∩ ∂{|u| > 0}.1 Let us define +Λ = Λu := +m +� +i=1 +λi. +1Observe that ui may be zero in some component of {|u| > 0}. + +11 +Theorem 5.1. For any K ⋐ Ω there exist constants c, C > 0 such that for any (local) +minimizer u +crn−1 ≤ +� +Br +dΛ ≤ Crn−1 +for any ball Br ⊂ K with x ∈ ∂{|u| > 0}. +Proof. Let 0 ≤ ζǫ ∈ C∞ +0 (Br+ǫ) be a suitable test function, such that ζǫ = 1 on Br and +|∇ζǫ| ≤ 2/ǫ. Then +� +ζǫ dλi = − +� +|∇ui|p−2∇ui · ∇ζǫ dx = − +� +Br+ǫ\Br +|∇ui|p−2∇ui · ∇ζǫ dx ≤ Crn−1, +where in the last inequality we have used that u is Lipschitz. Letting ǫ tend to zero, +we arrive at +� +Br +dλi ≤ Crn−1. +To prove the estimate from below, we argue indirectly. It also suffices to consider +the case r = 1. Assume there is a sequence of minimizers uk in the unit ball B1(0), +such that 0 ∈ ∂{|uk| > 0} and for the measures Λk := Λuk we have +εk := Λk(B1) → 0. +Since the functions uk are uniformly Lipschitz continuous, we may assume that +uk → u0 in B1/2, where u0 is Lipschitz continuous as well. We may also extract a +subsequence (still denote by uk) such that gi +k := |∇ui +k|p∇ui +k → gi +0 weakly-∗ in L∞(B1/2) +for all i = 1, . . ., m. We claim that +(9) +gi +0 = |∇ui +0|p∇ui +0, +in B1/2. +Suppose this is true, then for every positive test function ζ ∈ C∞ +0 (B1/2) one has +− +� +B1/2 +|∇ui +0|p−2∇ui +0 · ∇ζ dx = − lim +k→∞ +� +B1/2 +|∇ui +k|p−2∇ui +k · ∇ζ dx += lim +k→∞ +� +B1/2 +ζ dλi +k ≤ ∥ζ∥L∞(B1/2) lim +k→∞ εk = 0. +Thus λi +0 = 0 and ui +0 is p-harmonic for all i = 1, . . ., m (note that ui +0 is the limit of a se- +quence of p-subharmonic functions and we already know that it is p-subharmonic). +Since ui +0 ≥ 0 and ui +0(0) = 0, by the minimum principle, we have ui +0 ≡ 0 in B1/2. +On the other hand, by nondegeneracy property (Lemma 4.1) and that 0 ∈ ∂{|uk| > +0} we have +∥uk∥L∞(B1/4) ≥ c > 0. +Therefore, a similar inequality holds for u0 and we arrive at a contradiction. +To close the argument we need to prove (9). In fact, if Bρ = Bρ(y) ⊂ {|u0| > 0} +then Bρ ⊂ {|uk| > 0} for sufficiently large k and ui +k are p-harmonic in Bρ for all +i = 1, . . ., m, (see Lemma 2.3). Therefore, one can extract a subsequence of uk +locally converging to u0 in C1,α(Bρ). Hence, gi +0 = |∇ui +0|p∇ui +0 in Bρ for all i = 1, . . ., m. +Next, if Bρ ⊂ {|u0| = 0} then for any κ < 1 the nondegeneracy property entails that +Bκρ ⊂ {|uk| = 0} for sufficiently large k = k(κ). Thus gi +0 = 0 = |∇ui +0|p∇ui +0. We just +need to show Ln(∂{|u0| > 0} ∩ B1/2) = 0. If x0 ∈ ∂{|u0| > 0} ∩ B1/2, then u0(x0) = 0. + +12 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Choose xk ∈ ∂{|uk| > 0} ∩ B1 such that |xk − x0| = dist(x0, ∂{|uk| > 0}), then relation +(6) yields that |xk − x0| → 0. Apply Lemma 4.1 to obtain +∥uk∥L∞(B2r(x0)) ≥ ∥uk∥L∞(Br(xk)) ≥ cr, +for any ball Br(x0) ⊂ B1/2 and sufficiently large k. Passing to the limit we get the +same inequality for u0, +∥u0∥L∞(B2r(x0)) ≥ cr. +This along with the Lipschitz continuity of u0 is enough to prove that Ln(Br(x0) ∩ +{|u0| > 0}) ≥ cLn(Br) for some c > 0. (see the first part of the proof of Theorem 4.2). +This implies that Ln(∂{|u0| > 0} ∩ B1/2) = 0 (see Remark 4.3). +□ +The next theorem follows easily from Theorem 5.1. The proof is the same as the +proof of Theorem 4.5 in [2]. +Theorem 5.2. Let u be a (local) minimizer in Ω. Then +(i) For every K ⋐ Ω we have Hn−1(K ∩ ∂{|u| > 0}) < ∞. +(ii) There exist nonnegative Borel functions qi such that +∆pui = qiHn−1⌊ ∂{|u| > 0}, +that is for every ζ ∈ C∞ +0 (Ω) +− +� +Ω +|∇ui|p−2∇ui · ∇ζ dx = +� +Ω∩∂{|u|>0} +ζqidHn−1. +(iii) For any K ⋐ Ω there exist constants c, C > 0 such that +c ≤ +m +� +i=1 +qi ≤ C, +and for Br(x) ⊂ K with x ∈ ∂{|u| > 0} we have +crn−1 ≤ Hn−1(Br(x) ∩ ∂{|u| > 0}) ≤ Crn−1. +Remark 5.3. From (i) in Theorem 5.2 it follows that, locally, the set A = Ω∩{|u| > 0} has +finite perimeter in Ω in sense of that µu = −∇χA is a Borel measure and the total variation +|µu| is a Radon measure. We define the reduced boundary of A by +∂redA = {x ∈ Ω : |νu| = 1}, +where νu(x) is the unique unit vector with +� +Br(x) +���χA − χ{y:(y−x)·νu(x)<0} +��� = o(rn), +as r → 0, +if such a vector exists, and νu(x) = 0 otherwise. See [14], Chapter 4, for more details. +6. Local Analysis +To proceed, we will need some properties of the so-called blow-up limits. +Lemma 6.1. Let u be a (local) minimizer in Ω, K ⋐ Ω and Brk(xk) ⊂ K be a sequence of +balls with rk → 0, xk → x0 ∈ Ω, and u(xk) = 0. Consider the blow-up sequence +(10) +uk(x) = 1 +rk +u(xk + rkx). + +13 +For a subsequence, there is a limit u0 such that +uk → u0 in C0,α +loc(Rn; Rm) for every 0 < α < 1, +∇uk → ∇u0 a.e. in Rn, +(11) +∂{|uk| > 0} → ∂{|u0| > 0} locally in the Hausdorff distance, +(12) +χ{|uk|>0} → χ{|u0|>0} in L1 +loc(Rn; Rm), +(13) +if xk ∈ ∂{|uk| > 0} then 0 ∈ ∂{|u0| > 0}. +Proof. For the proof we refer to [2] and [3]. +□ +The following lemma shows that the blow-up limit is a minimizer in any ball. +Lemma 6.2. If u(xk) = 0 and xk → x0, then any blow-up limit u0 = limk uk (see (10)) +with respect to Brk(xk) is an absolute minimizer of J0 in any ball BR = BR(0), where +J0(v) := +� +BR +m +� +i=1 +|∇vi|p + Q(x0)pχ{|v|>0} dx. +Proof. Let w ∈ W1,p(BR; Rm) be such that wi ≥ 0 for i = 1, . . ., m and w = u0 on ∂BR. +To show J0(u0) ≤ J0(w), we choose a cut of function η ∈ C∞ +0 (BR) with 0 ≤ η ≤ 1 and +η = 1 in Br for some 0 < r < R, and define +wk = �w + (1 − η)(uk − u0)� ++ , +where the positive part is taken separately for each component. We also have +wk = uk on ∂BR. Since u is (local) minimizer, for sufficiently large k such that +BRrk(xk) ⋐ Ω, we have +� +BR +m +� +i=1 +|∇ui +k|p + Qp +kχ{|uk|>0} dx ≤ +� +BR +m +� +i=1 +|∇wi +k|p + Qp +kχ{|wk|>0} dx, +where Qk(x) := Q(xk + rkx). Since |∇uk| ≤ C (due to Lipschitz continuity of u) and +convergences (11) and (13), the limit of the left hand side will be J0(u0). Hence +J0(u0) ≤ lim inf +k→∞ +� +BR +m +� +i=1 +|∇wi +k|p + Qp +kχ{|wk|>0} dx +≤ +� +BR +m +� +i=1 +|∇wi|p dx + +� +Br +Q(x0)pχ{|w|>0} dx + lim inf +k→∞ +� +BR\Br +Qp +kχ{|wk|>0} dx +≤ +� +BR +m +� +i=1 +|∇wi|p dx + +� +Br +Q(x0)pχ{|w|>0} dx + Q(x0)p|BR \ Br|. +Now let r → R, we get J0(u0) ≤ J0(w). +□ +Lemma 6.3. Suppose ∇(Qp) ∈ L1(Ω) and u is an absolute minimizer. Then +� +|u|>0 +div + + +m +� +i=1 +� +|∇ui|pΨ − p|∇ui|p−2(∇ui · Ψ)∇ui� ++ QpΨ + + dx = 0, +for every Ψ ∈ C∞ +c (Ω; Rn). + +14 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Proof. Let us define +Φt(x) = x + tΨ(x) +and +ut(x) = u(Φt(x)). +One can show that for sufficiently small |t|, Φt : Ω → Ω is a diffeomorphism. We +have DΦt = I + tDΨ and for i = 1, . . ., m +∇ui +t = DΦt(x)∇ui(Φt(x)). +It follows that +|∇ui +t(x)|2 = (∇ui(Φt(x)))TAt(x)∇ui(Φt(x)), +where +At = (DΦt)TDΦt = I + t((DΨ)T + DΨ) + t2(DΨ)TDΨ. +By a change of variables, we have +J(ut) = +� +Ω +m +� +i=1 +|∇ui +t(x)|p + Qpχ{|ut|>0} dx += +� +Ω +m +� +i=1 +� +(∇ui(Φt(x)))TAt(x)∇ui(Φt(x)) +�p/2 + Qp(x)χ{|u|>0}(Φt(x)) dx += +� +Ω +m +� +i=1 +�� +(∇ui(y))TAt(Φ−1 +t (y))∇ui(y) +�p/2 + Qp(Φ−1 +t (y))χ{|u|>0}(y) +� ���det DyΦ−1 +t (y) +��� dy += +� +{|u|>0} +m +� +i=1 +�� +(∇ui(y))TAt(Φ−1 +t (y))∇ui(y) +�p/2 + Qp(Φ−1 +t (y)) +����det DyΦ−1 +t (y) +��� dy. +We also have +d +dtAt(Φ−1 +t (y)) +����t=0 = DΨ(y) + DΨ(y)T +and +d +dt +���det DyΦ−1 +t (y) +��� +����t=0 = −div Ψ(y). +Now differentiate J(ut) with respect to t and note that its minimum is attained at +t = 0, then +0 = d +dt J(ut) +����t=0 = +� +{|u|>0} +m +� +i=1 +p|∇ui|p−2(∇ui)TDΨ∇ui − Ψ · ∇(Qp)dx +− +� +{|u|>0} + + +m +� +i=1 +|∇ui|p + Qp + + div Ψ dx += − +� +{|u|>0} +div + + +m +� +i=1 +� +|∇ui|pΨ − p|∇ui|p−2(∇ui · Ψ)∇ui� ++ QpΨ + + ++ +m +� +i=1 +p(∇ui · Ψ)∆pui dx. +Since each ui is p-harmonic in {ui > 0}, see (2), we arrive at the desired claim, in the +lemma. +□ + +15 +Definition 6.4. The upper Hn−1-density at any point x0 ∈ ∂{|u| > 0} is defined as +Θ∗n−1 � +Hn−1� +∂{|u| > 0}, x0 +� +:= lim sup +r→0 +Hn−1(Br(x0) ∩ ∂{|u| > 0}) +ωn−1rn−1 +, +where ωn−1 denotes the volume of the unit sphere in Rn−1. We already know (see for example +Theorem 2.7 in [13]) that for Hn−1-a.e. point x0 ∈ ∂{|u| > 0}, their upper Hn−1-density +satisfy +Θ∗n−1 � +Hn−1� +∂{|u| > 0}, x0 +� +≤ 1. +Theorem 6.5. Let x0 ∈ ∂red{|u| > 0} and suppose that +Θ∗n−1 � +Hn−1� +∂{|u| > 0}, x0 +� +≤ 1. +Then Tan(∂{|u| > 0}, x0) = {x : x · ν(x0) = 0}. If, in addition, x0 is a Lebesgue point for +Radon measure qiHn−1� +∂{|u| > 0}, that is +(14) +� +Br(x0)∩∂{|u|>0} +|qi − qi(x0)| dHn−1 = o(rn−1), +as r → 0, +then qi(x0) = Q(x0) and +u(x0 + x) = (−x · ν(x0))+ ax0 + o(|x|), +as x → 0, +for some vector ax0 = (α1, . . ., αm) that +(15) +|ax0|p +p = (α1)p + · · · + (αm)p = +1 +p − 1Q(x0)p. +Proof. Without loss of generality assume that ν(x0) = en. +Let uk be a blow-up +sequence with respect to balls Brk(x0), with blow-up limit u0. Since ν(x0) is the +normal vector to ∂{|u| > 0} at x0, +� +Br(x0) +���χ{|u|>0|} − χ{x:(x−x0)·ν(x0)<0} +��� dx = o(rn), +as r → 0. +This along with (13) implies χ{|u0|>0} = χ{xn<0} almost every where in Rn. By Lemma +6.2 we know that u0 is an absolute minimizer of J0 and so continuous. +Then +{|u0| > 0} = {xn < 0}. This proves that {xn = 0} is the topological tangent plane to +∂{|u| > 0} at x0. Now let +φ(x) = min (1, max(0, 2 − |xn|)) η(x′), +where 0 ≤ η ∈ C∞ +0 (B′ +R) and B′ +R is (n−1)-dimensional ball with radius R (R is arbitrary +and fixed). Denote φk(x) := rkφ( x−x0 +rk ) and write +− +� +Rn |∇ui +0|p−2∇ui +0 · ∇φ dx = − lim +k→∞ +� +Rn |∇ui +k|p−2∇ui +k · ∇φ dx += − lim +k→∞ r−n +k +� +Rn |∇ui|p−2∇ui · ∇φk dx = lim +k→∞ r−n +k +� +Rn ∆puiφk dx += lim +k→∞ r−n +k +� +Rn qiφk dHn−1⌊ ∂{|u| > 0} = lim +k→∞ +� +Rn qi(x0 + rkx)φ(x)χ∂{|uk|>0} dHn−1 += lim +k→∞ +� +Rn qi(x0)φ(x)χ∂{|uk|>0} dHn−1 = qi(x0) +� +{xn=0} +η(x′) dx′, + +16 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +where we have used assumption (14) and property (12). Therefore, for any test +function ζ ∈ C∞ +0 (BR) we have +− +� +BR∩{xn<0} +|∇ui +0|p−2∇ui +0 · ∇ζ dx = qi(x0) +� +B′ +R +ζ(x′, 0) dx′. +Since ∆pui +0 = 0 in {xn < 0}, from boundary regularity it follows that +|∇ui +0|p−2∂nui +0 = −qi(x0) +on {xn = 0}, +in the classical sense. We need to show that +(16) +ui +0(x) = αi(−xn)+, +where αi := +� +qi(x0) +�1/(p−1) . +To see this, define w0 by +w0(x) := + +ui +0(x), +in xn ≤ 0, +−ui +0(x′, −xn), +in xn > 0. +It is obvious that w0 is p-harmonic in whole Rn as well as +∥∇w0∥L∞(Rn) = ∥∇ui +0∥L∞(Rn) ≤ ∥∇ui∥L∞(Br(x0)), +for any r > 0. +By Liouville’s theorem we conclude that w0 is a linear function. The boundary +value on xn = 0, (ui +0 = 0 and ∂nui +0 = −αi) shows that w0(x) = −αixn. This proves (16) +and shows that +ui(x0 + x) = αi(−xn)+ + o(|x|), +as x → 0. +We just have to show (15). To do this, note that u0 is an absolute minimizer of J0. +Apply Lemma 6.3 for u0 and some Ψ ∈ C∞ +c (Rn) +0 = +� +{xn<0} +div + + +m +� +i=1 +� +|∇ui +0|pΨ − p|∇ui +0|p−2(∇ui +0 · Ψ)∇ui +0 +� ++ Q(x0)pΨ + + dx += +� +{xn=0} +m +� +i=1 +� +|∇ui +0|p(Ψ · en) − p|∇ui +0|p−2(∇ui +0 · Ψ)∂nui +0 +� ++ Q(x0)p(Ψ · en) dHn−1 += +� +{xn=0} + + +m +� +i=1 +(1 − p)(αi)p + Q(x0)p + + (Ψ · en) dHn−1. +Thus (15) will be obtained. +□ +7. Regularity of free boundary +Definition 7.1. Let u ∈ C(Ω, Rm). We say that the boundary condition ∇|u|p = g on +∂{|u| > 0} holds in viscosity sense, if +◦ For every differentiable function φ : Rn → R that touches |u|p from below in some +x0 ∈ ∂{|u| > 0}, that is +|u(x0)|p = φ(x0), +and +|u|p ≥ φ in {|u| > 0} ∩ Br(x0) +for some r > 0, we have |∇φ(x0)| ≤ g(x0). + +17 +◦ For every differentiable function φ : Rn → R that touches |u|p from above in some +x0 ∈ ∂{|u| > 0}, that is +|u(x0)|p = φ(x0), +and +|u|p ≤ φ in {|u| > 0} ∩ Br(x0) +for some r > 0, we have |∇φ(x0)| ≥ g(x0). +Lemma 7.2. Let u be a (local) minimizer, then the boundary condition +∇|u|p = +1 +(p − 1)1/p Q, +on ∂{|u| > 0}. +holds in the viscosity sense. +Proof. We show that the boundary condition holds on every point x0 ∈ ∂{|u| > 0}. +Suppose φ touches |u|p from below at x0. Consider the blow-up sequences +uk(x) = u(x0 + rkx) +rk +and +φk(x) = φ(x0 + rkx) +rk +, +where rk ց 0. Observe that +φk(0) = |uk(0)|p = 0, +φk(x) ≤ |uk(x)|p, +and up to a subsequence we have +(17) +∇φ(x0) · x = lim +k→∞ φk(x) ≤ lim +k→∞ |uk(x)|p = |u0(x)|p +in Rn. +If∇φ(x0) = 0, theviscosityconditionholdstrivially. Otherwise, thenon-coincidence +set {|u0| > 0} contains half-space {x : ∇φ(x0) · x > 0}. +On the other hand, u0 +is minimizer of J0 (Lemma 6.2) and by Lemma 2.3 every nontrivial component +of u0, say ui +0, is positive in {x : ∇φ(x0) · x > 0}. +According to Lemma B.1, +ui +0(x) = αi(∇φ(x0) · x) + o(|x|) for some αi. Thus any blowup of u0 at x = 0 must be +of the form u00(x) = a0(∇φ(x0) · x) where a0 = (α1, · · · , αm). Again apply Lemma 6.2 +along with Lemma 6.3, we get +|a0|p|∇φ(x0)| = +1 +(p − 1)1/p Q(x0). +Thus (17) yields that +∇φ(x0) · x ≤ |u0(x)|p = |a0|p|∇φ(x0) · x| + o(|x|), +and so +|∇φ(x0)| ≤ +1 +(p − 1)1/p Q(x0). +The same argument holds when φ touches |u0|p from above. +□ +Definition 7.3. A domain Ω ⊂ Rn is called non-tangentially accessible domain (NTA) +with parameters M ≥ 1 and R0 > 0 if +(i) Ω satisfies the corkscrew condition, that is, for any x ∈ ∂Ω and r ∈ (0, R0) there +exists ar(x) ∈ Ω ∩ Br(x) such that M−1r < dist(ar(x), ∂Ω). +(ii) Rn \ Ω satisfies the corkscrew condition. +(iii) If x ∈ ∂Ω and x1, x2 ∈ Br(x) ∩ Ω for 0 < r < R0, then there exists a rectifiable curve +γ : [0, 1] → Ω with γ(0) = x1 and γ(1) = x2 such that H1(γ) ≤ M|x1 − x2| and +min +� +H1(γ([0, t])), H1(γ([t, 1])) +� +≤ Mdist(γ(t), ∂Ω), +for every t ∈ [0, 1], +where H1 denotes length or the one-dimensional Hausdorff measure. + +18 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Definition 7.4. We say that x0 ∈ ∂{|u| > 0} is a regular point of the free boundary if for +every ǫ > 0 there are r < 1 and a vector a ∈ Rm and unit vector ν ∈ Rn such that +∥ur,x0 − (x · ν)+a∥L∞(B1) ≤ ǫ. +We denote the set of all regular points by Ru. Theorem 6.5 proves that Hn−1(∂{|u| > +0} \ Ru) = 0. +Theorem 7.5. Let u = (u1, . . ., um) be a (local) minimizer and x0 ∈ Ru. Furthermore, +assume that Br0(x0) ∩ {|u| > 0} is NTA domain. Then Ru ∩ Br(x0), for some 0 < r ≤ r0, is +C1,α for a universal exponent 0 < α < 1. +Proof. We may assume that u1 > 0 in Br0(x0) ∩ {|u| > 0}. First we show that there is +a H¨older function g : Br(x0) ∩ ∂{|u| > 0} → [c, 1] for some 0 < r ≤ r0 and 0 < c ≤ 1 +such that u1 is a viscosity solution to the problem +∆pu1 = 0, +in {u1 > 0} ∩ Br, +|∇u1| = +gQ +(p − 1)1/p , +on ∂{u1 > 0} ∩ Br. +(18) +Since Br0(x0) ∩ {|u| > 0} is NTA domain, the boundary Harnack inequality (see +[18]) implies that gi := ui/u1 is H¨older continuous in {|u| > 0} ∩ Br for some r ≤ r0. +Define +g := (1 + gp +2 + · · · + gp +m)−1/p, +and observe that u1 = g|u|p. Suppose now the test function φ is touching u1 from +below in a point y ∈ ∂{|u| > 0}. For ρ small enough, choose a constant C > 0 such +that +1 +g(x) ≥ +1 +g(y) − C|x − y|µ ≥ 0, +for every x ∈ {|u| > 0} ∩ Bρ, +where µ is H¨older exponent of g. +Set ψ(x) = φ(x)(1/g(y) − C|x − y|µ), we get +ψ(y) = |u(y)|p = 0 (note that since φ(y) = 0, so ψ is differentiable at y) and +ψ(x) ≤ u1(x)( 1 +g(y) − C|x − y|µ) = g(x)( 1 +g(y) − C|x − y|µ)|u(x)|p ≤ |u(x)|p. +Therefore, ψ touches |u|p at y from below. By Lemma 7.2 +|∇ψ(y)| ≤ +Q(y) +(p − 1)1/p . +Note that ∇ψ(y) = +1 +g(y)∇φ(y) to see the boundary condition (18) in viscosity sense. +The regularity of free boundary follows by the known results on the regularity +of the one-phase scalar problem (18); see [15]. +□ +Our next result improves the theorem above, in the sense that we can remove +the NTA conditions, for p close to 2. The main reason for not being able to handle +the NTA, is the lack of ACF-monotonicity formula, use in classical paper; see proof +of Proposition A.3, to prove the connectivity argument for the NTA. +Theorem 7.6. Let u = (u1, . . ., um) be a (local) minimizer. Then there is ǫ0 > 0, such that +for any p ∈ (2 − ǫ0, 2 + ǫ0) we have +i) The regular set Ru, is locally C1,α. +ii) In dimensions 2, 3, 4 the free boundary is C1,α. +Here 0 < α < 1 is is a universal exponent. + +19 +Proof. To prove (i) it suffices, in virtue of Theorem 7.5, to show that the free bound- +ary is NTA, when ǫ0 is small enough. This, however, is a consequence of Proposition +A.5 in Appendix A, by choosing ǫ0 accordingly. +Turning to sstatement (ii) we recall that in these dimensions, and for p = 2, free +boundaries are locally C1,α; for n = 2 this was shown in [2], for n = 3 see [6], for +n = 4 see [12]. Now for p ≈ 2, the free boundary has to be close to that of the +case p = 2, and hence flat. More specifically, given a free boundary point x0 for the +p-problem, and p close enough to 2, we have that in Br(x0) the free boundaries of +both p and 2, have Huasdorff distance δ ≪ r, and in particular the p-free boundary +is (δ/2)-flat. +□ +Appendix A. Non-tangentially accessible domain (NTA) +We note that in Definition 7.3, the condition (iii) can be replaced by Harnack chain +condition (see [4]), i.e., +(iii)’ Given ε > 0, x1, x2 ∈ Ω such that dist(xi, ∂Ω) ≥ ε, i = 1, 2 and |x1 − x2| ≥ ˜Cε, +we can find points x1 = y1, y2, · · · , yℓ = x2 for which +(a) Bε(yi) ⊂ Ω for i = 1, · · · , ℓ. +(b) Bε(yi) ∩ Bε(yi+1) � ∅ for i = 1, · · · , ℓ − 1. +(c) The length of chain, ℓ, depends on ˜C but not on ε. +We need an analogue of Theorem 4.1 in [1] to show that the non-coincidence set +is NTA. +Lemma A.1. Let u be a minimizer and suppose 0 < |u(x0)|. Define δ = dist(x0, Γ), +δ1 = dist(x0, {|u| ≤ 1 +2|u(x0)|∞}), and suppose B(x0, δ) ⊂ Ω. Then, there exist universal +constants λ > 1 > σ such that +(i) σδ ≤ δ1 ≤ δ. +(ii) For some y ∈ ∂Bδ1(x0), |u(y)|∞ ≥ λ|u(x0)|∞. +Proof. According to relation (6), +c0δ ≤ |u(x0)| ≤ C0δ, +and if ui(x0) = max1≤j≤m uj(x0) = |u(x0)|∞, +c0δ +√m +≤ ui(x0) ≤ C0δ. +If z ∈ ∂{|u| ≤ 1 +2|u(x0)|∞} such that |z − x0| = δ1, then +1 +2ui(x0) = 1 +2|u(x0)|∞ ≤ |u(x0)| − |u(z)| ≤ |u(x0) − u(z)| ≤ C0δ1, +where we have used |u(z)| = 1 +2|u(x0)|∞ ≤ 1 +2|u(x0)|. Thus (i) holds for σ = +c0 +2C0 +√m. +To see (ii), note that v(x) := ui(x0 + δ1x)/ui(x0) is a p-harmonic function in Bδ/δ1 +such that v(0) = 1, v(ˆz) ≤ 1 +2 for some ˆz ∈ ∂B1. In addition, the Lipschitz constant of +v is bounded by C0δ1/ui(x0) ≤ C0δ/ui(x0) ≤ C0 +√m/c0 = 1/2σ. We claim that there +is a universal constant λ > 1 such that v( ˆy) ≥ λ for some ˆy ∈ ∂B1. Otherwise, we +find a sequence of p-harmonic functions vk with +∥vk∥L∞(B1) ≤ 1 + 1 +k, +∥∇vk∥L∞(B1) ≤ 1 +2σ, +vk(0) = 1, +vk(zk) ≤ 1 +2, + +20 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +for some |zk| = 1. Then there is a subsequence converging in C1,α(B1) to a p-harmonic +v0 where v0(0) = 1, v0(z0) ≤ 1/2, ∥v0∥L∞(B1) = 1. This is a contradiction with the +maximum principle. Therefore, we have found y ∈ ∂Bδ1(x0) such that +|u(y)|∞ ≥ ui(y) ≥ λui(x0) = λ|u(x0)|∞. +□ +Lemma A.2. There exists a universal constant ˜c0 such that if x0 ∈ ∂{|u| > 0}, x1 ∈ Br/2(x0) +and Ar is the connected component of {|u| > 1 +2|u(x1)|∞} ∩ Br(x0) containing x1, then +∥u∥L∞(Ar) ≥ ˜c0r. +Proof. WeuseLemma A.1 toinductivelydefinea sequenceofpointsx1, x2, . . ., xk, xk+1 +so that for j = 1, . . ., k, +(i) |xj+1 − xj| = δj = dist(xj, {|u| ≤ 1 +2|u(xj)|∞}), +(ii) |u(xj+1)|∞ ≥ λ|u(xj)|∞, +(iii) Bδj(xj) ⊂ {|u| > 1 +2|u(x1)|∞}. +By (ii), we know that this process cannot continue indefinitely without stepping +out of Br(x0). So, we stop at the first k for which Bδk+1(xk+1) � Br(x0). Also, by +Lemma A.1, we know that +δj ≤ dist(xj, Γ) ≤ +δj +σ , +and therefore by (6), +c0δj ≤ |u(xj)| ≤ +C0δj +σ . +Now applying (ii), we obtain (recall that σ = +c0 +2C0 +√m) +δj ≤ +√m +c0 +|u(xj)|∞ ≤ +√mλ−ℓ +c0 +|u(xj+ℓ)|∞ ≤ λ−ℓ +2σ2 δj+ℓ. +Therefore, +|xk − x0| ≤ |x1 − x0| + +k−1 +� +j=1 +|xj+1 − xj| ≤ r +2 + +k−1 +� +j=1 +δj ≤ r +2 + δk +2σ2 +k−1 +� +j=1 +λj−k ≤ r +2 + +δk +2σ2(λ − 1). +On the other hand, +δk+1 =dist(xk+1, {|u| ≤ 1 +2|u(xk+1)|∞}) +≤δk + dist(xk, {|u| ≤ 1 +2|u(xk+1)|∞}) +≤δk + dist(xk, {|u| ≤ 1 +2|u(xk)|∞}) = 2δk. +Since Bδk+1(xk+1) � Br(x0), we get +r ≤ |xk+1 − x0| + δk+1 ≤ |xk − x0| + 3δk ≤ r +2 + cδk. +Thus γr ≤ δk for the universal constant γ. It necessitates that Bγr(xk) ⊂ Ar and also +|u(xk)| ≥ c0dist(xk, Γ) ≥ c0γr. +In particular, +max +Ar |u| ≥ c0γr. +□ + +21 +Proposition A.3. There is ε0 > 0 such that for any p ∈ (2 − ε0, 2 + ε0) there is no global +minimizer of J0 (Lemma 6.2) that u(0) = 0 and {|u| > 0} ∩ BR is disconnected for every +R > 0. +Proof. First, for p = 2. We apply the monotonicity formula and Lemma 4.4 in [1]. +Let A1 and A2 two different connected components of {|u∞| > 0}. According to +Lemma 2.3, we may choose a positive component of vector u∞ for each set Ai, +i = 1, 2. Thus we find a function v which is harmonic in A1 ∪ A2 and vanishes in +{|u∞| = 0}. Also, since u∞ is a minimizer, we know that (Theorem 4.2) +|Br \ (A1 ∪ A2)| ≥ c|Br|. +From here we infer that Φ(r)/rβ is a non-decreasing function of r for some positive +constant β > 0 [1, Lemma 4.4], where +Φ(r) = 1 +r4 +�� +Br∩A1 +|∇v|2|x|2−n dx +� �� +Br∩A2 +|∇v|2|x|2−n dx +� +. +Since v is Lipschitz, say with constant C0, we have the bound +Φ(r) ≤ C4 +0, +and thus +Φ(1) ≤ r−βΦ(r) ≤ C4 +0r−β, +which can not be valid for sufficiently large value of r. The contradiction proves +the proposition when p = 2. +To prove the proposition for p close to 2 we argue by contradiction. Assume that +there is a sequence of global minimizers ui for pi → 2 that u(0) = 0 and {|ui| > 0}∩BR +is not connected for any R > 0. +We remark that all our results are stated in a slightly more general form, with +constants depending uniformly for all p ∈ [1/2, 3]; see for the details [9]. In fact, +we will have a uniform Lipschitz constant for all p in this compact interval as +well as the nondegeneracy constant. Also, the constant c in Theorem 4.2 will be +uniform. Therefore, we may choose a convergence subsequence ui → ˜u0. By the +same reasoning as the proof of Lemma 6.2 we get that ˜u0 is a minimizer for p = 2. +On the other hand, {|u0| > 0} ∩ BR is not connected for every R which contradicts +the later part of the proof. +□ +Lemma A.4. Let ε0 > 0 be the constant defined in Proposition A.3. +Then for any +p ∈ (2 − ε0, 2 + ε0) there are constants M ≥ 1 and R0 > 0 such that if x0 ∈ ∂{|u| > 0}, +y ∈ BR0(x0) ∩ ∂{|u| > 0} and x1, x2 ∈ Br(y) for some r < R0, then {|u| > d} ∩ BMr(y) has a +connected component containing x1 and x2, where d = 1 +2 min(|u(x1)|∞, |u(x2)|∞). +Proof. Fix constant p and assume the contrary there are sequences ∂{|u| > 0} ∋ +yi → x0, xi +1, xi +2 ∈ Bri(yi), ri → 0 and Mi → ∞ such that xi +1 and xi +2 are not connected +in {|u| > di} ∩ BMiri(yi) where di = 1 +2 min(|u(xi +1)|∞, |u(xi +2)|∞). Consider the blowup +sequence +u(yi + rix)/ri → u0(x), +di/ri → a, +Lemma A.2 yields that {|u0| > a} ∩ BR has at least two connected components for +any R > 0. Note that a < ∞ due to the Lipschitz regularity. Now consider a +blowdown of u0 +u0(Rix)/Ri → u∞(x), +Ri → ∞, + +22 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +then {|u∞| > 0}∩BR must have at least two connected components for any R > 0. On +the other hand, u∞ is a minimizer of J0; Lemma 6.2. This contradicts Proposition +A.3. +□ +Proposition A.5. Let u be a minimizer when p ∈ (2 − ε0, 2 + ε0) and ε0 is the constant +defined in Proposition A.3. +Suppose x0 ∈ ∂{|u| > 0}. +Then {|u| > 0} is NTA in a +neighborhood of x0. +Proof. Step 1: (Property (i), the corkscrew condition for {|u| > 0}.) +Assume that M > ∥∇u∥L∞(B1)/c where c is the nondegeneracy constant defined +in Lemma 4.1. If the condition fails at a point x ∈ ∂{|u| > 0}, then for any y ∈ +Br(x) ∩ {|u| > 0} we must have dist(y, ∂{|u| > 0}) ≤ M−1r. Thus +|u(y)| ≤ M−1r∥∇u∥L∞(B1) ≤ cr +and by nondegeneracy, Lemma 4.1, u = 0 in Bκr(x). It contradicts that x ∈ ∂{|u| > 0}. +Step 2: (Property (ii), the corkscrew condition for {|u| = 0}.) +Assume the contrary, {|u| = 0} does not satisfy the corkscrew condition in any +neighborhood of x0 for any constant M. Thus there is a sequence xj → x0 and +rj → 0 such that we can not find point arj(xj) with the desired property. On the +other hand, Theorem 4.2 infer that the interior of {|u| = 0} is nonempty. Let Bτj(yj) +be the biggest ball inside Brj(xj) ∩ {|u| = 0}, then we must have τj/rj → 0. Now +consider the blowup uj(x) := u(xj + rjx)/rj → u0(x), it will be a minimizer whose +coincidence set has no interior in B1. This contradicts Theorem 4.2. +Step 3: (Harnack chain condition.) +Suppose that x1 and x2 are such that for some ˜C > 0 and ε > 0 we have +|x1 − x2| < ˜Cε, +Bε(xi) ⊂ {|u| > 0}, i = 1, 2. +We may assume without loss of generality, dist(x1, ∂{|u| > 0}) ≤ dist(x2, ∂{|u| > 0}) = +δ0. If δ0 ≥ ˜Cε, then x1 ∈ B ˜Cε(x2) ⊂ {|u| > 0} and we can easily find the Harnack +chain. So, consider the case δ0 < ˜Cε and choose x ∈ ∂{|u| > 0} such that |x − x2| = δ0. +Then x1, x2 ∈ Br(x) for r = 2 ˜Cε. +By Lemma A.4, {|u| > d} ∩ BMr(x) has a connected component containing x1 and +x2, where d = 1 +2 min(|u(x1)|∞, |u(x2)|∞). Now we have a curve γ : [0, 1] → {|u| > +d} ∩ BMr(x) having x1 and x2 as end point. For every t ∈ [0, 1] we know that +|u(γ(t))| ≥ d ≥ +c0ε +2 √m +, +where c0 comes from (6). Hence, +dist(u(γ(t)), ∂{|u| > 0}) ≥ σε, +where σ = +c0 +2C0 +√m. Now we can find a sequence y1, · · · , yℓ on the image of γ such +that +γ[0, 1] ⊂ +ℓ +� +i=1 +Bσε(yi) ⊂ {|u| > 0} ∩ BMr+σε(x). +Since Mr + σε = (2M ˜C + σ)ε, the number of balls in covering, ℓ can be bounded by +a constant depending only on the dimension and (2M ˜C + σ)/σ, but not on x1, x2 or +ε. +□ + +23 +Appendix B. An approximation lemma +Here we prove a lemma, which we used in Lemma 7.2, and is generalization of +Lemma A.1 in [5] to any 1 < p < ∞ (see also Lemma A.1 in [10]). +Lemma B.1. Let u be a nonnegative Lipschitz function in B+ +1 and assume that it is +p-harmonic in {u > 0} and u(0) = 0 (1 < p < ∞). Then it has the asymptotic development +u(x) = αxn + o(|x|), +as x → 0, +for some α ≥ 0, if either +(i) u vanishes on {xn = 0}, or +(ii) {xn > 0} ⊂ {u > 0}. +Proof. Part (i) is Lemma A.1 in [10]. The proof of (ii) is also similar by a slight +modification. Let ℓk := sup{l : lxn ≤ u(x) in B+ +2−k}. Since ℓk is a nondecreasing +sequence and bounded by the Lipschitz constant of u. Suppose α = limk→∞ ℓk, +then +u(x) ≥ αxn + o(|x|). +If the claim fails there exists a sequence xk → 0 such that +u(xk) ≥ αxk +n + δ0|xk|, +for some δ0 > 0. Define uk(x) := u(rkx)/rk where rk = |xk| → 0. We may also assume +that rk ≤ 2−k. +Since uk are uniformly Lipschitz, we may consider the blowup +u0 = limk→∞ uk, as well as xk/rk → x0, |x0| = 1. From the construction we will have +αxn ≤ u0(x) in B+ +1 , and +δ0 +2 + αxn ≤ u0(x) +and +δ0 +2 + ℓkxn ≤ uk(x) +in Bε(x0), +for a sufficiently small ε > 0 and large k. Let now wk be a p-harmonic function in +B+ +1 with smooth boundary values +wk = ℓkxn +on ∂B+ +1 \ Bε/2(x0), +wk = ℓkxn + δ0 +4 +on ∂B+ +1 ∩ Bε/4(x0), +ℓkxn ≤ wk ≤ ℓkxn + δ0 +4 +on ∂B+ +1 ∩ Bε/2(x0), +wk = 0 +on {xn = 0} ∩ B1. +From the comparison principle we will have wk ≤ uk in B+ +1 . (Note that uk(x) ≥ ℓkxn, +since rk ≤ 2−k). Furthermore, wk → w0 in C1,σ(B+ +1/2) where w0 is p-harmonic with +boundary data w0 = 0 on {xn = 0} and w0 ≥ αxn on ∂B+ +1 . By Hopf boundary +principle, +w0(x) ≥ (α + µ)xn +in B+ +γ, +for some small µ and γ. Thus for x ∈ B+ +γ, +uk(x) ≥ wk(x) ≥ w0(x) − xn∥∇(wk − w0)∥L∞(B+ +1/2) +≥ +� +α + µ − ∥∇(wk − w0)∥L∞(B+ +1/2) +� +xn +≥ �α + µ/2� xn +Now returning to u we get u(x) ≥ (α + µ/2)xn in B+ +γ/rk. This is a contradiction with +the definition of ℓk when k is sufficiently large. +□ + +24 +MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN +Declarations +Data availability statement: All data needed are contained in the manuscript. +Funding and/or Conflicts of interests/Competing interests: The authors declare +that there are no financial, competing or conflict of interests. +References +[1] Aguilera, N. E., Caffarelli, L. A. and Spruck, J., An optimization problem in heat conduction, Annali +della Scuola Normale Superiore di Pisa-Classe di Scienze, 14 (1987), no. 3, 355–387. +[2] Alt, H. W. and Caffarelli, L. A., Existence and regularity for a minimum problem with free boundary, J. +Reine Angew. Math., 325 (1981), 105–144. +[3] Alt, H. W., Caffarelli, L. A. and Friedman, A., A free boundary problem for quasi-linear elliptic equations, +Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 11 (1984), no. 1, 1–44. +[4] Bennewitz, B. and Lewis, J., On the dimension of p-harmonic measure, Ann. Acad. Sci. Fenn. Math, 30 +(2005), no. 2, 459–505. +[5] Caffarelli, L. A., A harnack inequality approach to the regularity of free boundaries part ii: Flat free +boundaries are lipschitz, Communications on pure and applied mathematics, 42 (1989), no. 1, 55–78. +[6] Caffarelli, L.A., Jerison, D. and Kenig, C.E., Global energy minimizers for free boundary problems and +full regularity in three dimension, Contemp. Math., 350, Amer. Math. Soc., Providence, RI (2004), +83–97. +[7] Caffarelli, L. A., Shahgholian, H. and Yeressian, K., A minimization problem with free boundary related +to a cooperative system, Duke Mathematical Journal, 167 (2018), no. 10, 1825–1882. +[8] Danielli, D. and Petrosyan, A., A minimum problem with free boundary for a degenerate quasilinear +operator, Calculus of Variations and Partial Differential Equations, 23 (2005), no. 1, 97–124. +[9] Danielli, D. and Petrosyan, A., Full regularity of the free boundary in a Bernoulli-type problem in two +dimensions, Mathematical Research Letters, 13 (2006), no. 4, 667–681. +[10] Danielli, D. and Petrosyan, A., Shahgholian, H., A singular perturbation problem for the p-Laplace +operator, Indiana University mathematics journal, 52 (2003), no. 2, 457–476. +[11] De Silva, D. and Tortone, G., Improvement of flatness for vector valued free boundary problems, Mathe- +matics in Engineering, 2 (2020), no. 4, 598–613. +[12] Jerison, D., Savin, O., Some remarks on stability of cones for the one-phase free boundary problem, Geom. +Funct. Anal., 25 (2015), no. 4, 1240–1257. +[13] Evans, L. C. and Garzepy, R. F., Measure theory and fine properties of functions, Routledge, 2018. +[14] Federer, H., Geometric measure theory, Springer, 2014. +[15] Ferrari, F. and Lederman, C., Regularity of flat free boundaries for a p(x)-Laplacian problem with right +hand side, Nonlinear Analysis, 212 (2021), no. 112444, 25 pp. +[16] Heinonen, J., Kipelainen, T. and Martio, O., Nonlinear potential theory of degenerate elliptic +equations, Courier Dover Publications, 2018. +[17] Lewis, J. L., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana +University Mathematics Journal, 32 (1983), no. 6, 849–858. +[18] Lewis, J. L., Lundstr¨om, N. and Nystr¨om, K., Boundary Harnack inequalities for operators of p-Laplace +type in Reifenberg flat domains, Proc. Sympos. Pure Math., 79, Amer. Math. Soc., 2008. +[19] Mal`y, J. and Ziemer, W. P., Fine regularity of solutions of elliptic partial differential equations, +Mathematical Surveys and Monographs, 51, American Mathematical Society, 1997. +[20] Mazzoleni, D. and Terracini, S. and Velichkov, B., Regularity of the free boundary for the vectorial +Bernoulli problem, Analysis and PDE, 13 (2020), no. 3, 741–764. +Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran +Email address: fotouhi@sharif.edu +Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden +Email address: henriksh@kth.se + diff --git a/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/load_file.txt b/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f27d4aa20b28b838fa69e08d33fff541d7ee71ea --- /dev/null +++ b/OdE0T4oBgHgl3EQfTgAw/content/tmp_files/load_file.txt @@ -0,0 +1,800 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf,len=799 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='02236v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='AP] 5 Jan 2023 A MINIMIZATION PROBLEM WITH FREE BOUNDARY FOR p-LAPLACIAN WEAKLY COUPLED SYSTEM MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In this paper we consider a weakly coupled p-Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We prove various properties of any local minimizer and the corre- sponding free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For Ω ⊂ Rn (n ≥ 2), we consider the problem of minimizing the functional (1) J(u) = � Ω m � i=1 |∇ui|p + Qpχ{|u|>0}dx, 1 < p < ∞, in the class of vectorial functions K := {u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' , um) ∈ W1,p(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm) : u = g on ∂Ω and ui ≥ 0 for i = 1, · · · , m}, with a given boundary data g ∈ W1,p(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Here, Q is a H¨older function satisfying 0 < Qmin ≤ Q ≤ Qmax < ∞, for some constants Qmin and Qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We are interested in regularity properties of minimizers u, as well as the free boundary Γ = ∂{|u| > 0} ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Any local minimizer satisfies the p-Laplace equation div (|∇ui|p−2∇ui) = 0, in {ui > 0}, also denoted by ∆pui = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In fact, ∆pui is a nonnegative Radon measure with support on free boundary, Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The problem is to find a reasonable representation of this measure and put it into some pde-context for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This problem is referred to as Bernoulli-type free boundary problem, and is well studied in the literature, for the scalar case and for p = 2, starting with seminal work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Alt and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Caffarelli [2], and also for any 1 < p < ∞ in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' There are very few results for Bernoulli-type problems that involve systems (see [7, 11, 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In [7], the authors study the minimum problem (1) for p = 2 and show the smoothness of the regular part of free boundary as well as some partial result for Hausdorff dimension of singular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Indeed, they apply a reduction method to reduce the problem to its scalar counterpart and the same result for the scalar case can be extended to the vectorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, a vectorial Bernoulli problem with no Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' p-Laplacian, minimizers, free boundary regularity, system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This project was carried out during the program Geometric aspects of nonlinear PDE at Institute Mittag Leffler, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Shahgholian was supported by Swedish Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 1 2 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN sign assumption on the components is studied in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In [11], the same result has been obtained by the viscosity approach and improvement of flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In this paper, we deal with a (weakly coupled) cooperative system for p-laplacian version of Bernoulli-type problem, following similar procedure as that in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It should be remarked that our approaches in this paper, with some extra efforts, can be adapted to variable exponent case, as well as variable coefficient one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Similar types of results are then expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For clarity of exposition we shall introduce some notations and definitions, which are used frequently in this text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Throughout this paper, Rn will be equipped with the Euclidean inner product x · y and the induced norm |x|, Br(x0) will denote the open n-dimensional ball with center x0, radius r and its boundary with ∂Br(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In addition, Br = Br(0) and ∂Br = ∂Br(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For the target space, Rm, we use several norms |u|p = ((u1)p + · · · + (um)p)1/p, |u(x)|∞ = max 1≤j≤m |uj(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For convenience, we denote the Euclidean norm without the index, |u| = |u|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We also use the Euclidean norm in the definition of L∞-norm, that is ∥u∥L∞ := sup x |u(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Plan of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The paper is organized as follows: In Section 2, we study the existence of minimizer (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1) and show that minimizers are p-subharmonic (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Section 3 is devoted to the regularity property of solutions, includ- ing the H¨older regularity (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2) and the Lipschitz regularity (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Section 4 consists of the proof of nondegeneracy property (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 an estimate for the density of the free boundary is obtained which is enough to prove that the free boundary has zero Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The vector- valued measure ∆pu (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1) and (n−1)-Hausdorff dimension of free bound- ary (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2) are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The main result in Section 6 is the flatness of regular part of the free boundary (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We prove a partial result for the regularity of free boundary in Section 7 (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5), along with C1,α-regularity of the free boundary when p is sufficiently close to 2 (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In Appendix A we deal with NTA properties of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, in Appendix B we present an auxiliary lemma to study the asymptotic behaviour of p-harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Existence of a minimizer Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If J(g) < ∞, then there exists an absolute minimizer of J over class K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Obviously the functional is nonnegative, and hence it takes an infimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let uk be a minimizing sequence inf v∈K J(v) = lim k→∞ J(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 3 Then uk − g is bounded in W1,p 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm) and up to a subsequence we can assume that uk ⇀ u, weakly in W1,p(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm), uk → u, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' in Ω, for some u ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The latter convergence implies � Ω χ{|uk|>0}dx → � Ω χ{|u|>0}dx, and the weakly lower semicontinuity of the norm implies that � Ω m � i=1 |∇ui|pdx ≤ lim inf k→∞ � Ω m � i=1 |∇ui k|pdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' These together show that J(u) ≤ limk→∞ J(uk) and hence u ∈ K is an absolute minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We say u ∈ K is a local minimizer of J, if J(u) ≤ J(v) for any v ∈ K with ∥∇u − ∇v∥Lp(Ω) + ∥χ{|u|>0} − χ{|v|>0}∥L1(Ω) ≤ ε, for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Although, all results in this paper are proved for local minimizers, for the sake of convenience we argue with absolute minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If u is a (local) minimizer, then ui is p-subharmonic for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', � Ω |∇ui|p−2∇ui · ∇ϕ dx ≤ 0, for all ϕ ∈ C∞ 0 (Ω), ϕ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Moreover, in each component of {|u| > 0} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m, either ui is identically vanishing or it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence, (2) ∆pui = 0, in {|u| > 0}, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m, and consequently (by the maximum principle) ∥u∥L∞(Ω) ≤ ∥g∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let ϕ ∈ C∞ 0 (Ω), ϕ ≥ 0, t > 0, and define vj = uj for j � i and vi = (ui − tϕ)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then v ∈ K and we can choose v as a competitor, so 0 ≤ 1 t (J(v) − J(u)) =1 t � Ω |∇vi|p − |∇ui|p + Qp �χ{|v|>0} − χ{|u|>0} � dx ≤1 t � Ω |∇(ui − tϕ)|p − |∇ui|pdx → � Ω −p|∇ui|p−2∇ui · ∇ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The second statement in the lemma relies on the strong minimum principle for p-harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In fact, let x ∈ {|u| > 0} but ui(x) = 0 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Choose an index j such that uj > 0 inside Br(x) where r is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now denote by ˜ui the p-harmonic extension of ui inside Br(x) and consider the competitor ˜u by defining ˜uk = uk for k � i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since {|u| > 0} = {| ˜u| > 0}, we get J(˜u) < J(u) unless ˜ui = ui and ui is p-harmonic in Br(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' But this violates the strong minimum principle if ui(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 4 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Regularity of local minimizers Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer of J, and vi the harmonic replacement (majorant) for ui in B ⊂ Ω (for B a small ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then there is a universal constant C = C(n, p) such that � B |∇(ui − vi)|p dx ≤ CQp max|{|u| = 0} ∩ B|, when 2 ≤ p, � B |∇(ui − vi)|p dx ≤ C(Qmax)p2/2|{|u| = 0} ∩ B|p/2 �� B |∇ui|p dx �1−p/2 , when 1 < p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let vj = uj for all j � i, and extend vi by ui in Ω \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If B is small enough (when u is absolute minimizer, we do not need this assumption), then we have J(u) ≤ J(v) and consequently � B |∇ui|p − |∇vi|p dx ≤ Qp max |{|u| = 0} ∩ B| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Set now ws(x) = sui(x) + (1 − s)vi(x) for 0 ≤ s ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then � B |∇ui|p − |∇vi|p dx = � B |∇w1|p − |∇w0|p dx =p � 1 0 ds � B |∇ws|p−2∇ws · ∇(ui − vi) dx =p � 1 0 ds � B � |∇ws|p−2∇ws − |∇vi|p−2∇vi� ∇(ui − vi) dx =p � 1 0 ds s � B � |∇ws|p−2∇ws − |∇vi|p−2∇vi� ∇(ws − vi) dx, where for the third equality we have used ∆pvi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Next using � |b|p−2b − |a|p−2a � (b − a) ≥ γ \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 |b − a|2(|b| + |a|)p−2, 1 < p ≤ 2, |b − a|p, 2 ≤ p, we obtain for p ≥ 2 � B |∇ui|p − |∇vi|p dx ≥ γp � 1 0 ds s � B |∇(ws − vi)|p dx = γp � 1 0 sp−1ds � B |∇(ui − vi)|p dx, which shows the desired estimate for p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In case 1 < p ≤ 2, we have � B |∇ui|p − |∇vi|p dx ≥ γp � 1 0 ds s � B |∇(ws − vi)|2 � |∇ws| + |∇vi| �p−2 dx ≥ γp � 1 0 sds � B |∇(ui − vi)|2 � s|∇ui| + (2 − s)|∇vi| �p−2 dx, ≥ C � B |∇(ui − vi)|2 � |∇ui| + |∇vi| �p−2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, using the H¨older inequality, we have � B |∇(ui − vi)|p dx ≤ �� B |∇(ui − vi)|2 � |∇ui| + |∇vi| �p−2 dx �p/2 �� B � |∇ui| + |∇vi| �p�1−p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 5 We conclude the proof by applying � B |∇vi|p ≤ � B |∇ui|p (since vi is p-harmonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (H¨older regularity) Let u be a (local) minimizer of J in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then for some α = α(n, p) ∥u∥Cα(B3/4) ≤ C(n, p, ∥u∥L∞(B1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let M = ∥u∥L∞(B1) and Br = Br(y) for y ∈ B3/4 and r < 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ui is a p-subsolution, a Caccioppoli type inequality (see [16], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='27) implies that � Br |∇ui|p dx ≤ C rp � B2r (ui)p dx ≤ CMprn−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, if vi is the p-harmonic replacement of ui inside Br, we have the gradient estimate (see [17]) sup Br/2 |∇vi| ≤ � C rn � Br |∇vi|p dx �1/p ≤ CM r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now, let us take some ρ < r/2 which will be specified below and apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 in Br(y) ∥∇ui∥Lp(Bρ) ≤∥∇(ui − vi)∥Lp(Bρ) + ∥∇vi∥Lp(Bρ) ≤∥∇(ui − vi)∥Lp(Br) + Cρn/p∥∇vi∥L∞(Br/2) ≤C \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 rn/p + Mρn/pr−1 for 2 ≤ p, M1−p/2rn/p−1+p/2 + Mρn/pr−1 for 1 < p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus for r = ρ1−α, if we take α = α(n, p) sufficiently small, we obtain ∥∇ui∥Lp(Bρ) ≤ C(M, n, p, Qmax)ρn/p−1+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By virtue of Morrey’s theorem (see [19]) we conclude the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ The next lemma is essential to prove the Lipschitz regularity of the minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' , um) be a bounded minimizer in B1 and ui(0) = 0 for some 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then there exists a constant C = C(n, p, Qmax) > 0 such that ∥ui∥L∞(B1/4) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We need to remark that the constant C is independent of the boundary values of u on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In other words when going away from a free boundary, but staying uniformly inside the domain Ω, the minimizer cannot grow too large, regardless of the boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In other words, for large enough boundary values, the origin cannot be a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For the sake of convenience consider i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Towards a contradiction, assume that there is a sequence of bounded solutions uk in B1 such that ∥u1 k∥L∞(B1/4) > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Set dk(x) := dist(x, {u1 k = 0}) in B1, and define Ok := {x ∈ B1 : dk(x) ≤ (1 − |x|)/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 6 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Obviously, B1/4 ⊂ Ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We have also mk := sup Ok (1 − |x|)u1 k(x) ≥ 3 4 max B1/4 u1 k > 3 4k, Since u1 k is bounded (for fixed k), we get (1 − |x|)u1 k(x) → 0 as |x| → 1, and therefore mk is attained at some point xk ∈ Ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' So, u1 k(xk) = mk 1 − |xk| ≥ mk > 3 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now let yk ∈ ∂{u1 k > 0} ∩ B1 be such that |yk − xk| = dk(xk) =: δk, which satisfies δk ≤ (1 − |xk|)/3 due to xk ∈ Ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This implies that B2δk(yk) ⊂ B1 and Bδk/2(yk) ⊂ Ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Indeed, if z ∈ B2δk(yk), |z| ≤ |z − yk| + |yk − xk| + |xk| ≤ 2δk + δk + |xk| ≤ 1, and if z ∈ Bδk/2(yk), 1 − |z| ≥ 1 − |xk| − |xk − yk| − |yk − z| ≥ 1 − |xk| − δk − δk/2 ≥ 3δk/2 ≥ 3|z − yk| ≥ 3dk(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, we have 1 − |z| ≥ (1 − |xk|)/2 for any z ∈ Bδk/2(yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then 1 − |xk| 2 max Bδk/2(yk) u1 k ≤ max z∈Bδk/2(yk)(1 − |z|)u1 k(z) ≤ max z∈Ok (1 − |z|)u1 k(z) = (1 − |xk|)u1 k(xk) or max Bδk/2(yk) u1 k ≤ 2u1 k(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since Bδk(xk) ⊂ {u1 k > 0}, then u1 k is p-harmonic inside Bδk(xk), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' ∆pu1 k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By the Harnack inequality for p-harmonic functions, there is a constant c = c(n, p) such that min B4δk/5(xk) u1 k ≥ cu1 k(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In particular, max Bδk/4(yk) u1 k ≥ cu1 k(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We define the sequence wk(x) := uk(yk + (δk/2)x) u1 k(xk) , whose first component satisfies (3) max B1 w1 k ≤ 2, max B1/2 w1 k ≥ c > 0, w1 k(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Moreover, wk is a minimizer of Jk(w) = � B1 m � i=1 |∇wi|p + Qp kχ{|w|>0}dx, where Qk(x) = δkQ(yk+(δk/2)x) 2u1 k(xk) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now consider v1 k to be p-harmonic replacement of u1 k in B3/4 and apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 (4) � B3/4 |∇(w1 k − v1 k)|p dx ≤ C(max Qk)p → 0, 7 when 2 ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Similar statement holds for 1 < p ≤ 2, we just need to note that ∥∇w1 k∥Lp is uniformly bounded (w1 k is p-subsolution and uniformly bounded in B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Furthermore, w1 k and v1 k are uniformly Cα in B5/8 and we can extract a subsequence (still denoted by w1 k and v1 k) such that w1 k → w0 and v1 k → v0 uniformly in B5/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Observe that ∆pv0 = 0 in B5/8 and (4) implies that w0 = v0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence, w0 is also p-harmonic and by the strong maximum principle, w0 ≡ 0 in B5/8, since w0 ≥ 0 and w0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, (3) necessitates max B1/2 w0 ≥ c > 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ A direct consequence of the above lemma is the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If dist(x0, {ui = 0}) < 1 5dist(x0, ∂Ω) then ui(x0) ≤ 4Cdist(x0, {ui = 0}), where C is the constant defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Choose y0 ∈ {ui = 0} such that dist(x0, {ui = 0}) = |x0 − y0| = d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3 to v(x) := u(y0 + 4d0x) 4d0 to get ui(x0) ≤ 4Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ With the above two results we will obtain uniform Lipschitz regularity for minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer in Ω, then u is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Moreover, for every K ⋐ Ω such that K ∩ ∂{ui > 0} � ∅ for some 1 ≤ i ≤ m, there is a constant C = C(n, p, Qmax, dist(K, ∂Ω), Ω) > 0 such that ∥∇ui∥L∞(K) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Once again we remark that the constant C does not depend on the boundary values of the minimizer, as long as we stay uniformly inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 1: We show that ui is bounded in K with a universal constant C depend- ing on the following ingredients n, p, Qmax, dist(K, ∂Ω), Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let r0 = 1 5dist(K, ∂Ω) and for any arbitrary point x ∈ K there is a sequence of points x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', xk ∈ K with (we can assume K is connected, otherwise replace it with a bigger one which is connected) xj ∈ Br0/2(xj−1), for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', k, Br0(xj) ⊂ {ui > 0} for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', k − 1 and Br0(xk) ∩ {ui = 0} � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Note that k, the number of points, only depends on Ω and dist(K, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4, we get ui(xk) ≤ 4Cr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ui is p-harmonic in Br0(xj), j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', k − 1, by virtue of Harnack’s inequality, there is a constant c such that ui(xj+1) ≥ cui(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 8 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Thus ui(x) ≤ 4c−kCr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 2: Here we find a control on ∇ui at points close to {ui = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If d = dist(y, {ui = 0}) < 1 11dist(y, ∂Ω), every points x0 ∈ Bd(y) satisfy condition Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then ui(x0) ≤ 4Cdist(x0, {ui = 0}) ≤ 8Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let us define v(x) := ui(y + dx) d which is a p-harmonic in B1 and ∥v∥L∞(B1) ≤ 8C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By p-Laplacian estimate for gradient, we obtain |∇v(0)| ≤ ˜C(n, p, Qmax), that is |∇ui(y)| ≤ C(n, p, Qmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 3: Let r1 = 1 11dist(K, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If dist(x, {ui = 0}) ≤ r1, by the result of Step 2 we have already |∇ui(x)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If dist(x, {ui = 0}) > r1, then ui is p-harmonic inside Br1(x) and ∥ui∥L∞(Br0) is universally bounded by the result of Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus |∇ui(x)| will be universally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ A straightforward corollary to this theorem, that can be useful later, is the following Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer for our functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For every K ⋐ Ω there exists constant C = C(n, p, Qmax, dist(K, ∂Ω), Ω) such that 1 r − � ∂Br ui dx > C implies ui > 0 in Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If Br ⊂ K contains a free boundary point, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5, ui ≤ Cr on ∂Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Nondegeneracy Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For any 0 < κ < 1 there exists a constant c = c(κ, n, m, p, Qmin) > 0 such that for every minimizer u and for any (small) ball Br ⊂ Ω ∥u∥L∞(Br) < cr implies u = 0 in Bκr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Without loss of generality, we may assume r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let M = ∥u∥L∞(B √κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let φ(x) = φκ(|x|) be the solution of ∆pφ = 0, in B √κ \\ Bκ, φ = 0 on ∂Bκ, φ = 1 on ∂B √κ and extend φ = 0 in Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Set v = M √κφ and wi = min(ui, v) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since v ≥ ui on ∂B √κ, so wi = ui on ∂B √κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore J(u) ≤ J(w), or equivalently � B √κ m � i=1 |∇ui|p + Qpχ{|u|>0} dx ≤ � B √κ\\Bκ m � i=1 |∇wi|p + Qpχ{|w|>0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 9 Since {|w| > 0} ⊂ {|u| > 0}, we get � Bκ m � i=1 |∇ui|p + Qpχ{|u|>0} dx ≤ � B √κ\\Bκ m � i=1 � |∇wi|p − |∇ui|p� dx ≤ p � B √κ\\Bκ m � i=1 |∇wi|p−2∇wi · ∇(wi − ui) dx = −p � ∂Bκ m � i=1 |∇wi|p−2(wi − ui)(∇wi · ν) dHn−1 = p � ∂Bκ m � i=1 |∇v|p−2ui(∇v · ν) dHn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since |∇v| ≤ C(p, κ, n)M on ∂Bκ, we find out that (5) � Bκ m � i=1 |∇ui|p + Qpχ{|u|>0} dx ≤ CMp−1 m � i=1 � ∂Bκ ui dHn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, � ∂Bκ ui dHn−1 ≤ C(n, κ) � Bκ ui + |∇ui| dx = C(n, κ) � Bκ � ui + |∇ui| � χ{ui>0} dx ≤ C(n, κ, p, Qmin) � Bκ MQpχ{ui>0} + |∇ui|p + Qpχ{ui>0} dx ≤ C(n, κ, p, Qmin)(1 + M) � Bκ |∇ui|p + Qpχ{ui>0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Comparing with (5), we we arrive at � Bκ m � i=1 |∇ui|p + Qpχ{|u|>0} dx ≤ CMp−1(1 + M) � Bκ m � i=1 |∇ui|p + Qpχ{|u|>0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, if M is small enough, we obtain that u = 0 in Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ An immediate consequence of the above lemma is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For any K ⋐ Ω there are positive constants c0, C0 such that if Br(x) ⊂ K ∩ {|u| > 0} touches ∂{|u| > 0} then (6) c0r ≤ |u(x)| ≤ C0r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For K ⋐ Ω there exists constant 0 < c = c(n, m, p, K, Ω) < 1 such that for any (local) minimizer u and for any (small) ball Br(x) ⊂ K with x ∈ ∂{|u| > 0}, (7) c < Ln(Br(x) ∩ {|u| > 0}) Ln(Br(x)) < 1 − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, there exists y ∈ Br/2 such that |u(y)| ≥ cr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Using Lipschitz continuity we get − � ∂Bκr(y) |u| ≥ cr 2 , 10 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN provided κ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence 1 κr− � ∂Bκr(y) |u| ≥ c 2κ, and also for at least one component ui 1 κr− � ∂Bκr(y) ui ≥ c 2κm, which by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='6 implies |u| > 0 in Bκr(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This gives the lower estimate in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To prove the estimate from above we assume, for simplicity, r = 1 and suppose (towards a contradiction) that there is a sequence of minimizers uk in B1(0) such that 0 ∈ ∂{|uk| > 0} and Ln({|uk| = 0}) =: εk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let vi k be a p-harmonic function in B1/2 with boundary data vi k = ui k on ∂B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, we obtain that (8) � B1/2 |∇(vi k − ui k)|p dx ≤ C(εk) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ui k and vi k are both uniformly Lipschitz in B1/4, we may assume that ui k → ui 0 and vi k → vi 0 uniformly in B1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Observe that ∆pvi 0 = 0 and (8) implies that ui 0 = vi 0+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus ∆pui 0 = 0 in B1/4 and from the strong minimum principle (since ui 0(0) = 0) it follows ui 0 ≡ 0 in B1/4, since ui 0 ≥ 0 and ui 0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand form nondegeneracy property, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, we know ∥uk∥L∞(B1/2) ≥ c > 0, which implies a similar inequality for u0, and hence a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2, along with the Lebesgue density theorem implies that the free boundary has zero Lebesgue measure Ln(∂{|u| > 0}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The vector-valued measure ∆pu Let 0 ≤ ζ ∈ C∞ 0 (Ω) be a test function, and define the measure λi by � ζ dλi = − � |∇ui|p−2∇ui · ∇ζ dx, which in virtue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3 is a bounded non-negative measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' a Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Obviously λi is the formal way of expressing ∆pui in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since each ui is p-subharmonic in Ω and ui ≥ 0 we have that λi is a positive Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Because ui is also p-harmonic in {ui > 0} we have that the support of λi is in Ω ∩ ∂{ui > 0} ⊆ Ω ∩ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 Let us define Λ = Λu := m � i=1 λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 1Observe that ui may be zero in some component of {|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 11 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For any K ⋐ Ω there exist constants c, C > 0 such that for any (local) minimizer u crn−1 ≤ � Br dΛ ≤ Crn−1 for any ball Br ⊂ K with x ∈ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let 0 ≤ ζǫ ∈ C∞ 0 (Br+ǫ) be a suitable test function, such that ζǫ = 1 on Br and |∇ζǫ| ≤ 2/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then � ζǫ dλi = − � |∇ui|p−2∇ui · ∇ζǫ dx = − � Br+ǫ\\Br |∇ui|p−2∇ui · ∇ζǫ dx ≤ Crn−1, where in the last inequality we have used that u is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Letting ǫ tend to zero, we arrive at � Br dλi ≤ Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To prove the estimate from below, we argue indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It also suffices to consider the case r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Assume there is a sequence of minimizers uk in the unit ball B1(0), such that 0 ∈ ∂{|uk| > 0} and for the measures Λk := Λuk we have εk := Λk(B1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since the functions uk are uniformly Lipschitz continuous, we may assume that uk → u0 in B1/2, where u0 is Lipschitz continuous as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We may also extract a subsequence (still denote by uk) such that gi k := |∇ui k|p∇ui k → gi 0 weakly-∗ in L∞(B1/2) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We claim that (9) gi 0 = |∇ui 0|p∇ui 0, in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose this is true, then for every positive test function ζ ∈ C∞ 0 (B1/2) one has − � B1/2 |∇ui 0|p−2∇ui 0 · ∇ζ dx = − lim k→∞ � B1/2 |∇ui k|p−2∇ui k · ∇ζ dx = lim k→∞ � B1/2 ζ dλi k ≤ ∥ζ∥L∞(B1/2) lim k→∞ εk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus λi 0 = 0 and ui 0 is p-harmonic for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m (note that ui 0 is the limit of a se- quence of p-subharmonic functions and we already know that it is p-subharmonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ui 0 ≥ 0 and ui 0(0) = 0, by the minimum principle, we have ui 0 ≡ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, by nondegeneracy property (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1) and that 0 ∈ ∂{|uk| > 0} we have ∥uk∥L∞(B1/4) ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, a similar inequality holds for u0 and we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To close the argument we need to prove (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In fact, if Bρ = Bρ(y) ⊂ {|u0| > 0} then Bρ ⊂ {|uk| > 0} for sufficiently large k and ui k are p-harmonic in Bρ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m, (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, one can extract a subsequence of uk locally converging to u0 in C1,α(Bρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence, gi 0 = |∇ui 0|p∇ui 0 in Bρ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Next, if Bρ ⊂ {|u0| = 0} then for any κ < 1 the nondegeneracy property entails that Bκρ ⊂ {|uk| = 0} for sufficiently large k = k(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus gi 0 = 0 = |∇ui 0|p∇ui 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We just need to show Ln(∂{|u0| > 0} ∩ B1/2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If x0 ∈ ∂{|u0| > 0} ∩ B1/2, then u0(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 12 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Choose xk ∈ ∂{|uk| > 0} ∩ B1 such that |xk − x0| = dist(x0, ∂{|uk| > 0}), then relation (6) yields that |xk − x0| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 to obtain ∥uk∥L∞(B2r(x0)) ≥ ∥uk∥L∞(Br(xk)) ≥ cr, for any ball Br(x0) ⊂ B1/2 and sufficiently large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Passing to the limit we get the same inequality for u0, ∥u0∥L∞(B2r(x0)) ≥ cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This along with the Lipschitz continuity of u0 is enough to prove that Ln(Br(x0) ∩ {|u0| > 0}) ≥ cLn(Br) for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (see the first part of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This implies that Ln(∂{|u0| > 0} ∩ B1/2) = 0 (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ The next theorem follows easily from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The proof is the same as the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then (i) For every K ⋐ Ω we have Hn−1(K ∩ ∂{|u| > 0}) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (ii) There exist nonnegative Borel functions qi such that ∆pui = qiHn−1⌊ ∂{|u| > 0}, that is for every ζ ∈ C∞ 0 (Ω) − � Ω |∇ui|p−2∇ui · ∇ζ dx = � Ω∩∂{|u|>0} ζqidHn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (iii) For any K ⋐ Ω there exist constants c, C > 0 such that c ≤ m � i=1 qi ≤ C, and for Br(x) ⊂ K with x ∈ ∂{|u| > 0} we have crn−1 ≤ Hn−1(Br(x) ∩ ∂{|u| > 0}) ≤ Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From (i) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 it follows that, locally, the set A = Ω∩{|u| > 0} has finite perimeter in Ω in sense of that µu = −∇χA is a Borel measure and the total variation |µu| is a Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We define the reduced boundary of A by ∂redA = {x ∈ Ω : |νu| = 1}, where νu(x) is the unique unit vector with � Br(x) ���χA − χ{y:(y−x)·νu(x)<0} ��� = o(rn), as r → 0, if such a vector exists, and νu(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' See [14], Chapter 4, for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Local Analysis To proceed, we will need some properties of the so-called blow-up limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer in Ω, K ⋐ Ω and Brk(xk) ⊂ K be a sequence of balls with rk → 0, xk → x0 ∈ Ω, and u(xk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Consider the blow-up sequence (10) uk(x) = 1 rk u(xk + rkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 13 For a subsequence, there is a limit u0 such that uk → u0 in C0,α loc(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm) for every 0 < α < 1, ∇uk → ∇u0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' in Rn, (11) ∂{|uk| > 0} → ∂{|u0| > 0} locally in the Hausdorff distance, (12) χ{|uk|>0} → χ{|u0|>0} in L1 loc(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm), (13) if xk ∈ ∂{|uk| > 0} then 0 ∈ ∂{|u0| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For the proof we refer to [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ The following lemma shows that the blow-up limit is a minimizer in any ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If u(xk) = 0 and xk → x0, then any blow-up limit u0 = limk uk (see (10)) with respect to Brk(xk) is an absolute minimizer of J0 in any ball BR = BR(0), where J0(v) := � BR m � i=1 |∇vi|p + Q(x0)pχ{|v|>0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let w ∈ W1,p(BR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rm) be such that wi ≥ 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m and w = u0 on ∂BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To show J0(u0) ≤ J0(w), we choose a cut of function η ∈ C∞ 0 (BR) with 0 ≤ η ≤ 1 and η = 1 in Br for some 0 < r < R, and define wk = �w + (1 − η)(uk − u0)� + , where the positive part is taken separately for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We also have wk = uk on ∂BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since u is (local) minimizer, for sufficiently large k such that BRrk(xk) ⋐ Ω, we have � BR m � i=1 |∇ui k|p + Qp kχ{|uk|>0} dx ≤ � BR m � i=1 |∇wi k|p + Qp kχ{|wk|>0} dx, where Qk(x) := Q(xk + rkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since |∇uk| ≤ C (due to Lipschitz continuity of u) and convergences (11) and (13), the limit of the left hand side will be J0(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence J0(u0) ≤ lim inf k→∞ � BR m � i=1 |∇wi k|p + Qp kχ{|wk|>0} dx ≤ � BR m � i=1 |∇wi|p dx + � Br Q(x0)pχ{|w|>0} dx + lim inf k→∞ � BR\\Br Qp kχ{|wk|>0} dx ≤ � BR m � i=1 |∇wi|p dx + � Br Q(x0)pχ{|w|>0} dx + Q(x0)p|BR \\ Br|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now let r → R, we get J0(u0) ≤ J0(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose ∇(Qp) ∈ L1(Ω) and u is an absolute minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then � |u|>0 div \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 m � i=1 � |∇ui|pΨ − p|∇ui|p−2(∇ui · Ψ)∇ui� + QpΨ \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb dx = 0, for every Ψ ∈ C∞ c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 14 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let us define Φt(x) = x + tΨ(x) and ut(x) = u(Φt(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' One can show that for sufficiently small |t|, Φt : Ω → Ω is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We have DΦt = I + tDΨ and for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', m ∇ui t = DΦt(x)∇ui(Φt(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It follows that |∇ui t(x)|2 = (∇ui(Φt(x)))TAt(x)∇ui(Φt(x)), where At = (DΦt)TDΦt = I + t((DΨ)T + DΨ) + t2(DΨ)TDΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By a change of variables, we have J(ut) = � Ω m � i=1 |∇ui t(x)|p + Qpχ{|ut|>0} dx = � Ω m � i=1 � (∇ui(Φt(x)))TAt(x)∇ui(Φt(x)) �p/2 + Qp(x)χ{|u|>0}(Φt(x)) dx = � Ω m � i=1 �� (∇ui(y))TAt(Φ−1 t (y))∇ui(y) �p/2 + Qp(Φ−1 t (y))χ{|u|>0}(y) � ���det DyΦ−1 t (y) ��� dy = � {|u|>0} m � i=1 �� (∇ui(y))TAt(Φ−1 t (y))∇ui(y) �p/2 + Qp(Φ−1 t (y)) ����det DyΦ−1 t (y) ��� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We also have d dtAt(Φ−1 t (y)) ����t=0 = DΨ(y) + DΨ(y)T and d dt ���det DyΦ−1 t (y) ��� ����t=0 = −div Ψ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now differentiate J(ut) with respect to t and note that its minimum is attained at t = 0, then 0 = d dt J(ut) ����t=0 = � {|u|>0} m � i=1 p|∇ui|p−2(∇ui)TDΨ∇ui − Ψ · ∇(Qp)dx − � {|u|>0} \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 m � i=1 |∇ui|p + Qp \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb div Ψ dx = − � {|u|>0} div \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 m � i=1 � |∇ui|pΨ − p|∇ui|p−2(∇ui · Ψ)∇ui� + QpΨ \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + m � i=1 p(∇ui · Ψ)∆pui dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since each ui is p-harmonic in {ui > 0}, see (2), we arrive at the desired claim, in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 15 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The upper Hn−1-density at any point x0 ∈ ∂{|u| > 0} is defined as Θ∗n−1 � Hn−1� ∂{|u| > 0}, x0 � := lim sup r→0 Hn−1(Br(x0) ∩ ∂{|u| > 0}) ωn−1rn−1 , where ωn−1 denotes the volume of the unit sphere in Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We already know (see for example Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='7 in [13]) that for Hn−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' point x0 ∈ ∂{|u| > 0}, their upper Hn−1-density satisfy Θ∗n−1 � Hn−1� ∂{|u| > 0}, x0 � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let x0 ∈ ∂red{|u| > 0} and suppose that Θ∗n−1 � Hn−1� ∂{|u| > 0}, x0 � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then Tan(∂{|u| > 0}, x0) = {x : x · ν(x0) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If, in addition, x0 is a Lebesgue point for Radon measure qiHn−1� ∂{|u| > 0}, that is (14) � Br(x0)∩∂{|u|>0} |qi − qi(x0)| dHn−1 = o(rn−1), as r → 0, then qi(x0) = Q(x0) and u(x0 + x) = (−x · ν(x0))+ ax0 + o(|x|), as x → 0, for some vector ax0 = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', αm) that (15) |ax0|p p = (α1)p + · · · + (αm)p = 1 p − 1Q(x0)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Without loss of generality assume that ν(x0) = en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let uk be a blow-up sequence with respect to balls Brk(x0), with blow-up limit u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ν(x0) is the normal vector to ∂{|u| > 0} at x0, � Br(x0) ���χ{|u|>0|} − χ{x:(x−x0)·ν(x0)<0} ��� dx = o(rn), as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This along with (13) implies χ{|u0|>0} = χ{xn<0} almost every where in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 we know that u0 is an absolute minimizer of J0 and so continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then {|u0| > 0} = {xn < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This proves that {xn = 0} is the topological tangent plane to ∂{|u| > 0} at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now let φ(x) = min (1, max(0, 2 − |xn|)) η(x′), where 0 ≤ η ∈ C∞ 0 (B′ R) and B′ R is (n−1)-dimensional ball with radius R (R is arbitrary and fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Denote φk(x) := rkφ( x−x0 rk ) and write − � Rn |∇ui 0|p−2∇ui 0 · ∇φ dx = − lim k→∞ � Rn |∇ui k|p−2∇ui k · ∇φ dx = − lim k→∞ r−n k � Rn |∇ui|p−2∇ui · ∇φk dx = lim k→∞ r−n k � Rn ∆puiφk dx = lim k→∞ r−n k � Rn qiφk dHn−1⌊ ∂{|u| > 0} = lim k→∞ � Rn qi(x0 + rkx)φ(x)χ∂{|uk|>0} dHn−1 = lim k→∞ � Rn qi(x0)φ(x)χ∂{|uk|>0} dHn−1 = qi(x0) � {xn=0} η(x′) dx′, 16 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN where we have used assumption (14) and property (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, for any test function ζ ∈ C∞ 0 (BR) we have − � BR∩{xn<0} |∇ui 0|p−2∇ui 0 · ∇ζ dx = qi(x0) � B′ R ζ(x′, 0) dx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ∆pui 0 = 0 in {xn < 0}, from boundary regularity it follows that |∇ui 0|p−2∂nui 0 = −qi(x0) on {xn = 0}, in the classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We need to show that (16) ui 0(x) = αi(−xn)+, where αi := � qi(x0) �1/(p−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To see this, define w0 by w0(x) := \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 ui 0(x), in xn ≤ 0, −ui 0(x′, −xn), in xn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It is obvious that w0 is p-harmonic in whole Rn as well as ∥∇w0∥L∞(Rn) = ∥∇ui 0∥L∞(Rn) ≤ ∥∇ui∥L∞(Br(x0)), for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Liouville’s theorem we conclude that w0 is a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The boundary value on xn = 0, (ui 0 = 0 and ∂nui 0 = −αi) shows that w0(x) = −αixn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This proves (16) and shows that ui(x0 + x) = αi(−xn)+ + o(|x|), as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We just have to show (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To do this, note that u0 is an absolute minimizer of J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3 for u0 and some Ψ ∈ C∞ c (Rn) 0 = � {xn<0} div \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 m � i=1 � |∇ui 0|pΨ − p|∇ui 0|p−2(∇ui 0 · Ψ)∇ui 0 � + Q(x0)pΨ \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb dx = � {xn=0} m � i=1 � |∇ui 0|p(Ψ · en) − p|∇ui 0|p−2(∇ui 0 · Ψ)∂nui 0 � + Q(x0)p(Ψ · en) dHn−1 = � {xn=0} \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed m � i=1 (1 − p)(αi)p + Q(x0)p \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (Ψ · en) dHn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus (15) will be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Regularity of free boundary Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u ∈ C(Ω, Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We say that the boundary condition ∇|u|p = g on ∂{|u| > 0} holds in viscosity sense, if For every differentiable function φ : Rn → R that touches |u|p from below in some x0 ∈ ∂{|u| > 0}, that is |u(x0)|p = φ(x0), and |u|p ≥ φ in {|u| > 0} ∩ Br(x0) for some r > 0, we have |∇φ(x0)| ≤ g(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 17 For every differentiable function φ : Rn → R that touches |u|p from above in some x0 ∈ ∂{|u| > 0}, that is |u(x0)|p = φ(x0), and |u|p ≤ φ in {|u| > 0} ∩ Br(x0) for some r > 0, we have |∇φ(x0)| ≥ g(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a (local) minimizer, then the boundary condition ∇|u|p = 1 (p − 1)1/p Q, on ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' holds in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We show that the boundary condition holds on every point x0 ∈ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose φ touches |u|p from below at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Consider the blow-up sequences uk(x) = u(x0 + rkx) rk and φk(x) = φ(x0 + rkx) rk , where rk ց 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Observe that φk(0) = |uk(0)|p = 0, φk(x) ≤ |uk(x)|p, and up to a subsequence we have (17) ∇φ(x0) · x = lim k→∞ φk(x) ≤ lim k→∞ |uk(x)|p = |u0(x)|p in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If∇φ(x0) = 0, theviscosityconditionholdstrivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Otherwise, thenon-coincidence set {|u0| > 0} contains half-space {x : ∇φ(x0) · x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, u0 is minimizer of J0 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2) and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3 every nontrivial component of u0, say ui 0, is positive in {x : ∇φ(x0) · x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' According to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, ui 0(x) = αi(∇φ(x0) · x) + o(|x|) for some αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus any blowup of u0 at x = 0 must be of the form u00(x) = a0(∇φ(x0) · x) where a0 = (α1, · · · , αm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Again apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 along with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3, we get |a0|p|∇φ(x0)| = 1 (p − 1)1/p Q(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus (17) yields that ∇φ(x0) · x ≤ |u0(x)|p = |a0|p|∇φ(x0) · x| + o(|x|), and so |∇φ(x0)| ≤ 1 (p − 1)1/p Q(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The same argument holds when φ touches |u0|p from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' A domain Ω ⊂ Rn is called non-tangentially accessible domain (NTA) with parameters M ≥ 1 and R0 > 0 if (i) Ω satisfies the corkscrew condition, that is, for any x ∈ ∂Ω and r ∈ (0, R0) there exists ar(x) ∈ Ω ∩ Br(x) such that M−1r < dist(ar(x), ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (ii) Rn \\ Ω satisfies the corkscrew condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (iii) If x ∈ ∂Ω and x1, x2 ∈ Br(x) ∩ Ω for 0 < r < R0, then there exists a rectifiable curve γ : [0, 1] → Ω with γ(0) = x1 and γ(1) = x2 such that H1(γ) ≤ M|x1 − x2| and min � H1(γ([0, t])), H1(γ([t, 1])) � ≤ Mdist(γ(t), ∂Ω), for every t ∈ [0, 1], where H1 denotes length or the one-dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 18 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We say that x0 ∈ ∂{|u| > 0} is a regular point of the free boundary if for every ǫ > 0 there are r < 1 and a vector a ∈ Rm and unit vector ν ∈ Rn such that ∥ur,x0 − (x · ν)+a∥L∞(B1) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We denote the set of all regular points by Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5 proves that Hn−1(∂{|u| > 0} \\ Ru) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', um) be a (local) minimizer and x0 ∈ Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Furthermore, assume that Br0(x0) ∩ {|u| > 0} is NTA domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then Ru ∩ Br(x0), for some 0 < r ≤ r0, is C1,α for a universal exponent 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We may assume that u1 > 0 in Br0(x0) ∩ {|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' First we show that there is a H¨older function g : Br(x0) ∩ ∂{|u| > 0} → [c, 1] for some 0 < r ≤ r0 and 0 < c ≤ 1 such that u1 is a viscosity solution to the problem ∆pu1 = 0, in {u1 > 0} ∩ Br, |∇u1| = gQ (p − 1)1/p , on ∂{u1 > 0} ∩ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (18) Since Br0(x0) ∩ {|u| > 0} is NTA domain, the boundary Harnack inequality (see [18]) implies that gi := ui/u1 is H¨older continuous in {|u| > 0} ∩ Br for some r ≤ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Define g := (1 + gp 2 + · · · + gp m)−1/p, and observe that u1 = g|u|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose now the test function φ is touching u1 from below in a point y ∈ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For ρ small enough, choose a constant C > 0 such that 1 g(x) ≥ 1 g(y) − C|x − y|µ ≥ 0, for every x ∈ {|u| > 0} ∩ Bρ, where µ is H¨older exponent of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Set ψ(x) = φ(x)(1/g(y) − C|x − y|µ), we get ψ(y) = |u(y)|p = 0 (note that since φ(y) = 0, so ψ is differentiable at y) and ψ(x) ≤ u1(x)( 1 g(y) − C|x − y|µ) = g(x)( 1 g(y) − C|x − y|µ)|u(x)|p ≤ |u(x)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, ψ touches |u|p at y from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 |∇ψ(y)| ≤ Q(y) (p − 1)1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Note that ∇ψ(y) = 1 g(y)∇φ(y) to see the boundary condition (18) in viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The regularity of free boundary follows by the known results on the regularity of the one-phase scalar problem (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Our next result improves the theorem above, in the sense that we can remove the NTA conditions, for p close to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The main reason for not being able to handle the NTA, is the lack of ACF-monotonicity formula, use in classical paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' see proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3, to prove the connectivity argument for the NTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', um) be a (local) minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then there is ǫ0 > 0, such that for any p ∈ (2 − ǫ0, 2 + ǫ0) we have i) The regular set Ru, is locally C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' ii) In dimensions 2, 3, 4 the free boundary is C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Here 0 < α < 1 is is a universal exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To prove (i) it suffices, in virtue of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5, to show that the free bound- ary is NTA, when ǫ0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This, however, is a consequence of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5 in Appendix A, by choosing ǫ0 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Turning to sstatement (ii) we recall that in these dimensions, and for p = 2, free boundaries are locally C1,α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' for n = 2 this was shown in [2], for n = 3 see [6], for n = 4 see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now for p ≈ 2, the free boundary has to be close to that of the case p = 2, and hence flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' More specifically, given a free boundary point x0 for the p-problem, and p close enough to 2, we have that in Br(x0) the free boundaries of both p and 2, have Huasdorff distance δ ≪ r, and in particular the p-free boundary is (δ/2)-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Non-tangentially accessible domain (NTA) We note that in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3, the condition (iii) can be replaced by Harnack chain condition (see [4]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', (iii)’ Given ε > 0, x1, x2 ∈ Ω such that dist(xi, ∂Ω) ≥ ε, i = 1, 2 and |x1 − x2| ≥ ˜Cε, we can find points x1 = y1, y2, · · · , yℓ = x2 for which (a) Bε(yi) ⊂ Ω for i = 1, · · · , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (b) Bε(yi) ∩ Bε(yi+1) � ∅ for i = 1, · · · , ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (c) The length of chain, ℓ, depends on ˜C but not on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We need an analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 in [1] to show that the non-coincidence set is NTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a minimizer and suppose 0 < |u(x0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Define δ = dist(x0, Γ), δ1 = dist(x0, {|u| ≤ 1 2|u(x0)|∞}), and suppose B(x0, δ) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then, there exist universal constants λ > 1 > σ such that (i) σδ ≤ δ1 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (ii) For some y ∈ ∂Bδ1(x0), |u(y)|∞ ≥ λ|u(x0)|∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' According to relation (6), c0δ ≤ |u(x0)| ≤ C0δ, and if ui(x0) = max1≤j≤m uj(x0) = |u(x0)|∞, c0δ √m ≤ ui(x0) ≤ C0δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If z ∈ ∂{|u| ≤ 1 2|u(x0)|∞} such that |z − x0| = δ1, then 1 2ui(x0) = 1 2|u(x0)|∞ ≤ |u(x0)| − |u(z)| ≤ |u(x0) − u(z)| ≤ C0δ1, where we have used |u(z)| = 1 2|u(x0)|∞ ≤ 1 2|u(x0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus (i) holds for σ = c0 2C0 √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To see (ii), note that v(x) := ui(x0 + δ1x)/ui(x0) is a p-harmonic function in Bδ/δ1 such that v(0) = 1, v(ˆz) ≤ 1 2 for some ˆz ∈ ∂B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In addition, the Lipschitz constant of v is bounded by C0δ1/ui(x0) ≤ C0δ/ui(x0) ≤ C0 √m/c0 = 1/2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We claim that there is a universal constant λ > 1 such that v( ˆy) ≥ λ for some ˆy ∈ ∂B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Otherwise, we find a sequence of p-harmonic functions vk with ∥vk∥L∞(B1) ≤ 1 + 1 k, ∥∇vk∥L∞(B1) ≤ 1 2σ, vk(0) = 1, vk(zk) ≤ 1 2, 20 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN for some |zk| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then there is a subsequence converging in C1,α(B1) to a p-harmonic v0 where v0(0) = 1, v0(z0) ≤ 1/2, ∥v0∥L∞(B1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This is a contradiction with the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, we have found y ∈ ∂Bδ1(x0) such that |u(y)|∞ ≥ ui(y) ≥ λui(x0) = λ|u(x0)|∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' There exists a universal constant ˜c0 such that if x0 ∈ ∂{|u| > 0}, x1 ∈ Br/2(x0) and Ar is the connected component of {|u| > 1 2|u(x1)|∞} ∩ Br(x0) containing x1, then ∥u∥L∞(Ar) ≥ ˜c0r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' WeuseLemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 toinductivelydefinea sequenceofpointsx1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', xk, xk+1 so that for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=', k, (i) |xj+1 − xj| = δj = dist(xj, {|u| ≤ 1 2|u(xj)|∞}), (ii) |u(xj+1)|∞ ≥ λ|u(xj)|∞, (iii) Bδj(xj) ⊂ {|u| > 1 2|u(x1)|∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By (ii), we know that this process cannot continue indefinitely without stepping out of Br(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' So, we stop at the first k for which Bδk+1(xk+1) � Br(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, we know that δj ≤ dist(xj, Γ) ≤ δj σ , and therefore by (6), c0δj ≤ |u(xj)| ≤ C0δj σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now applying (ii), we obtain (recall that σ = c0 2C0 √m) δj ≤ √m c0 |u(xj)|∞ ≤ √mλ−ℓ c0 |u(xj+ℓ)|∞ ≤ λ−ℓ 2σ2 δj+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, |xk − x0| ≤ |x1 − x0| + k−1 � j=1 |xj+1 − xj| ≤ r 2 + k−1 � j=1 δj ≤ r 2 + δk 2σ2 k−1 � j=1 λj−k ≤ r 2 + δk 2σ2(λ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, δk+1 =dist(xk+1, {|u| ≤ 1 2|u(xk+1)|∞}) ≤δk + dist(xk, {|u| ≤ 1 2|u(xk+1)|∞}) ≤δk + dist(xk, {|u| ≤ 1 2|u(xk)|∞}) = 2δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since Bδk+1(xk+1) � Br(x0), we get r ≤ |xk+1 − x0| + δk+1 ≤ |xk − x0| + 3δk ≤ r 2 + cδk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus γr ≤ δk for the universal constant γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It necessitates that Bγr(xk) ⊂ Ar and also |u(xk)| ≥ c0dist(xk, Γ) ≥ c0γr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In particular, max Ar |u| ≥ c0γr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 21 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' There is ε0 > 0 such that for any p ∈ (2 − ε0, 2 + ε0) there is no global minimizer of J0 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2) that u(0) = 0 and {|u| > 0} ∩ BR is disconnected for every R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' First, for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We apply the monotonicity formula and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let A1 and A2 two different connected components of {|u∞| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3, we may choose a positive component of vector u∞ for each set Ai, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus we find a function v which is harmonic in A1 ∪ A2 and vanishes in {|u∞| = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, since u∞ is a minimizer, we know that (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2) |Br \\ (A1 ∪ A2)| ≥ c|Br|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From here we infer that Φ(r)/rβ is a non-decreasing function of r for some positive constant β > 0 [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4], where Φ(r) = 1 r4 �� Br∩A1 |∇v|2|x|2−n dx � �� Br∩A2 |∇v|2|x|2−n dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since v is Lipschitz, say with constant C0, we have the bound Φ(r) ≤ C4 0, and thus Φ(1) ≤ r−βΦ(r) ≤ C4 0r−β, which can not be valid for sufficiently large value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The contradiction proves the proposition when p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' To prove the proposition for p close to 2 we argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Assume that there is a sequence of global minimizers ui for pi → 2 that u(0) = 0 and {|ui| > 0}∩BR is not connected for any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We remark that all our results are stated in a slightly more general form, with constants depending uniformly for all p ∈ [1/2, 3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' see for the details [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' In fact, we will have a uniform Lipschitz constant for all p in this compact interval as well as the nondegeneracy constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Also, the constant c in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 will be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Therefore, we may choose a convergence subsequence ui → ˜u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By the same reasoning as the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 we get that ˜u0 is a minimizer for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, {|u0| > 0} ∩ BR is not connected for every R which contradicts the later part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let ε0 > 0 be the constant defined in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then for any p ∈ (2 − ε0, 2 + ε0) there are constants M ≥ 1 and R0 > 0 such that if x0 ∈ ∂{|u| > 0}, y ∈ BR0(x0) ∩ ∂{|u| > 0} and x1, x2 ∈ Br(y) for some r < R0, then {|u| > d} ∩ BMr(y) has a connected component containing x1 and x2, where d = 1 2 min(|u(x1)|∞, |u(x2)|∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Fix constant p and assume the contrary there are sequences ∂{|u| > 0} ∋ yi → x0, xi 1, xi 2 ∈ Bri(yi), ri → 0 and Mi → ∞ such that xi 1 and xi 2 are not connected in {|u| > di} ∩ BMiri(yi) where di = 1 2 min(|u(xi 1)|∞, |u(xi 2)|∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Consider the blowup sequence u(yi + rix)/ri → u0(x), di/ri → a, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 yields that {|u0| > a} ∩ BR has at least two connected components for any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Note that a < ∞ due to the Lipschitz regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now consider a blowdown of u0 u0(Rix)/Ri → u∞(x), Ri → ∞, 22 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN then {|u∞| > 0}∩BR must have at least two connected components for any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, u∞ is a minimizer of J0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This contradicts Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a minimizer when p ∈ (2 − ε0, 2 + ε0) and ε0 is the constant defined in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose x0 ∈ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then {|u| > 0} is NTA in a neighborhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 1: (Property (i), the corkscrew condition for {|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=') Assume that M > ∥∇u∥L∞(B1)/c where c is the nondegeneracy constant defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If the condition fails at a point x ∈ ∂{|u| > 0}, then for any y ∈ Br(x) ∩ {|u| > 0} we must have dist(y, ∂{|u| > 0}) ≤ M−1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus |u(y)| ≤ M−1r∥∇u∥L∞(B1) ≤ cr and by nondegeneracy, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1, u = 0 in Bκr(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' It contradicts that x ∈ ∂{|u| > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 2: (Property (ii), the corkscrew condition for {|u| = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=') Assume the contrary, {|u| = 0} does not satisfy the corkscrew condition in any neighborhood of x0 for any constant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus there is a sequence xj → x0 and rj → 0 such that we can not find point arj(xj) with the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' On the other hand, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2 infer that the interior of {|u| = 0} is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let Bτj(yj) be the biggest ball inside Brj(xj) ∩ {|u| = 0}, then we must have τj/rj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now consider the blowup uj(x) := u(xj + rjx)/rj → u0(x), it will be a minimizer whose coincidence set has no interior in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This contradicts Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Step 3: (Harnack chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=') Suppose that x1 and x2 are such that for some ˜C > 0 and ε > 0 we have |x1 − x2| < ˜Cε, Bε(xi) ⊂ {|u| > 0}, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We may assume without loss of generality, dist(x1, ∂{|u| > 0}) ≤ dist(x2, ∂{|u| > 0}) = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If δ0 ≥ ˜Cε, then x1 ∈ B ˜Cε(x2) ⊂ {|u| > 0} and we can easily find the Harnack chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' So, consider the case δ0 < ˜Cε and choose x ∈ ∂{|u| > 0} such that |x − x2| = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then x1, x2 ∈ Br(x) for r = 2 ˜Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='4, {|u| > d} ∩ BMr(x) has a connected component containing x1 and x2, where d = 1 2 min(|u(x1)|∞, |u(x2)|∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now we have a curve γ : [0, 1] → {|u| > d} ∩ BMr(x) having x1 and x2 as end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' For every t ∈ [0, 1] we know that |u(γ(t))| ≥ d ≥ c0ε 2 √m , where c0 comes from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Hence, dist(u(γ(t)), ∂{|u| > 0}) ≥ σε, where σ = c0 2C0 √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Now we can find a sequence y1, · · · , yℓ on the image of γ such that γ[0, 1] ⊂ ℓ � i=1 Bσε(yi) ⊂ {|u| > 0} ∩ BMr+σε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since Mr + σε = (2M ˜C + σ)ε, the number of balls in covering, ℓ can be bounded by a constant depending only on the dimension and (2M ˜C + σ)/σ, but not on x1, x2 or ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 23 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' An approximation lemma Here we prove a lemma, which we used in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='2, and is generalization of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 in [5] to any 1 < p < ∞ (see also Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let u be a nonnegative Lipschitz function in B+ 1 and assume that it is p-harmonic in {u > 0} and u(0) = 0 (1 < p < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Then it has the asymptotic development u(x) = αxn + o(|x|), as x → 0, for some α ≥ 0, if either (i) u vanishes on {xn = 0}, or (ii) {xn > 0} ⊂ {u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Part (i) is Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content='1 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' The proof of (ii) is also similar by a slight modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let ℓk := sup{l : lxn ≤ u(x) in B+ 2−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since ℓk is a nondecreasing sequence and bounded by the Lipschitz constant of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Suppose α = limk→∞ ℓk, then u(x) ≥ αxn + o(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' If the claim fails there exists a sequence xk → 0 such that u(xk) ≥ αxk n + δ0|xk|, for some δ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Define uk(x) := u(rkx)/rk where rk = |xk| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' We may also assume that rk ≤ 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Since uk are uniformly Lipschitz, we may consider the blowup u0 = limk→∞ uk, as well as xk/rk → x0, |x0| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From the construction we will have αxn ≤ u0(x) in B+ 1 , and δ0 2 + αxn ≤ u0(x) and δ0 2 + ℓkxn ≤ uk(x) in Bε(x0), for a sufficiently small ε > 0 and large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Let now wk be a p-harmonic function in B+ 1 with smooth boundary values wk = ℓkxn on ∂B+ 1 \\ Bε/2(x0), wk = ℓkxn + δ0 4 on ∂B+ 1 ∩ Bε/4(x0), ℓkxn ≤ wk ≤ ℓkxn + δ0 4 on ∂B+ 1 ∩ Bε/2(x0), wk = 0 on {xn = 0} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' From the comparison principle we will have wk ≤ uk in B+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' (Note that uk(x) ≥ ℓkxn, since rk ≤ 2−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Furthermore, wk → w0 in C1,σ(B+ 1/2) where w0 is p-harmonic with boundary data w0 = 0 on {xn = 0} and w0 ≥ αxn on ∂B+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' By Hopf boundary principle, w0(x) ≥ (α + µ)xn in B+ γ, for some small µ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Thus for x ∈ B+ γ, uk(x) ≥ wk(x) ≥ w0(x) − xn∥∇(wk − w0)∥L∞(B+ 1/2) ≥ � α + µ − ∥∇(wk − w0)∥L∞(B+ 1/2) � xn ≥ �α + µ/2� xn Now returning to u we get u(x) ≥ (α + µ/2)xn in B+ γ/rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' This is a contradiction with the definition of ℓk when k is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' □ 24 MORTEZA FOTOUHI AND HENRIK SHAHGHOLIAN Declarations Data availability statement: All data needed are contained in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' Funding and/or Conflicts of interests/Competing interests: The authors declare that there are no financial, competing or conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE0T4oBgHgl3EQfTgAw/content/2301.02236v1.pdf'} +page_content=' References [1] Aguilera, N.' metadata={'source': 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b/UtE2T4oBgHgl3EQfXgct/content/tmp_files/2301.03844v1.pdf.txt @@ -0,0 +1,1343 @@ +Look Beyond Bias with Entropic Adversarial Data +Augmentation +Thomas Duboudin 1 +Emmanuel Dellandr´ea 1 +Corentin Abgrall 2 +Gilles H´enaff 2 +Liming Chen 1 +1 Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA Lyon, +Univ Claude Bernard Lyon 1, Univ Louis Lumi`ere Lyon 2, +LIRIS, UMR5205, 69134 Ecully, France +{thomas.duboudin, emmanuel.dellandrea, liming.chen}@ec-lyon.fr +2 Thales LAS France +{corentin.abgrall, gilles.henaff}@fr.thalesgroup.com +Abstract—Deep neural networks do not discriminate between +spurious and causal patterns, and will only learn the most +predictive ones while ignoring the others. This shortcut learning +behaviour is detrimental to a network’s ability to generalize to an +unknown test-time distribution in which the spurious correlations +do not hold anymore. Debiasing methods were developed to make +networks robust to such spurious biases but require to know in +advance if a dataset is biased and make heavy use of minority +counter-examples that do not display the majority bias of their +class. In this paper, we argue that such samples should not be +necessarily needed because the ”hidden” causal information is +often also contained in biased images. To study this idea, we +propose 3 publicly released synthetic classification benchmarks, +exhibiting predictive classification shortcuts, each of a different +and challenging nature, without any minority samples acting +as counter-examples. First, we investigate the effectiveness of +several state-of-the-art strategies on our benchmarks and show +that they do not yield satisfying results on them. Then, we propose +an architecture able to succeed on our benchmarks, despite +their unusual properties, using an entropic adversarial data +augmentation training scheme. An encoder-decoder architecture +is tasked to produce images that are not recognized by a classifier, +by maximizing the conditional entropy of its outputs, and keep +as much as possible of the initial content. A precise control of +the information destroyed, via a disentangling process, enables +us to remove the shortcut and leave everything else intact. +Furthermore, results competitive with the state-of-the-art on the +BAR dataset ensure the applicability of our method in real-life +situations. +I. INTRODUCTION +Deep neural networks are now the preferred method in +computer vision when facing classification, object detection, +or semantic segmentation tasks, as they exhibit human-level +performances on such kind of tasks [1], [2]. However, a +mismatch between the training and testing data distribution +often leads to a sharp drop in performance [3]. One of +the reasons behind the inability of deep networks to ensure +good performance on unseen data distribution is the frequent +presence of biases in the training dataset [4]–[7]. If these +biases are the most predictive patterns for the task, a network +will learn to use them, and ignore the less effective ones, to +make its decision (a behaviour called shortcut learning [8], +a bias is a spurious shortcut). On a new data distribution, +with different biases, the network will not be able to use the +previously learned patterns to take proper decisions. +Fig. 1. Samples of our benchmarks. First and second rows are our Colored- +MNIST training and test data. Third and fourth rows are training and test +data for the Colored-Patch CIFAR10 benchmark, fifth and last rows are the +training and test data for the Located-Patch CIFAR10 benchmark +Several strategies have been designed to prevent deep +networks from overly focusing on the biases. A first family +of works assumes that the bias is given as an auxiliary label +(such as in [9]) on the images, other works [8], [10]–[14] +consider that the bias is what is naturally learned by a model. +Both aims to make the network invariant to the bias. The +second family of works still requires to know that the dataset +is strongly biased and that what is naturally learned can be +safely ignored, without losing information. Most of these +debiasing methods rely on particular training samples that do +not exhibit the majority bias and increase their importance in +the training procedure to prevent the model from learning the +biases. +We argue that it is desirable to design methods that do not +rely on such few portion of samples. In real-life situations, +it could lead to overestimate the importance of outliers that +should be ignored, such as annotation errors (Northcutt et al. +[15] estimated that the average annotation errors percentage +in usual computer vision datasets was 3.3%). It may also +be optimistic to find counter-examples of the biases in some +situations, e.g. with synthetic datasets where there is often a +arXiv:2301.03844v1 [cs.LG] 10 Jan 2023 + +Fig. 2. Overview of our architecture. Full lines denote the algorithm pipeline for the original images, dashed lines the pipeline for the reconstructed images +and the dotted lines for the entropic images. CE denotes a cross-entropy loss, H the conditional entropy of the final classifier, and L1 an L1 distance loss. +The 2 encoders displayed refer to the unique one used, but were displayed twice for readability. +low diversity of situations. Furthermore, most of the time, the +causal patterns are still present in the biased images, they are +simply ”hidden” behind the more effective biases. Finding +the causal patterns in the biased samples should therefore +be possible, and probably leads to better generalization +than counter-examples based debiasing methods due to the +reliance on a vastly larger amount of data. Besides, knowing +in advance that a dataset is biased is often an unrealistic +hypothesis (spurious biases can be noticeable background or +context, as in [16], but also inconspicuous hospital specific +clues, as in [17]). +To study debiasing without the help of minority samples, +we first design 3 synthetic benchmark datasets, for a +classification task, based on the CIFAR-10 [18] and MNIST +[19] datasets, without any counter-examples. The nature +of the biases in the datasets is crafted to be diverse and +challenging. On our Colored-MNIST, the bias is spatially +mingled with the underlying causal patterns, on our Colored- +Patch CIFAR10, the bias is a very local one, and for our +Located-Patch CIFAR10, the biases are not texture-based +biases but positional ones. Samples of our dataset can be +found in Figure 1. Experiments on state-of-the-art strategies +show that the benchmarks are indeed challenging since no +methods yield satisfying results on them. Furthermore, to +solve the shortcut issue without using particular data samples, +we draw inspiration from methods coming from the domain +generalization research field, in which the goal is to make +networks robust to unknown domain shifts. From a biased +image, we aim to create an image from which the shortcut, +or bias, is ”removed”, but for which everything else is kept +intact. Using an encoder-decoder architecture, we transform +biased images into images that maximize the entropy of a +classifier (trained on both original and transformed images). +With a precise control of the amount of information destroyed, +via a disentanglement process, we are able to generate images +where only the shortcuts are noticeably altered and replaced +by patterns that not recognized by the classifier. A classifier +trained on these transformed images learn the previously +missed patterns since the most obvious ones are now missing. +Since we do not assume that the shortcuts were spurious (in +a real-life situation, what is naturally learned by a network +will be a complex mix of causal and spurious features), we +train a classifier on both original and transformed images to +learn both the shortcuts and the less effective patterns. +To summarize, our contributions are threefold : +• We design 3 synthetic biased benchmark datasets, for a +classification task, based on the CIFAR10 and MNIST +datasets. +• We show that existing works only very partially mitigate +the accuracy drop at test-time on our datasets. +• We propose a novel method relying on an adversarial +encoder-decoder model that removes the bias from the +images via entropy maximization. We demonstrate the +effectiveness of our strategy on our datasets and on the +more realistic BAR [10] dataset. +II. RELATED WORKS +A. Debiasing +Deep neural networks learn correlations between patterns +and labels without any regard to how spurious they are [9], + +if 1 += 2 +Eq. 2, line 3 +if 1 2 +Encoder + Eq. 1, ine 2 +Eq. 1, line 1 +Encoder +1 + DR,DH +Decoder +Eq. 2, line 2 +◆ if +C2 +First Classifier +(features extractor) +CE +CE.H +Final Classifier +Eq. 2, line 1 +First Classifier +CE +CE +FC +if 1 +≠ 2 +Eq. 1, line 3[10]. In a situation where a dataset is heavily biased, this +behaviour will hinder generalization but this issue might be +mitigated by having the network ignore the biased patterns +even though they are predictive. Kim et al. [9], are able to +make a network invariant to biased patterns, but require the +bias to be given as an auxiliary label. This is tedious from +a human annotation perspective and restrict the application +of the method to a bias that can be spotted with the human +eye. Based on this limitation, another line of work [8], [10]– +[14] focuses on finding counter-examples i.e., samples that +do not share the majority bias of their class, often via a loss- +based criterion. Once such samples are found, their importance +during the training is increased via an over-sampling scheme +as in Just-Train-Twice (JTT) [13] or via a loss re-weighting +scheme as in Learning-from-Failure (LfF) [10]. Closest to our +work is the work of Kim et al. [14] that allows the creation +of a debiased dataset from a biased one, by generating hybrid +images in which biases and labels are decorrelated. A model +is then simply trained on this decorrelated dataset and does +not learn the biases as they are no longer predictive. Iterating +over this idea, in [12] (LDD), Lee et al. propose to train on +”virtual” hybrids by disentangling bias and causal patterns at +features level and creating combinations unseen in the training +set before training on them. +B. Domain Generalization +Domain generalization is a research field aiming to make +deep networks robust to unknown and unforeseen domain +shifts between training and test-time, without having any +information about the test-time domain (such as non-annotated +samples from the target domain, as in unsupervised domain +adaptation [20], [21]). Most algorithms assume to have access +to data coming from several identified different domains and +aim to learn more robust features by learning features that +are shared among all source domains [21]–[23]. In single- +source domain generalization, however, only one domain is +available at training time. In such situation, it is no longer +possible to find domain-invariant features, and some meth- +ods rely instead on finding more diverse and semantically +different patterns than normal. Carlucci et al. [24] (Jigsaw), +for instance, used a self-supervised objective alongside the +classification, based on solving jigsaw puzzles: having an +auxiliary task not related to classification enables the network +to learn less domain specific patterns. Representation Self- +Challenging (RSC) [25] is a dropout strategy in which the +muted coefficients in the intermediary features are the ones +most responsible for the prediction and not randomly chosen +ones. This therefore forces the network to use less strongly +correlated patterns. Spectral Decoupling, a method introduced +by Pezeshki et al. [26], proposes an L2 regularization on the +output logits of the network (instead of the weights), to combat +the ”gradient starvation” phenomenon and prevent the network +from learning only a subset of the useful patterns. Finally, +adversarial data augmentation [27] has been used to generate +samples outside of the training distribution by maximizing the +cross-entropy classification loss. In this context, Zhao et al. +[28] used entropy maximization as a secondary regularizer to +further push samples away from the training manifold. Our +method only uses entropy maximization and a strict control +mechanism to only alter the shortcuts. +C. Disentanglement +Disentanglement is extensively used in the image-to-image +translation field where the content of an image is often +divided between domain-specific scene-invariant and scene- +specific domain-invariant information [29]–[31]. Our method +is inspired by image-to-image translation strategies, such as +MUNIT [29], but applied to a different content division. We +disentangle the content into semantic (what is learned by a +network solving a task), and non-semantic (the remaining +information contained in the image), which may contain +information useful for the task but only what is normally +overlooked by a network. +III. BENCHMARK DATASETS +We introduce 3 benchmark datasets based on the MNIST, +and CIFAR-10 datasets. The datasets are publicly available on +github 1. +A. Colored-MNIST +In the training set of this biased dataset, every digit is +colored with its particular class color. All the colors chosen +are fairly different, and there are no counter-examples. The +validation dataset is colored with the same colors as the +training set. In the test set however, all the digits are colored +with the same color no matter the class so that the accuracy +on the test set reflects how well the network learned the less +efficient patterns, i.e. the shapes of the digits. The test color +is the average of all training colors so that the network can’t +rely on color anymore to classify the images. Other works +have introduced color-biased MNIST datasets [11], [26], [32]. +Ours differs from theirs most notably by the lack of counter- +examples, but also by having a single domain for training, and +by having an unbiased dataset for test rather than a dataset +differently biased e.g. with different class-color combinations. +One particularity of the colored MNIST datasets is the fact that +the shape and color information are spatially mingled, which +prevents a simple training with cropping or cutout [33] data +augmentation to find the shape information. +B. Colored-Patch CIFAR10 +Inspired by the work of [8], we also design a more complex +benchmark dataset based on CIFAR10. For each of the training +images, a 5x5 pixels colored patch is added in the top left +corner. The color of the patch is the image’s class color, again +with no counter-examples. In the test set all the images have +the same patch color, which is the average of the training +colors. For this dataset, the bias is spatially located in a tiny +part of the images instead of being global as with the MNIST +dataset. The colors used are the same than the ones used for +the previous dataset. +1https://github.com/liris-tduboudin/Look-Beyond-Bias + +C. Located-Patch CIFAR10 +Finally, to experiment on a biased dataset for which the bias +is not texture-based, we create a CIFAR10-based benchmark +where it is the position of a 5x5 pixels patch that is absolutely +correlated with the label e.g. top left corner patch for planes, +bottom right corner for horses. The color of the added patch +is red, for all classes. For the test dataset, the average patch +is added to all the images no matter their class. +IV. ENTROPIC ADVERSARIAL DATA AUGMENTATION +A. Proposed Method +Our method uses 4 distinctly trained neural networks: +2 classifiers: a first classifier C0, with its convolutional +features extractor F0, and a final classifier Cf, with the same +architecture, one encoder E, and one decoder D. The decoder +D takes as input the output of the encoder E(x), and the +output of the features extractor F0(x). These 2 feature maps +are simply resized and concatenated in the channel dimension +before being sent to the decoder. We chose this fusion strategy +against AdaIN [34] based methods, such as [14], because +it is simpler and makes it easier for the decoder to learn +spatial information. The decoder outputs simultaneously 2 +images (the output image has 6 channels, the first 3 being the +reconstructed image DR, and the remaining 3 the entropic +images image DH). +The first classifier C0 is trained to minimize the cross- +entropy on the original images, without alteration to a standard +training procedure. The encoder, through a latent space recon- +struction loss, is made invariant to the shortcuts learned by the +first classifier C0. The decoder DR is conditioned to output the +encoder’s input content with the features extractor’s F0 input +shortcut. This architecture aims to produce a disentangled +representation of an image between the features used by +the first classifier C0 and the remaining information E(x) +needed to reconstruct the original images. The remaining +3 channels of the decoder (DH) are used to generate the +entropic images via an adversarial training scheme. Samples +of such hybrid and entropic images can found in Figure +3. The final classifier is simply trained to classify both the +original and the entropic images coming from DH to learn +both the shortcuts (not necessarily spurious) and the ”hidden” +patterns. All the networks are simultaneously trained with their +corresponding losses, though the first classifier can be trained +offline beforehand and frozen during the training of the other +networks. A full schema of the proposed method is available +in Figure 2. +LR(DR, E) = αEx∼px[||DR(E(x), F0(x)) − x||1] ++ βE(x1,x2)∼px2[||E(DR(E(x1), F0(x2))) − E(x1)||1] ++ γE(x1,x2)∼px2[− +� +i +δi(C0(x2))× +logδi(C0(DR(E(x1), F0(x2))))] +(1) +To properly condition the encoder and the decoder DR to +yield a disentangled representation, several training objectives +are required: a reconstruction loss in the image space be- +tween the original image and the reconstructed one (Eq.1, +line 1), an encoder latent space reconstruction loss (Eq.1, +line 2), and a classifier prediction consistency loss (Eq.1, +line 3). The classifier consistency loss is a cross-entropy +between the prediction on an original image x2, C0(x2), +and the prediction on a hybrid image created from E(x1) +and F0(x2): C0(DR(E(x1), F0(x2))), with δi being the i-th +softmax coefficient: δi(y) = eyi/� +j eyj. The last 2 losses +require the sampling of 2 images simultaneously (x2 can be +obtained by applying a permutation on the current batch along +the sample dimension). These losses enable the encoder to +learn all necessary patterns for the reconstruction task but +the shortcuts already provided by the features extractor F0, +and prevent the decoder from inferring the shortcuts from the +encoder representation, as the shortcut of its hybrid output +DR(E(x1), F0(x2)) must be the one of x2. The first and +the final classifiers are not optimized with regards to these +constraints. +LH(DH) = εEx∼px[− +� +i +qlogδi(Cf(DH(E(x), F0(x))))] ++ µEx∼px[||DH(E(x), F0(x)) − Ef0∼pf0 [DH(E(x), f0)||1] ++ νEx∼px[||E(DH(E(x), F0(x))) − E(x)||1] +(2) +The entropic output of the decoder DH is trained to maxi- +mize the entropy of the final classifier, that is trained on both +the original images and the entropic ones. The rational behind +this adversarial loss is that high-entropy images do not contain +patterns that can be preferentially linked to a class, and hence +are more likely to be devoid of the original shortcuts. Maxi- +mizing the entropy of the first classifier only leads to changes +in the bias, and not to the complete removal we are aiming +for. Alongside the entropy maximization (Eq. 2, line 1, with +q the uniform probability density: q = 1/Nc where Nc is the +number of classes), the entropic images are subject to several +constraints to avoid the destruction of all information. The first +constraint lies in the decoder itself: up until the last layer, the +weights are shared for both the entropic images generation +and the reconstruction task. We also use the encoder latent +space reconstruction loss (Eq.2, line 3) on the ground that the +entropic images should precisely not modify what is extracted +by the encoder (everything but the shortcut). Finally, we use an +encoder-conditioned expected image reconstruction loss (Eq.2, +line 2). This loss aims to drive the entropic images toward an +image that should already be confusing for the classifier, while +keeping the information not used in classification intact. It is +implemented by minimizing the L1 loss between the entropic +image DH(E(x1), F0(x1)) and an hybrid image generated +from the encoding of x1: DR(E(x1), F0(x2)). Because the +semantic image x2 is randomly sampled every iteration, mini- +mizing this loss will eventually yield the average-biased image +Ef0∼pf0 [DR(E(x1), f0)]. + +B. Baselines for comparison +We compare ourselves with several state-of-the-art strate- +gies from either the debiasing or the domain generalization +community. A change of biases between training and test is a +domain shift, and even though domain generalization methods +were not developed with such shift in mind, it is interesting to +see how they perform. We also compare ourselves with simple +baselines as a reality check: first, with the standard training +procedure (stochastic gradient descent with momentum, with +the cross-entropy loss), then, with dropout [35]. Dropout is +a naive way of finding more useful patterns, by preventing +the network to use all the available information at disposal. +Since nothing prevents the network from learning the same +patterns in several filters, we experiment with an orthogonality +constraints on the weights (inspired by [36]) to force neurons +to be different, and with a constraint over the covariance +matrix of the intermediary activations (inspired by [37]), on the +ground that filters that activate very often together are likely +to check for the same patterns. Finally, we compare ourselves +with the single-source domain generalization methods (Jigsaw, +Spectral Decoupling, and RSC) and debiasing methods (LfF, +JTT and LDD) reviewed in the related works section. +V. EXPERIMENTS AND DISCUSSION +A. Experimental setup +Our architecture exists in 2 different flavors : large-scale, +and small-scale. The large-scale version uses a ResNet18 [1] +(adapted to the CIFAR10 and MNIST datasets as in [38]) +as classifier, and a UNet [39] as encoder and decoder. The +features extractor F0 is the classifier without its last layer. The +UNet is divided into a multi-output encoder and a multi-input +decoder to account for the skip-connections. Each encoder +output is taken as input by the corresponding decoder input. +The semantic information from the features extractor F0 is +concatenated to the deepest encoder output before being given +to the deepest decoder input. The small-scale version uses +a LeNet as classifier (as used in [24] for the MNIST-based +experiments), a 4-layers convolutional network as encoder, +and a 4-layers decoder with transposed convolutions. For both +architecture, the optimizer used is Adam [40] with the same +learning rates for all the networks. We evaluate our approach +on our synthetic benchmarks and on the more realistic Biased- +Action-Recognition (BAR) dataset. BAR images are divided +into 6 activity classes and exhibit a strong (but not absolute) +background bias in the training data: climbing often takes +place in on grey rocky background, throwing on a green +baseball field, etc. The test set however is mostly made of +images taken from unusual circumstances. This dataset is a +common benchmark for debiasing methods that make use of +the training minority samples. No data augmentation is used +for the experiments as we want to assess the effectiveness +of our method without adding any predefined invariances to +the training procedure. Hyper-parameters settings and model +selection are done without using the test set, to mimic a +situation where the test-time data distribution is unknown. +For our architecture, we use the entropy loss curve to set +the sensitive ε hyper-parameters: it should, in fact, increase +continuously during training. A diminishing entropy means +that the final classifier cannot keep up with the decoder’s +entropic images and that there is a destruction of information. +The goal is to aim for the slowest increasing entropy curve +possible. For the model selection with our method, we use the +model at the final epoch, on the ground that training on the +entropic images is much slower than the training on the normal +images. Training is considered complete when the entropy +curve is no longer increases. Finally, because the effect of the +random initialization of the network is greater than usual in a +domain shift situation [41], results are averaged over 3 runs. +More details about the architecture and the training hyper- +parameters are available in the supplementary material. +B. Results and analysis +Fig. 3. Samples of generated images. First & second rows: original images. +Third row: hybrid images with first row content and second row shortcuts. +Fourth row: hybrid images with second row content and first row shortcuts. +Last row: entropic images of the first row. +Our main results are available in Table I for the large-scale +version. Small-scale results are available in the supplementary +material. The numbers displayed are the average accuracy +± the standard deviation. Our method yields a significant +accuracy improvement over the previous works on our +synthetic benchmarks on the test sets, while retaining a +very high accuracy on the validation sets. A high accuracy +reached in both validation and test indicates that both the +shortcuts and the ”hidden” patterns are learned: if the +shortcuts are completely ignored, we expect the accuracy to +be similar for the biased and unbiased datasets. The drop +in validation for Colored-Patch CIFAR10 is most likely +due to non-optimal hyper-parameters: we used the same +hyper-parameters for all 3 synthetic datasets. Final accuracy +seems to be moderately sensitive to loss weights, except for +the entropy maximization weight. Other strategies perform +only marginally better than the standard training procedure. +It is not surprising for debiasing methods that require explicit +counter-examples. Furthermore, our architecture enables the +final classifier to find all the possible patterns ”hidden” +behind the shortcut: the accuracy of our method on the +test dataset is roughly equal to the accuracy we get when + +TABLE I +MAIN RESULTS +Large-Scale Experiments (Resnet18 as classifier) +Dataset → +Colored-MNIST +Colored-Patch CIFAR10 +Located-Patch CIFAR10 +BAR +Method ↓ +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Standard Training Procedure +100 ± 0.0 +18.8 ± 7.4 +100 ± 0.0 +10.3 ± 0.5 +100 ± 0.0 +10.0 ± 0.0 +97.5 ± 0.6 +49.1 ± 1.9 +Dropout +100 ± 0.0 +14.6 ± 6.2 +100 ± 0.0 +11.0 ± 1.8 +100 ± 0.0 +10.1 ± 0.1 +98.9 ± 0.3 +49.4 ± 1.0 +Dropout & Orthogonality [36] +100 ± 0.0 +10.2 ± 0.1 +100 ± 0.0 +10.0 ± 0.0 +100 ± 0.0 +10.0 ± 0.0 +98.8 ± 0.6 +48.0 ± 1.4 +Dropout & Covariance [37] +93.3 ± 3.8 +27.2 ± 5.4 +86.4 ± 10.5 +21.6 ± 2.4 +67.9 ± 9.5 +26.3 ± 2.2 +90.6 ± 3.1 +28.0 ± 3.2 +Jigsaw Puzzle [24] +99.8 ± 0.3 +21.5 ± 4.9 +99.9 ± 0.0 +17.8 ± 0.9 +100 ± 0.0 +12.1 ± 0.5 +97.5 ± 0.3 +49.8 ± 1.8 +Spectral Decoupling [26] +99.9 ± 0.0 +24.5 ± 2.9 +100 ± 0.0 +10.4 ± 0.35 +100 ± 0.0 +10.2 ± 0.1 +96.1 ± 0.6 +44.5 ± 1.8 +RSC [25] +100 ± 0.0 +12.5 ± 1.9 +96.7 ± 5.7 +10.0 ± 0.1 +100 ± 0.0 +10.0 ± 0.0 +96.8 ± 0.9 +50.2 ± 1.4 +LfF [10] +100 ± 0.0 +9.8 ± 1.9 +100 ± 0.0 +10.2 ± 0.3 +100 ± 0.0 +10.6 ± 0.7 +97.4 ± 0.3 +54.3 ± 2.3 +JTT [13] +100 ± 0.0 +12.5 ± 4.3 +100 ± 0.0 +10.4 ± 0.4 +100 ± 0.0 +10.0 ± 0.02 +97.3 ± 0.8 +50.2 ± 2.9 +LDD [12] +100 ± 0.0 +14.8 ± 5.3 +100 ± 0.0 +10.0 ± 0.2 +100 ± 0.0 +10.2 ± 0.3 +98.3 ± 0.8 +53.61 ± 2.7 +Ours +99.8 ± 0.2 +97.3 ± 0.84 +93.8 ± 1.3 +78.9 ± 0.34 +98.0 ± 0.6 +75.6 ± 3.8 +97.1 ± 0.8 +54.4 ± 1.1 +Standard Training Procedure +99.4 ± 0.04 +99.4 ± 0.04 +77.4 ± 0.08 +77.4 ± 0.08 +77.4 ± 0.08 +77.4 ± 0.08 +- +- +on the original datasets +training the same network normally on the original MNIST or +CIFAR10 datasets (without biases). On the BAR dataset, our +method performs on par with the state-of-the-art debiasing +methods, without explicitly relying on the unusual samples. +Discrepancies between original LfF results and ours is +mostly due to the resizing of the images for computational +convenience (128x128 in our experiments, 224x224 in the +original ones). Samples of our entropic images can found +in Figure 3 and are effectively devoid of the original shortcuts. +We also conduct an ablation study of the small-scale version +of our architecture on the synthetic benchmark datasets. Our +study was conducted to shed the light on 2 questions: 1 +- is the disentangling part of the architecture needed i.e. +can’t an entropy maximization and a L1 loss between the +transformed image and the original one be sufficient to yield +unbiased images ? 2 - is the entropy maximization constraint +needed i.e. can’t the disentangling part with the encoder- +conditioned expected image L1 reconstruction loss be enough +? Results of the ablation study are available in Table II. +The reported numbers are the average accuracy on the test +sets of the datasets. For the first experiment, we conduct a +study with varying entropy maximization loss weight ε, the +weight for the identity loss α is fixed to 1.0. For the second +experiment, all the used weights (α, β, γ, µ) are fixed to 1, all +the others to 0.0. Our study shows that, while either simplified +architecture can yield satisfying results on a certain dataset, +for a strategy to work well on all benchmarks it has to use all +the proposed constraints. Without the disentangling part (Eq. +1), the accuracy suffers on all datasets but especially on the +Colored-MNIST dataset, no matter the ε used. We hypothesize +that this is due to the spatially widespread bias in the dataset. +Without the entropy maximization loss, the architecture is not +able to learn anything on the CIFAR10-based benchmarks. For +the Located-Patch CIFAR10, this is due to the positional nature +of the bias: without the entropy maximization constraint, the +patch is simply replaced by a brown patch (similar to what can +be seen in Figure 3 with the hybrid images). A classification +model trained on these samples will simply make use of a +differently colored patch to take its decisions. There are no +constraints to push the encoder-decoder to realistically inpaint +the missing patch (such as a co-occurrence discriminator loss +[31]) as the entropy maximization is effective enough. For the +Colored-Patch CIFAR10, imperfections in the disentanglement +process prevents the decoder from completely removing the +original color of the patch. +TABLE II +ABLATION STUDY +Dataset → +Colored- +MNIST +Colored-Patch +CIFAR10 +Located-Patch +CIFAR10 +1 - no +disentanglement +ε = 10−4 +67.1 +63.5 +12.1 +ε = 10−3 +63.6 +69.3 +53.5 +ε = 10−2 +31.2 +62.7 +60.3 +ε = 10−1 +49.1 +35.7 +22.4 +ε = 1.0 +25.6 +14.4 +16.0 +2 - no entropy +maximization +97.7 +16.0 +17.3 +Full Method +98.4 +69.2 +67.3 +VI. CONCLUSION +In this paper, we created 3 different benchmarks to study +the behaviour of domain generalization and debiasing methods +when facing a dataset where the bias is shared by all the +samples without exception. No existing work yield satisfying +results on it. 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Courville, “Out-of-distribution generalization via risk +extrapolation (rex),” arXiv preprint arXiv:2003.00688, 2020. + +ACKNOWLEDGEMENT +This work was in part supported by the 4D Vision project +funded by the Partner University Fund (PUF), a FACE pro- +gram, as well as the French Research Agency, l’Agence +Nationale de Recherche (ANR), through the projects Learn +Real (ANR-18-CHR3-0002-01), Chiron (ANR-20-IADJ-0001- +01), Aristotle (ANR-21-FAI1-0009-01), and the joint support +of the French national program of investment of the future and +the regions through the PSPC FAIR Waste project. +SUPPLEMENTARY MATERIAL +This supplementary material presents additional results and +details about the experiments conducted in the paper, that were +not directly included due to the 6 pages limit. +A. Debiasing and label noise +We want to study the behaviour of debiasing methods when +they are applied to a dataset containing annotation errors, or +label noise, as their impact might be a potential drawback. To +do so, we create another synthetic dataset based on MNIST. +The training images are colorized as in our original Colored- +MNIST dataset (every images of a particular class are colored +the same), but the image label is replaced by a random one +(chosen uniformly among all labels) with a certain probability +p. We used p = 0.01, which gives 1% of randomly labeled +samples. This is the order of magnitude of label noise that is +encountered in usual datasets [15]. The test set is from the +same data distribution, with label noise. There is no domain +shift in this situation. We conducted this experiment with a +ResNet18 as classifier for the debiasing methods, and with +the large-scale version of our architecture. Results of debiasing +methods and of our approach on this dataset are available in +Table III. +TABLE III +DEBIASING AND LABEL NOISE +Method +Colored-MNIST w. label noise +LDD [12] +99.1 +JTT [13] +99.3 +LfF [10] +63.9 +Standard Training +99.2 +Ours +99.0 +The only method that does not succeed in dealing with the +label noise is LfF: training collapses after a few iterations +(see Figure 4), and does not recover. The best test accuracy +is reached at the very beginning of the training, before the +minority samples are noticeably over-weighted, and even then +it is far below the other works. All the other methods yield +perfect results. Training debiasing methods on a dataset with +wrongly labeled samples might have a adverse effect on the +resulting accuracy, depending on the precise strategy used. +Fig. 4. Test accuracy over training iterations for LfF, on our Colored-MNIST +with label noise. +B. Ablation study +1 - for the first ablation experiment, the reconstruction output +of the decoder DR is no longer trained with any objectives. +The entropic output DH and the encoder E are trained with +both an entropy maximization and an image reconstruction +loss, between the entropic image and the original one. Samples +of generated entropic images for this ablation are available in +Figure 6. The visualization of the samples confirms the quan- +titative results: too much information starts to be destroyed +for ε = 10−2, and too little for ε = 10−4, hence the drops +in test-time accuracy at both ends of the weight range for +Located-Patch CIFAR10, and Colored-MNIST. +Fig. 5. +Samples of generated encoder-conditioned expected images for the +second ablation study. +2 - for the second ablation experiment, the only modification +of the architecture lies in the weights used for the different +losses: all the introduced losses are used for the reconstruction +output DR, but for the entropic output DH only the encoder- +conditioned image reconstruction loss remains. All other losses +are removed. Samples of generated images for this study +are available in Figure 5. The encoder-conditioned image +reconstruction loss is not sufficient to ensure the complete +removal of the shortcut, as can be seen with the samples of +the Located-Patch CIFAR10 dataset. +C. Experimental details for the large-scale experiments +Synthetic Datasets: the MNIST images are colorized and +resized to 32x32 pixels, resulting in 3x32x32 pixel images, + +0.6 +0.4 +0.2 +0 +5k +5k +15k +25k +35k +45k +55kFig. 6. Samples of generated entropic images for the first ablation study. The first row corresponds to an entropy maximization objective weight of ε = 10−4. +This weight is increased by a factor 10 between each row. For every dataset, the first 3 columns are the original images, and the remaining 3 the corresponding +entropic images. +as the CIFAR10-based benchmarks images. For all synthetic +experiments, the images are normalized with a mean and a +standard deviation both of [0.5,0.5,0.5]. The validation set +and the test set are both made with the images from the +original MNIST or CIFAR10 test sets, but biased differently. +There is no pretraining of the classifiers for these datasets. +BAR: the BAR images are resized to 128x128 pixels, +for computational convenience, and normalized with the +following means and standard deviations: [0.485, 0.456, +0.406], and [0.229, 0.224, 0.225]. The validation set is made +of 300 images that are removed from the original training +set, keeping ∼ 1700 images for training. For all experiments +on BAR, whether for our approach or existing methods, the +ResNet18 is pretrained on ImageNet. Samples of the original +BAR dataset can be found in Figure 7 alongside with their +hybrid and entropic counterparts. +Standard Training Procedure: for all datasets, we used a +learning rate of 10−3 with the stochastic gradient descent +(SGD) with a momentum of 0.9 and trained for 100 epochs, +with a batch size of 128 for all datasets (all the experiments +are trained with this batch size). The test-time model is +selected by best accuracy on the validation set. In the case +where the best accuracy is reached several time during +training (a common phenomenon since it is easy to reach +100% accuracy on the biased datasets), the model retained is +the first to reach it. +Jigsaw: images are divided into a 2x2 grid, whose patches are +then shuffled. The weight for the permutation classification +loss is fixed to 1.0 for BAR, 1.0 for the synthetic datasets. +Early stopping was applied as soon as the validation accuracy +reached 99%. Subsequently, the model selected was the last +one. Other training hyper-parameters are the same as for the +standard training procedure and this will also be the case for +the next experiments, unless specified otherwise. +Dropout: dropout was applied at the end of the features +extractor +part +of +the +classifiers. +It +is +before +the +last +fully-connected layer for the ResNet18, and after the 2 +convolutional layers for the LeNet. The zeroing probability is +chosen randomly at each iteration. +Dropout & Covariance Constraint: the penalty is calculated +with the L2 norm of the covariance matrix of the features +extractor activations computed over the current batch. The +diagonal of the covariance matrix is fixed to 0 beforehand. +The weight for the covariance penalty was fixed to 10−4 for +all experiments. +Dropout & Orthogonality Constraint: the orthogonality +constraint is calculated for all the layers, and the final +constraint used is the average penalty over all layers. A +particular layer’s penalty is the L2 norm of the dot product +between a layer’s weights and its transpose. The weight of the +penalty is fixed to 1.0 for all experiments. Overall, the effects +of these additional constraints compared to a simple dropout +are negligible. Nonetheless, due to the apparent simplistic +nature of our synthetic datasets, we deemed necessary to try +naive approaches first to ensure that a more complex one was +indeed needed to reach satisfying results. +RSC: we used channel-wise dropout with a batch percentage +of 100% and the amount of channels dropped is randomly +chosen at every step. Channels are sorted with respect to their +usefulness for the classification on each samples in the batch, +and a varying number of the most effective ones are dropped. +Spectral Decoupling: the weight for the L2 on the raw +logits of the network (before the softmax) is fixed to 0.01 +for BAR, and to 1.0 for the synthetic benchmarks. Early +stopping is applied when the validation accuracy reaches 99%. +LfF: the architecture was trained with Adam and a learning +rate of 10−4, for 100 epochs, with an amplification factor +q = 0.7, for all the experiments. + +8LDD: the architecture was trained with Adam and a learning +rate of 10−4, for 100 epochs. The amplification factor +is the same as for LfF. The hyper-parameters specific to +LDD (λdis, λswapb, λswap) were kept to the default value: +[1.0,1.0,1.0], used in the original paper for datasets similar +to ours. The bias-conflicting augmentation is scheduled to be +applied after the first epoch for BAR and after the default +value (10k iterations) for the synthetic datasets. Without +counter-examples, this parameter has no impact on the results. +JTT: one of the core principle in JTT is to train the first +network only for a limited amount of epochs, to avoid +overfitting on the train set and keep a few number of +misclassified train samples. For our synthetic experiments, +the perfect classifier was reached before the end of the first +epoch. To properly adapt the method we stopped the training +after a 99% accuracy on the current batch was reached for +the 10-th time since the beginning. On BAR, the first network +was trained for 1 epoch, before switching to training on the +over-sampled dataset. +Ours: The architecture was trained for 100 epochs, with Adam +and a learning rate of 10−4. The hyper-parameters used were: +α = 1.0, β = γ = 0.1, ε = 10−3, µ = 1.0, ν = 0.1, for both +the experiments on the synthetic datasets and BAR. +D. Small-scale experiments +To demonstrate the wide range of applicability of our +method, we experiment with a small-scale version of our +architecture. The classifier used is a LeNet (taken from [24]), +the encoder (respectively the decoder) is a custom network +with 4 convolutional (respectively transposed convolutional) +layers. The architecture details are available in Table IV. The +decoder has 256 input channels while the encoder only has 128 +output channels, because the LeNet features extractor (layers +parts of the features extractor are marked in bold) output also +has 128 channels. There are no skip-connections, and the +encoder and decoder are single-output and single-input. The +hyper-parameters used for the small-scale experiments are +exactly the same as before for our approach and the debiasing +methods. Some domain generalization algorithms however +required different hyper-parameters: they were only trained +for 20 epochs, the weight for the logits norm minimization +in Spectral Decoupling was fixed to 5.0, the weight for the +permutation classification loss in Jigsaw was fixed to 10.0 +and the covariance norm minimization weight used was 10−3. +The results of our small-scale architecture, alongside with the +debiasing and domain generalization methods (also evaluated +with our LeNet), are available in Table V. +Our method is still effective, but the difference between +the results on the original datasets and ours is shortened +compared to the large-scale version. The debiasing methods +perform as bad as with the large-scale version, which was to +be expected again. Interestingly, the small-scale version of +the domain generalization methods perform better than their +TABLE IV +SMALL-SCALE NETWORKS ARCHITECTURE +Syntax follows PyTorch format +LeNet +Encoder +Decoder +Conv2d(3,64,5) +Conv2d(3,32,3) +ConvTranspose2d(256,64,4,2) +ReLU +ReLU +ReLU +MaxPool2d(2,2) +Conv2d(32,64,3) +ConvTranspose2d(64,64,3) +Conv2d(64,128,5) +ReLU +ReLU +ReLU +Conv2d(64,64,3) +ConvTranspose2d(64,32,3) +MaxPool2d(2,2) +ReLU +ReLU +Linear(3200, 1024) +Conv2d(64,128,3,2) +ConvTranspose2d(32,6,3) +ReLU +ReLU +Linear(1024,1024) +ReLU +Linear(1024, 10) +large-scale counterparts, especially on the Colored-MNIST +dataset. While for the large-scale setting all of these performed +similarly on all 3 benchmarks, there are clear differences in +the small-scale setting: most of the strategies now perform +better on the Colored-MNIST benchmark. Dropout-based +methods and Jigsaw yield an important increase of accuracy +on the Colored-MNIST, but not on the CIFAR10-based +benchmarks. +Notably, +Spectral +Decoupling +produces +an +increase of accuracy in all 3 benchmarks, even if not the best +on Colored-MNIST. +To compare our method with the debiasing strategies in +a situation where they should be able to perform well, we +design a version of our benchmarks with counter-examples. +For 1% of the samples, the shortcut e.g. the red patch +position, is chosen randomly (with equal probability for each +shortcuts) and not based on the image label. The test set is the +same as the original one, with an average shortcut applied to +each image. The small-scale results of these experiments are +available in Table VI. The hyper-parameters used are the same +as the ones used for the previous small-scale experiments, +except for the starting bias-conflicting augmentation iteration +in LDD. In a situation with counter-examples, the scheduling +is important. We start the bias-conflicting augmentation after +the first training epoch for all datasets. +The difference of behaviour between Colored-MNIST and +the other benchmarks has widened compared to the situation +without counter-examples: standard training reaches satisfying +accuracy on Colored-MNIST, while having no positive impact +on the others. Likewise, domain generalization methods only +yield good results on the Colored-MNIST. As expected, the +behaviour of debiasing methods changes completely. They all +perform almost perfectly on the MNIST-based benchmark, +and produce a noticeable accuracy increase on the others +benchmarks. LDD’s performance are, on average, on par with +our method, except in the Located-CIFAR10 dataset where +it reaches better test accuracy at the cost of a large drop +in validation accuracy. Our small-scale experiments show the +necessity of trying the various methods on diverse benchmarks, +as the strength of the correlation between the shortcuts and the +labels is not the only factor of success for a method. + +TABLE V +SMALL-SCALE EXPERIMENTS +Dataset → +Colored-MNIST +Colored-Patch CIFAR10 +Located-Patch CIFAR10 +Method ↓ +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Standard Training Procedure +100 ± 0.0 +27.2 ± 3.3 +100 ± 0.0 +13.2 ± 3.7 +100 ± 0.0 +15.5 ± 0.5 +Dropout +100 ± 0.0 +43.0 ± 5.0 +100 ± 0.0 +15.8 ± 3.5 +100 ± 0.0 +15.2 ± 0.8 +Dropout & Orthogonality [36] +100 ± 0.0 +36.5 ± 1.7 +100 ± 0.0 +15.6 ± 1.5 +100 ± 0.0 +24.0 ± 0.5 +Dropout & Covariance [37] +100 ± 0.0 +34.9 ± 3.9 +100 ± 0.0 +14.8 ± 2.0 +100 ± 0.0 +20.9 ± 1.8 +Jigsaw Puzzle [24] +98.3 ± 0.2 +65.9 ± 4.8 +97.7 ± 0.3 +22.0 ± 1.0 +99.7 ± 0.1 +21.3 ± 0.4 +Spectral Decoupling [26] +99.6 ± 0.1 +49.1 ± 2.5 +95.4 ± 0.2 +30.5 ± 1.0 +96.9 ± 1.8 +29.1 ± 1.2 +RSC [25] +99.7 ± 0.0 +45.0 ± 0.6 +95.8 ± 2.0 +14.5 ± 2.0 +100 ± 0.0 +11.4 ± 0.1 +LfF [10] +100 ± 0.0 +23.9 ± 5 +100 ± 0.0 +14.8 ± 1.8 +100 ± 0.0 +15.3 ± 2.1 +JTT [13] +100 ± 0.0 +29.2 ± 6.7 +100 ± 0.0 +15.4 ± 2.8 +100 ± 0.0 +16.1 ± 0.4 +LDD [12] +100 ± 0.0 +12.2 ± 2.7 +100 ± 0.0 +14.2 ± 4.1 +100 ± 0.0 +10.0 ± 0.0 +Ours +99.9 ± 0.0 +98.4 ± 1.5 +94.3 ± 0.2 +69.2 ± 2.0 +97.9 ± 0.5 +67.3 ± 0.3 +Standard Training Procedure +99.2 ± 0.04 +99.2 ± 0.04 +72.8 ± 0.4 +72.8 ± 0.4 +72.8 ± 0.4 +72.8 ± 0.4 +on the original datasets +TABLE VI +SMALL-SCALE EXPERIMENTS ON DATASETS WITH COUNTER-EXAMPLES +Dataset → +Colored-MNIST +Colored-Patch CIFAR10 +Located-Patch CIFAR10 +Method ↓ +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Val. Acc. +Test Acc. +Standard Training Procedure +99.4 ± 0.1 +65.9 ± 4.6 +99.1 ± 0.0 +18.1 ± 3.8 +99.2 ± 0.0 +27.1 ± 2.1 +Dropout +99.4 ± 0.0 +67.3 ± 6.1 +99.2 ± 0.0 +17.3 ± 4.9 +99.3 ± 0.0 +22.9 ± 2.9 +Dropout & Orthogonality [36] +99.4 ± 0.0 +68.0 ± 2.0 +99.0 ± 0.1 +26.8 ± 2.0 +99.1 ± 0.1 +27.5 ± 0.2 +Dropout & Covariance [37] +99.4 ± 0.0 +59.6 ± 5.0 +99.14 ± 0.0 +22.9 ± 1.6 +99.2 ± 0.0 +17.0 ± 5.4 +Jigsaw Puzzle [24] +98.0 ± 0.2 +72.9 ± 0.9 +95.8 ± 0.5 +23.6 ± 0.6 +98.5 ± 0.1 +20.8 ± 1.8 +Spectral Decoupling [26] +98.8 ± 0.2 +49.6 ± 1.3 +95.3 ± 0.2 +30.4 ± 1.1 +96.9 ± 1.1 +29.0 ± 2.4 +RSC [25] +98.6 ± 0.0 +72.8 ± 5.5 +96.3 ± 0.4 +17.7 ± 2.3 +99.2 ± 0.0 +14.9 ± 0.8 +LfF [10] +98.7 ± 0.4 +94.8 ± 0.8 +89.8 ± 0.8 +37.73 ± 0.2 +87.3 ± 3.1 +40.4 ± 1.3 +JTT [13] +99.8 ± 0.0 +94.1 ± 0.1 +98.6 ± 0.0 +30.5 ± 0.7 +98.7 ± 0.0 +40.7 ± 0.5 +LDD [12] +99.5 ± 0.1 +98.2 ± 0.2 +86.1 ± 1.0 +67.6 ± 0.6 +86.5 ± 1.2 +69.4 ± 0.7 +Ours +99.8 ± 0.1 +97.6 ± 0.1 +88.0 ± 0.2 +68.9 ± 0.2 +95.9 ± 0.3 +64.7 ± 0.3 +Standard Training Procedure +99.2 ± 0.0 +99.2 ± 0.0 +72.7 ± 0.4 +72.7 ± 0.4 +72.7 ± 0.4 +72.7 ± 0.4 +on the original datasets +Fig. 7. Samples of original BAR images (first 2 rows) and their generated hybrid (middle 2 rows) and entropic (last 2 rows) versions. The entropic images +are almost devoid of bright colors, showing that a classifier rely heavily on them instead on the causal patterns. The blue color is not removed as it is not +strongly correlated with a particular class: diving, pole vaulting and fishing images exhibit large amount of blue. + diff --git a/UtE2T4oBgHgl3EQfXgct/content/tmp_files/load_file.txt b/UtE2T4oBgHgl3EQfXgct/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a926a74ce57a1d5ff470587405d28420af52ad17 --- /dev/null +++ b/UtE2T4oBgHgl3EQfXgct/content/tmp_files/load_file.txt @@ -0,0 +1,1168 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf,len=1167 +page_content='Look Beyond Bias with Entropic Adversarial Data Augmentation Thomas Duboudin 1 Emmanuel Dellandr´ea 1 Corentin Abgrall 2 Gilles H´enaff 2 Liming Chen 1 1 Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA Lyon, Univ Claude Bernard Lyon 1, Univ Louis Lumi`ere Lyon 2, LIRIS, UMR5205, 69134 Ecully, France {thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='duboudin, emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='dellandrea, liming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='chen}@ec-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='fr 2 Thales LAS France {corentin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='abgrall, gilles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='henaff}@fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='thalesgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='com Abstract—Deep neural networks do not discriminate between spurious and causal patterns, and will only learn the most predictive ones while ignoring the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This shortcut learning behaviour is detrimental to a network’s ability to generalize to an unknown test-time distribution in which the spurious correlations do not hold anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Debiasing methods were developed to make networks robust to such spurious biases but require to know in advance if a dataset is biased and make heavy use of minority counter-examples that do not display the majority bias of their class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In this paper, we argue that such samples should not be necessarily needed because the ”hidden” causal information is often also contained in biased images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To study this idea, we propose 3 publicly released synthetic classification benchmarks, exhibiting predictive classification shortcuts, each of a different and challenging nature, without any minority samples acting as counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' First, we investigate the effectiveness of several state-of-the-art strategies on our benchmarks and show that they do not yield satisfying results on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Then, we propose an architecture able to succeed on our benchmarks, despite their unusual properties, using an entropic adversarial data augmentation training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' An encoder-decoder architecture is tasked to produce images that are not recognized by a classifier, by maximizing the conditional entropy of its outputs, and keep as much as possible of the initial content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A precise control of the information destroyed, via a disentangling process, enables us to remove the shortcut and leave everything else intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Furthermore, results competitive with the state-of-the-art on the BAR dataset ensure the applicability of our method in real-life situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' INTRODUCTION Deep neural networks are now the preferred method in computer vision when facing classification, object detection, or semantic segmentation tasks, as they exhibit human-level performances on such kind of tasks [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' However, a mismatch between the training and testing data distribution often leads to a sharp drop in performance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' One of the reasons behind the inability of deep networks to ensure good performance on unseen data distribution is the frequent presence of biases in the training dataset [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' If these biases are the most predictive patterns for the task, a network will learn to use them, and ignore the less effective ones, to make its decision (a behaviour called shortcut learning [8], a bias is a spurious shortcut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' On a new data distribution, with different biases, the network will not be able to use the previously learned patterns to take proper decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of our benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' First and second rows are our Colored- MNIST training and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Third and fourth rows are training and test data for the Colored-Patch CIFAR10 benchmark, fifth and last rows are the training and test data for the Located-Patch CIFAR10 benchmark Several strategies have been designed to prevent deep networks from overly focusing on the biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A first family of works assumes that the bias is given as an auxiliary label (such as in [9]) on the images, other works [8], [10]–[14] consider that the bias is what is naturally learned by a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Both aims to make the network invariant to the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The second family of works still requires to know that the dataset is strongly biased and that what is naturally learned can be safely ignored, without losing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Most of these debiasing methods rely on particular training samples that do not exhibit the majority bias and increase their importance in the training procedure to prevent the model from learning the biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We argue that it is desirable to design methods that do not rely on such few portion of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In real-life situations, it could lead to overestimate the importance of outliers that should be ignored, such as annotation errors (Northcutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [15] estimated that the average annotation errors percentage in usual computer vision datasets was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' It may also be optimistic to find counter-examples of the biases in some situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' with synthetic datasets where there is often a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='03844v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='LG] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Overview of our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Full lines denote the algorithm pipeline for the original images, dashed lines the pipeline for the reconstructed images and the dotted lines for the entropic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' CE denotes a cross-entropy loss, H the conditional entropy of the final classifier, and L1 an L1 distance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The 2 encoders displayed refer to the unique one used, but were displayed twice for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' low diversity of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Furthermore, most of the time, the causal patterns are still present in the biased images, they are simply ”hidden” behind the more effective biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Finding the causal patterns in the biased samples should therefore be possible, and probably leads to better generalization than counter-examples based debiasing methods due to the reliance on a vastly larger amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Besides, knowing in advance that a dataset is biased is often an unrealistic hypothesis (spurious biases can be noticeable background or context, as in [16], but also inconspicuous hospital specific clues, as in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To study debiasing without the help of minority samples, we first design 3 synthetic benchmark datasets, for a classification task, based on the CIFAR-10 [18] and MNIST [19] datasets, without any counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The nature of the biases in the datasets is crafted to be diverse and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' On our Colored-MNIST, the bias is spatially mingled with the underlying causal patterns, on our Colored- Patch CIFAR10, the bias is a very local one, and for our Located-Patch CIFAR10, the biases are not texture-based biases but positional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of our dataset can be found in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Experiments on state-of-the-art strategies show that the benchmarks are indeed challenging since no methods yield satisfying results on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Furthermore, to solve the shortcut issue without using particular data samples, we draw inspiration from methods coming from the domain generalization research field, in which the goal is to make networks robust to unknown domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' From a biased image, we aim to create an image from which the shortcut, or bias, is ”removed”, but for which everything else is kept intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Using an encoder-decoder architecture, we transform biased images into images that maximize the entropy of a classifier (trained on both original and transformed images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' With a precise control of the amount of information destroyed, via a disentanglement process, we are able to generate images where only the shortcuts are noticeably altered and replaced by patterns that not recognized by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A classifier trained on these transformed images learn the previously missed patterns since the most obvious ones are now missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Since we do not assume that the shortcuts were spurious (in a real-life situation, what is naturally learned by a network will be a complex mix of causal and spurious features), we train a classifier on both original and transformed images to learn both the shortcuts and the less effective patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To summarize, our contributions are threefold : We design 3 synthetic biased benchmark datasets, for a classification task, based on the CIFAR10 and MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We show that existing works only very partially mitigate the accuracy drop at test-time on our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We propose a novel method relying on an adversarial encoder-decoder model that removes the bias from the images via entropy maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We demonstrate the effectiveness of our strategy on our datasets and on the more realistic BAR [10] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Debiasing Deep neural networks learn correlations between patterns and labels without any regard to how spurious they are [9], if 1 = 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2, line 3 if 1 2 Encoder Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1, ine 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1, line 1 Encoder 1 DR,DH Decoder Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2, line 2 ◆ if C2 First Classifier (features extractor) CE CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='H Final Classifier Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2, line 1 First Classifier CE CE FC if 1 ≠ 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1, line 3[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In a situation where a dataset is heavily biased, this behaviour will hinder generalization but this issue might be mitigated by having the network ignore the biased patterns even though they are predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [9], are able to make a network invariant to biased patterns, but require the bias to be given as an auxiliary label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This is tedious from a human annotation perspective and restrict the application of the method to a bias that can be spotted with the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Based on this limitation, another line of work [8], [10]– [14] focuses on finding counter-examples i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=', samples that do not share the majority bias of their class, often via a loss- based criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Once such samples are found, their importance during the training is increased via an over-sampling scheme as in Just-Train-Twice (JTT) [13] or via a loss re-weighting scheme as in Learning-from-Failure (LfF) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Closest to our work is the work of Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [14] that allows the creation of a debiased dataset from a biased one, by generating hybrid images in which biases and labels are decorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A model is then simply trained on this decorrelated dataset and does not learn the biases as they are no longer predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Iterating over this idea, in [12] (LDD), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' propose to train on ”virtual” hybrids by disentangling bias and causal patterns at features level and creating combinations unseen in the training set before training on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Domain Generalization Domain generalization is a research field aiming to make deep networks robust to unknown and unforeseen domain shifts between training and test-time, without having any information about the test-time domain (such as non-annotated samples from the target domain, as in unsupervised domain adaptation [20], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Most algorithms assume to have access to data coming from several identified different domains and aim to learn more robust features by learning features that are shared among all source domains [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In single- source domain generalization, however, only one domain is available at training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In such situation, it is no longer possible to find domain-invariant features, and some meth- ods rely instead on finding more diverse and semantically different patterns than normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Carlucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [24] (Jigsaw), for instance, used a self-supervised objective alongside the classification, based on solving jigsaw puzzles: having an auxiliary task not related to classification enables the network to learn less domain specific patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Representation Self- Challenging (RSC) [25] is a dropout strategy in which the muted coefficients in the intermediary features are the ones most responsible for the prediction and not randomly chosen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This therefore forces the network to use less strongly correlated patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Spectral Decoupling, a method introduced by Pezeshki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [26], proposes an L2 regularization on the output logits of the network (instead of the weights), to combat the ”gradient starvation” phenomenon and prevent the network from learning only a subset of the useful patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Finally, adversarial data augmentation [27] has been used to generate samples outside of the training distribution by maximizing the cross-entropy classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In this context, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' [28] used entropy maximization as a secondary regularizer to further push samples away from the training manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our method only uses entropy maximization and a strict control mechanism to only alter the shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Disentanglement Disentanglement is extensively used in the image-to-image translation field where the content of an image is often divided between domain-specific scene-invariant and scene- specific domain-invariant information [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our method is inspired by image-to-image translation strategies, such as MUNIT [29], but applied to a different content division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We disentangle the content into semantic (what is learned by a network solving a task), and non-semantic (the remaining information contained in the image), which may contain information useful for the task but only what is normally overlooked by a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' BENCHMARK DATASETS We introduce 3 benchmark datasets based on the MNIST, and CIFAR-10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The datasets are publicly available on github 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Colored-MNIST In the training set of this biased dataset, every digit is colored with its particular class color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' All the colors chosen are fairly different, and there are no counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The validation dataset is colored with the same colors as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In the test set however, all the digits are colored with the same color no matter the class so that the accuracy on the test set reflects how well the network learned the less efficient patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' the shapes of the digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The test color is the average of all training colors so that the network can’t rely on color anymore to classify the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Other works have introduced color-biased MNIST datasets [11], [26], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Ours differs from theirs most notably by the lack of counter- examples, but also by having a single domain for training, and by having an unbiased dataset for test rather than a dataset differently biased e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' with different class-color combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' One particularity of the colored MNIST datasets is the fact that the shape and color information are spatially mingled, which prevents a simple training with cropping or cutout [33] data augmentation to find the shape information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Colored-Patch CIFAR10 Inspired by the work of [8], we also design a more complex benchmark dataset based on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For each of the training images, a 5x5 pixels colored patch is added in the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The color of the patch is the image’s class color, again with no counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In the test set all the images have the same patch color, which is the average of the training colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For this dataset, the bias is spatially located in a tiny part of the images instead of being global as with the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The colors used are the same than the ones used for the previous dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='com/liris-tduboudin/Look-Beyond-Bias C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Located-Patch CIFAR10 Finally, to experiment on a biased dataset for which the bias is not texture-based, we create a CIFAR10-based benchmark where it is the position of a 5x5 pixels patch that is absolutely correlated with the label e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' top left corner patch for planes, bottom right corner for horses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The color of the added patch is red, for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the test dataset, the average patch is added to all the images no matter their class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' ENTROPIC ADVERSARIAL DATA AUGMENTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Proposed Method Our method uses 4 distinctly trained neural networks: 2 classifiers: a first classifier C0, with its convolutional features extractor F0, and a final classifier Cf, with the same architecture, one encoder E, and one decoder D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The decoder D takes as input the output of the encoder E(x), and the output of the features extractor F0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' These 2 feature maps are simply resized and concatenated in the channel dimension before being sent to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We chose this fusion strategy against AdaIN [34] based methods, such as [14], because it is simpler and makes it easier for the decoder to learn spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The decoder outputs simultaneously 2 images (the output image has 6 channels, the first 3 being the reconstructed image DR, and the remaining 3 the entropic images image DH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The first classifier C0 is trained to minimize the cross- entropy on the original images, without alteration to a standard training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The encoder, through a latent space recon- struction loss, is made invariant to the shortcuts learned by the first classifier C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The decoder DR is conditioned to output the encoder’s input content with the features extractor’s F0 input shortcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This architecture aims to produce a disentangled representation of an image between the features used by the first classifier C0 and the remaining information E(x) needed to reconstruct the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The remaining 3 channels of the decoder (DH) are used to generate the entropic images via an adversarial training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of such hybrid and entropic images can found in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The final classifier is simply trained to classify both the original and the entropic images coming from DH to learn both the shortcuts (not necessarily spurious) and the ”hidden” patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' All the networks are simultaneously trained with their corresponding losses, though the first classifier can be trained offline beforehand and frozen during the training of the other networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A full schema of the proposed method is available in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' LR(DR, E) = αEx∼px[||DR(E(x), F0(x)) − x||1] + βE(x1,x2)∼px2[||E(DR(E(x1), F0(x2))) − E(x1)||1] + γE(x1,x2)∼px2[− � i δi(C0(x2))× logδi(C0(DR(E(x1), F0(x2))))] (1) To properly condition the encoder and the decoder DR to yield a disentangled representation, several training objectives are required: a reconstruction loss in the image space be- tween the original image and the reconstructed one (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1, line 1), an encoder latent space reconstruction loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1, line 2), and a classifier prediction consistency loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1, line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The classifier consistency loss is a cross-entropy between the prediction on an original image x2, C0(x2), and the prediction on a hybrid image created from E(x1) and F0(x2): C0(DR(E(x1), F0(x2))), with δi being the i-th softmax coefficient: δi(y) = eyi/� j eyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The last 2 losses require the sampling of 2 images simultaneously (x2 can be obtained by applying a permutation on the current batch along the sample dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' These losses enable the encoder to learn all necessary patterns for the reconstruction task but the shortcuts already provided by the features extractor F0, and prevent the decoder from inferring the shortcuts from the encoder representation, as the shortcut of its hybrid output DR(E(x1), F0(x2)) must be the one of x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The first and the final classifiers are not optimized with regards to these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' LH(DH) = εEx∼px[− � i qlogδi(Cf(DH(E(x), F0(x))))] + µEx∼px[||DH(E(x), F0(x)) − Ef0∼pf0 [DH(E(x), f0)||1] + νEx∼px[||E(DH(E(x), F0(x))) − E(x)||1] (2) The entropic output of the decoder DH is trained to maxi- mize the entropy of the final classifier, that is trained on both the original images and the entropic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The rational behind this adversarial loss is that high-entropy images do not contain patterns that can be preferentially linked to a class, and hence are more likely to be devoid of the original shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Maxi- mizing the entropy of the first classifier only leads to changes in the bias, and not to the complete removal we are aiming for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Alongside the entropy maximization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2, line 1, with q the uniform probability density: q = 1/Nc where Nc is the number of classes), the entropic images are subject to several constraints to avoid the destruction of all information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The first constraint lies in the decoder itself: up until the last layer, the weights are shared for both the entropic images generation and the reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We also use the encoder latent space reconstruction loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2, line 3) on the ground that the entropic images should precisely not modify what is extracted by the encoder (everything but the shortcut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Finally, we use an encoder-conditioned expected image reconstruction loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2, line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This loss aims to drive the entropic images toward an image that should already be confusing for the classifier, while keeping the information not used in classification intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' It is implemented by minimizing the L1 loss between the entropic image DH(E(x1), F0(x1)) and an hybrid image generated from the encoding of x1: DR(E(x1), F0(x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Because the semantic image x2 is randomly sampled every iteration, mini- mizing this loss will eventually yield the average-biased image Ef0∼pf0 [DR(E(x1), f0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Baselines for comparison We compare ourselves with several state-of-the-art strate- gies from either the debiasing or the domain generalization community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A change of biases between training and test is a domain shift, and even though domain generalization methods were not developed with such shift in mind, it is interesting to see how they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We also compare ourselves with simple baselines as a reality check: first, with the standard training procedure (stochastic gradient descent with momentum, with the cross-entropy loss), then, with dropout [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Dropout is a naive way of finding more useful patterns, by preventing the network to use all the available information at disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Since nothing prevents the network from learning the same patterns in several filters, we experiment with an orthogonality constraints on the weights (inspired by [36]) to force neurons to be different, and with a constraint over the covariance matrix of the intermediary activations (inspired by [37]), on the ground that filters that activate very often together are likely to check for the same patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Finally, we compare ourselves with the single-source domain generalization methods (Jigsaw, Spectral Decoupling, and RSC) and debiasing methods (LfF, JTT and LDD) reviewed in the related works section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' EXPERIMENTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Experimental setup Our architecture exists in 2 different flavors : large-scale, and small-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The large-scale version uses a ResNet18 [1] (adapted to the CIFAR10 and MNIST datasets as in [38]) as classifier, and a UNet [39] as encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The features extractor F0 is the classifier without its last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The UNet is divided into a multi-output encoder and a multi-input decoder to account for the skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Each encoder output is taken as input by the corresponding decoder input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The semantic information from the features extractor F0 is concatenated to the deepest encoder output before being given to the deepest decoder input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The small-scale version uses a LeNet as classifier (as used in [24] for the MNIST-based experiments), a 4-layers convolutional network as encoder, and a 4-layers decoder with transposed convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For both architecture, the optimizer used is Adam [40] with the same learning rates for all the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We evaluate our approach on our synthetic benchmarks and on the more realistic Biased- Action-Recognition (BAR) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' BAR images are divided into 6 activity classes and exhibit a strong (but not absolute) background bias in the training data: climbing often takes place in on grey rocky background, throwing on a green baseball field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The test set however is mostly made of images taken from unusual circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This dataset is a common benchmark for debiasing methods that make use of the training minority samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' No data augmentation is used for the experiments as we want to assess the effectiveness of our method without adding any predefined invariances to the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Hyper-parameters settings and model selection are done without using the test set, to mimic a situation where the test-time data distribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For our architecture, we use the entropy loss curve to set the sensitive ε hyper-parameters: it should, in fact, increase continuously during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A diminishing entropy means that the final classifier cannot keep up with the decoder’s entropic images and that there is a destruction of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The goal is to aim for the slowest increasing entropy curve possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the model selection with our method, we use the model at the final epoch, on the ground that training on the entropic images is much slower than the training on the normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Training is considered complete when the entropy curve is no longer increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Finally, because the effect of the random initialization of the network is greater than usual in a domain shift situation [41], results are averaged over 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' More details about the architecture and the training hyper- parameters are available in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Results and analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' First & second rows: original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Third row: hybrid images with first row content and second row shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Fourth row: hybrid images with second row content and first row shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Last row: entropic images of the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our main results are available in Table I for the large-scale version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Small-scale results are available in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The numbers displayed are the average accuracy ± the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our method yields a significant accuracy improvement over the previous works on our synthetic benchmarks on the test sets, while retaining a very high accuracy on the validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A high accuracy reached in both validation and test indicates that both the shortcuts and the ”hidden” patterns are learned: if the shortcuts are completely ignored, we expect the accuracy to be similar for the biased and unbiased datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The drop in validation for Colored-Patch CIFAR10 is most likely due to non-optimal hyper-parameters: we used the same hyper-parameters for all 3 synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Final accuracy seems to be moderately sensitive to loss weights, except for the entropy maximization weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Other strategies perform only marginally better than the standard training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' It is not surprising for debiasing methods that require explicit counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Furthermore, our architecture enables the final classifier to find all the possible patterns ”hidden” behind the shortcut: the accuracy of our method on the test dataset is roughly equal to the accuracy we get when TABLE I MAIN RESULTS Large-Scale Experiments (Resnet18 as classifier) Dataset → Colored-MNIST Colored-Patch CIFAR10 Located-Patch CIFAR10 BAR Method ↓ Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Standard Training Procedure 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 Dropout 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 Dropout & Orthogonality [36] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 Dropout & Covariance [37] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 Jigsaw Puzzle [24] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 Spectral Decoupling [26] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='35 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 RSC [25] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 LfF [10] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 JTT [13] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='02 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 LDD [12] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 Ours 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='84 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='34 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 Standard Training Procedure 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='04 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='08 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='08 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='08 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='08 on the original datasets training the same network normally on the original MNIST or CIFAR10 datasets (without biases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' On the BAR dataset, our method performs on par with the state-of-the-art debiasing methods, without explicitly relying on the unusual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Discrepancies between original LfF results and ours is mostly due to the resizing of the images for computational convenience (128x128 in our experiments, 224x224 in the original ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of our entropic images can found in Figure 3 and are effectively devoid of the original shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We also conduct an ablation study of the small-scale version of our architecture on the synthetic benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our study was conducted to shed the light on 2 questions: 1 is the disentangling part of the architecture needed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' can’t an entropy maximization and a L1 loss between the transformed image and the original one be sufficient to yield unbiased images ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2 - is the entropy maximization constraint needed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' can’t the disentangling part with the encoder- conditioned expected image L1 reconstruction loss be enough ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Results of the ablation study are available in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The reported numbers are the average accuracy on the test sets of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the first experiment, we conduct a study with varying entropy maximization loss weight ε, the weight for the identity loss α is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the second experiment, all the used weights (α, β, γ, µ) are fixed to 1, all the others to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our study shows that, while either simplified architecture can yield satisfying results on a certain dataset, for a strategy to work well on all benchmarks it has to use all the proposed constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Without the disentangling part (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1), the accuracy suffers on all datasets but especially on the Colored-MNIST dataset, no matter the ε used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We hypothesize that this is due to the spatially widespread bias in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Without the entropy maximization loss, the architecture is not able to learn anything on the CIFAR10-based benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the Located-Patch CIFAR10, this is due to the positional nature of the bias: without the entropy maximization constraint, the patch is simply replaced by a brown patch (similar to what can be seen in Figure 3 with the hybrid images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A classification model trained on these samples will simply make use of a differently colored patch to take its decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' There are no constraints to push the encoder-decoder to realistically inpaint the missing patch (such as a co-occurrence discriminator loss [31]) as the entropy maximization is effective enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For the Colored-Patch CIFAR10, imperfections in the disentanglement process prevents the decoder from completely removing the original color of the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' TABLE II ABLATION STUDY Dataset → Colored- MNIST Colored-Patch CIFAR10 Located-Patch CIFAR10 1 - no disentanglement ε = 10−4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ε = 10−3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ε = 10−2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ε = 10−1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 2 - no entropy maximization 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 Full Method 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' CONCLUSION In this paper, we created 3 different benchmarks to study the behaviour of domain generalization and debiasing methods when facing a dataset where the bias is shared by all the samples without exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' No existing work yield satisfying results on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We then proposed a generative architecture rely- ing on entropic adversarial data augmentation and on disentan- gling a representation between shortcuts and remaining useful patterns and showed that it performed as well as possible considering the classifiers used, on our 3 benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Further experiments on the BAR dataset yielded results competitive with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This is an indication that the explicit search for counter-examples might not be necessarily needed: the information usually overlook by neural networks is contained even in samples that exhibit the shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The future works will be dedicated to the study of more real-life like situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Courville, “Out-of-distribution generalization via risk extrapolation (rex),” arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='00688, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was in part supported by the 4D Vision project funded by the Partner University Fund (PUF), a FACE pro- gram, as well as the French Research Agency, l’Agence Nationale de Recherche (ANR), through the projects Learn Real (ANR-18-CHR3-0002-01), Chiron (ANR-20-IADJ-0001- 01), Aristotle (ANR-21-FAI1-0009-01), and the joint support of the French national program of investment of the future and the regions through the PSPC FAIR Waste project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL This supplementary material presents additional results and details about the experiments conducted in the paper, that were not directly included due to the 6 pages limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Debiasing and label noise We want to study the behaviour of debiasing methods when they are applied to a dataset containing annotation errors, or label noise, as their impact might be a potential drawback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To do so, we create another synthetic dataset based on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The training images are colorized as in our original Colored- MNIST dataset (every images of a particular class are colored the same), but the image label is replaced by a random one (chosen uniformly among all labels) with a certain probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We used p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='01, which gives 1% of randomly labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This is the order of magnitude of label noise that is encountered in usual datasets [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The test set is from the same data distribution, with label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' There is no domain shift in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We conducted this experiment with a ResNet18 as classifier for the debiasing methods, and with the large-scale version of our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Results of debiasing methods and of our approach on this dataset are available in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' TABLE III DEBIASING AND LABEL NOISE Method Colored-MNIST w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' label noise LDD [12] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 JTT [13] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 LfF [10] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 Standard Training 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 Ours 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 The only method that does not succeed in dealing with the label noise is LfF: training collapses after a few iterations (see Figure 4), and does not recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The best test accuracy is reached at the very beginning of the training, before the minority samples are noticeably over-weighted, and even then it is far below the other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' All the other methods yield perfect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Training debiasing methods on a dataset with wrongly labeled samples might have a adverse effect on the resulting accuracy, depending on the precise strategy used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test accuracy over training iterations for LfF, on our Colored-MNIST with label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Ablation study 1 - for the first ablation experiment, the reconstruction output of the decoder DR is no longer trained with any objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The entropic output DH and the encoder E are trained with both an entropy maximization and an image reconstruction loss, between the entropic image and the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of generated entropic images for this ablation are available in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The visualization of the samples confirms the quan- titative results: too much information starts to be destroyed for ε = 10−2, and too little for ε = 10−4, hence the drops in test-time accuracy at both ends of the weight range for Located-Patch CIFAR10, and Colored-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of generated encoder-conditioned expected images for the second ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 2 - for the second ablation experiment, the only modification of the architecture lies in the weights used for the different losses: all the introduced losses are used for the reconstruction output DR, but for the entropic output DH only the encoder- conditioned image reconstruction loss remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' All other losses are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of generated images for this study are available in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The encoder-conditioned image reconstruction loss is not sufficient to ensure the complete removal of the shortcut, as can be seen with the samples of the Located-Patch CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Experimental details for the large-scale experiments Synthetic Datasets: the MNIST images are colorized and resized to 32x32 pixels, resulting in 3x32x32 pixel images, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 0 5k 5k 15k 25k 35k 45k 55kFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of generated entropic images for the first ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The first row corresponds to an entropy maximization objective weight of ε = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' This weight is increased by a factor 10 between each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For every dataset, the first 3 columns are the original images, and the remaining 3 the corresponding entropic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' as the CIFAR10-based benchmarks images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For all synthetic experiments, the images are normalized with a mean and a standard deviation both of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The validation set and the test set are both made with the images from the original MNIST or CIFAR10 test sets, but biased differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' There is no pretraining of the classifiers for these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' BAR: the BAR images are resized to 128x128 pixels, for computational convenience, and normalized with the following means and standard deviations: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='485, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='456, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='406], and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='229, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='224, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The validation set is made of 300 images that are removed from the original training set, keeping ∼ 1700 images for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For all experiments on BAR, whether for our approach or existing methods, the ResNet18 is pretrained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of the original BAR dataset can be found in Figure 7 alongside with their hybrid and entropic counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Standard Training Procedure: for all datasets, we used a learning rate of 10−3 with the stochastic gradient descent (SGD) with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 and trained for 100 epochs, with a batch size of 128 for all datasets (all the experiments are trained with this batch size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The test-time model is selected by best accuracy on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In the case where the best accuracy is reached several time during training (a common phenomenon since it is easy to reach 100% accuracy on the biased datasets), the model retained is the first to reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Jigsaw: images are divided into a 2x2 grid, whose patches are then shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The weight for the permutation classification loss is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 for BAR, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 for the synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Early stopping was applied as soon as the validation accuracy reached 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Subsequently, the model selected was the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Other training hyper-parameters are the same as for the standard training procedure and this will also be the case for the next experiments, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Dropout: dropout was applied at the end of the features extractor part of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' It is before the last fully-connected layer for the ResNet18, and after the 2 convolutional layers for the LeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The zeroing probability is chosen randomly at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Dropout & Covariance Constraint: the penalty is calculated with the L2 norm of the covariance matrix of the features extractor activations computed over the current batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The diagonal of the covariance matrix is fixed to 0 beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The weight for the covariance penalty was fixed to 10−4 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Dropout & Orthogonality Constraint: the orthogonality constraint is calculated for all the layers, and the final constraint used is the average penalty over all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' A particular layer’s penalty is the L2 norm of the dot product between a layer’s weights and its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The weight of the penalty is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Overall, the effects of these additional constraints compared to a simple dropout are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Nonetheless, due to the apparent simplistic nature of our synthetic datasets, we deemed necessary to try naive approaches first to ensure that a more complex one was indeed needed to reach satisfying results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' RSC: we used channel-wise dropout with a batch percentage of 100% and the amount of channels dropped is randomly chosen at every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Channels are sorted with respect to their usefulness for the classification on each samples in the batch, and a varying number of the most effective ones are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Spectral Decoupling: the weight for the L2 on the raw logits of the network (before the softmax) is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='01 for BAR, and to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 for the synthetic benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Early stopping is applied when the validation accuracy reaches 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' LfF: the architecture was trained with Adam and a learning rate of 10−4, for 100 epochs, with an amplification factor q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7, for all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 8LDD: the architecture was trained with Adam and a learning rate of 10−4, for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The amplification factor is the same as for LfF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The hyper-parameters specific to LDD (λdis, λswapb, λswap) were kept to the default value: [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0], used in the original paper for datasets similar to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The bias-conflicting augmentation is scheduled to be applied after the first epoch for BAR and after the default value (10k iterations) for the synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Without counter-examples, this parameter has no impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' JTT: one of the core principle in JTT is to train the first network only for a limited amount of epochs, to avoid overfitting on the train set and keep a few number of misclassified train samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For our synthetic experiments, the perfect classifier was reached before the end of the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To properly adapt the method we stopped the training after a 99% accuracy on the current batch was reached for the 10-th time since the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' On BAR, the first network was trained for 1 epoch, before switching to training on the over-sampled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Ours: The architecture was trained for 100 epochs, with Adam and a learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The hyper-parameters used were: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0, β = γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1, ε = 10−3, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1, for both the experiments on the synthetic datasets and BAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Small-scale experiments To demonstrate the wide range of applicability of our method, we experiment with a small-scale version of our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The classifier used is a LeNet (taken from [24]), the encoder (respectively the decoder) is a custom network with 4 convolutional (respectively transposed convolutional) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The architecture details are available in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The decoder has 256 input channels while the encoder only has 128 output channels, because the LeNet features extractor (layers parts of the features extractor are marked in bold) output also has 128 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' There are no skip-connections, and the encoder and decoder are single-output and single-input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The hyper-parameters used for the small-scale experiments are exactly the same as before for our approach and the debiasing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Some domain generalization algorithms however required different hyper-parameters: they were only trained for 20 epochs, the weight for the logits norm minimization in Spectral Decoupling was fixed to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0, the weight for the permutation classification loss in Jigsaw was fixed to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 and the covariance norm minimization weight used was 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The results of our small-scale architecture, alongside with the debiasing and domain generalization methods (also evaluated with our LeNet), are available in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our method is still effective, but the difference between the results on the original datasets and ours is shortened compared to the large-scale version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The debiasing methods perform as bad as with the large-scale version, which was to be expected again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Interestingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' the small-scale version of the domain generalization methods perform better than their TABLE IV SMALL-SCALE NETWORKS ARCHITECTURE Syntax follows PyTorch format LeNet Encoder Decoder Conv2d(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5) Conv2d(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) ConvTranspose2d(256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2) ReLU ReLU ReLU MaxPool2d(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2) Conv2d(32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) ConvTranspose2d(64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) Conv2d(64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5) ReLU ReLU ReLU Conv2d(64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) ConvTranspose2d(64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) MaxPool2d(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2) ReLU ReLU Linear(3200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 1024) Conv2d(64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2) ConvTranspose2d(32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3) ReLU ReLU Linear(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1024) ReLU Linear(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 10) large-scale counterparts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' especially on the Colored-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' While for the large-scale setting all of these performed similarly on all 3 benchmarks, there are clear differences in the small-scale setting: most of the strategies now perform better on the Colored-MNIST benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Dropout-based methods and Jigsaw yield an important increase of accuracy on the Colored-MNIST, but not on the CIFAR10-based benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Notably, Spectral Decoupling produces an increase of accuracy in all 3 benchmarks, even if not the best on Colored-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' To compare our method with the debiasing strategies in a situation where they should be able to perform well, we design a version of our benchmarks with counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' For 1% of the samples, the shortcut e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' the red patch position, is chosen randomly (with equal probability for each shortcuts) and not based on the image label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The test set is the same as the original one, with an average shortcut applied to each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The small-scale results of these experiments are available in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The hyper-parameters used are the same as the ones used for the previous small-scale experiments, except for the starting bias-conflicting augmentation iteration in LDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' In a situation with counter-examples, the scheduling is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' We start the bias-conflicting augmentation after the first training epoch for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The difference of behaviour between Colored-MNIST and the other benchmarks has widened compared to the situation without counter-examples: standard training reaches satisfying accuracy on Colored-MNIST, while having no positive impact on the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Likewise, domain generalization methods only yield good results on the Colored-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' As expected, the behaviour of debiasing methods changes completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' They all perform almost perfectly on the MNIST-based benchmark, and produce a noticeable accuracy increase on the others benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' LDD’s performance are, on average, on par with our method, except in the Located-CIFAR10 dataset where it reaches better test accuracy at the cost of a large drop in validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Our small-scale experiments show the necessity of trying the various methods on diverse benchmarks, as the strength of the correlation between the shortcuts and the labels is not the only factor of success for a method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' TABLE V SMALL-SCALE EXPERIMENTS Dataset → Colored-MNIST Colored-Patch CIFAR10 Located-Patch CIFAR10 Method ↓ Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Standard Training Procedure 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 Dropout 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 Dropout & Orthogonality [36] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 Dropout & Covariance [37] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 Jigsaw Puzzle [24] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 Spectral Decoupling [26] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 RSC [25] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 LfF [10] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 5 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 JTT [13] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 LDD [12] 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 Ours 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 Standard Training Procedure 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='04 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 on the original datasets TABLE VI SMALL-SCALE EXPERIMENTS ON DATASETS WITH COUNTER-EXAMPLES Dataset → Colored-MNIST Colored-Patch CIFAR10 Located-Patch CIFAR10 Method ↓ Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Test Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Standard Training Procedure 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 Dropout 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 Dropout & Orthogonality [36] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 Dropout & Covariance [37] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 Jigsaw Puzzle [24] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 Spectral Decoupling [26] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 RSC [25] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='0 72.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content='4 on the original datasets Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' Samples of original BAR images (first 2 rows) and their generated hybrid (middle 2 rows) and entropic (last 2 rows) versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The entropic images are almost devoid of bright colors, showing that a classifier rely heavily on them instead on the causal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} +page_content=' The blue color is not removed as it is not strongly correlated with a particular class: diving, pole vaulting and fishing images exhibit large amount of blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfXgct/content/2301.03844v1.pdf'} diff --git a/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/2301.04297v1.pdf.txt b/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/2301.04297v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e892904cc250211e05e0b61f5f3ba4e793b6277 --- /dev/null +++ b/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/2301.04297v1.pdf.txt @@ -0,0 +1,1291 @@ + +Optothermal rotation of micro-/nano-objects in liquids + +Authors: Hongru Ding,1 Zhihan Chen,2 Carolina Ponce,1 and Yuebing Zheng1,2,* + +Affiliations: +1Walker Department of Mechanical Engineering, The University of Texas at Austin, +Austin, TX 78712, USA. +2Materials Science & Engineering Program and Texas Materials Institute, The University +of Texas at Austin, Austin, TX 78712, USA. +*Correspondence to: zheng@austin.utexas.edu. + +Abstract +Controllable rotation of micro-/nano-objects provides tremendous opportunities for cellular +biology, three-dimensional (3D) imaging, and micro/nanorobotics. Among different rotation +techniques, optical rotation is particularly attractive due to its contactless and fuel-free operation. +However, optical rotation precision is typically impaired by the intrinsic optical heating of the +target objects. Optothermal rotation, which harnesses light-modulated thermal effects, features +simpler optics, lower operational power, and higher applicability to various objects. In this +Feature Article, we discuss the recent progress of optothermal rotation with a focus on work from +our research group. We categorize the various rotation techniques based on distinct physical +mechanisms, including thermophoresis, thermoelectricity, thermo-electrokinetics, thermo- +osmosis, thermal convection, and thermo-capillarity. Benefiting from the different rotation modes +(i.e., in-plane and out-of-plane rotation), diverse applications in single-cell mechanics, 3D bio- +imaging, and micro/nanomotors are demonstrated. We conclude the article with our perspectives +on the operating guidelines, existing challenges, and future directions of optothermal rotation. + + + +Introduction +Awarded the Nobel Prize in Physics 2018, optical tweezers have proved to be an effective +instrument for non-contact spatial manipulation of micro/nanoparticles as well as living +biological samples.1-8 On the basis of optical tweezers, various optical rotation platforms +have been developed for exquisite control of micro/nanoscale targets over their rotational +degrees of freedom using light (e.g., laser beams).9-11 The light-powered rotation, termed +optical rotation, holds significant application potential in nano-torque sensing,12-14 +nanosurgery,15, 16 micro/nanofluidic systems,17, 18 and so on. To achieve optical rotation, +dynamic intensity profile,19, 20 nonlinear polarization,12, 21, 22 optical angular momenta,23, 24 +or radiation pressure25 is typically utilized to produce asymmetric light-matter interactions +for the generation of optical torques. In addition, sophisticated designs are also required on +the rotating targets, which typically possess on-demand geometry, optical birefringence, or +specific compositions.26-29 The rigorous requirements for light beams and properties of +particles largely restrict the broader applications of optical rotation. Moreover, the optical +rotation of nanoparticles is extremely challenging as optical forces and torques decrease +substantially with particle size.6 Though high-power light beams can be used for rotation +at the nanoscale level, the resultant optical heating will lead to strong Brownian motions +that significantly reduce the frequency stability of the rotation.30 +To overcome the bottlenecks of complex optics, limited applicability, and heat-induced +instability of optical rotation, heat-mediated optical rotation, i.e., optothermal rotation, has +been developed for rotary control of micro-/nano-objects by leveraging optical heating to +generate thermal forces and torques. Over the past two decades, researchers have +established multiple theories to describe the translational migration of colloids and living +objects under temperature gradient fields.31-36 Consequently, different optothermal +manipulation techniques have been developed,37-52 leading to applications in +nanofabrication,53-58 chemical and biological sensing,59-61 and cargo delivery.62, 63 Based +on this, optothermal rotation has been further developed recently for precise and versatile +rotations of colloidal particles and living cells with striking advantages such as low +operation power, high applicability, and long working distance. Through the synergy of +diverse light-induced thermal forces and/or optical forces, researchers, including us, + +manage to rotate objects of diverse sizes, materials, and shapes with low-power laser beams +and simple optics for the growing demands in sophisticated biological measurements, +imaging, and the development of nano-engines. This Feature Article focuses on the +development of the optothermal rotation of micro/nanoscale objects in liquids (Fig. 1). We +first introduce the fundamentals of different thermal forces that can drive stable +optothermal rotation. Then, different approaches to achieving in-plane and out-of-plane +rotation in respect of the substrates are reviewed based on their working mechanisms and +various potential applications. Finally, we provide our perspectives on existing challenges +and future directions of optothermal rotation. +2. Fundamental mechanisms +Different physical mechanisms for optothermal rotation in liquids are introduced in +this section, including thermophoresis, thermoelectricity, thermo-electrokinetics, +thermo-osmosis, thermal convection, and thermo-capillarity. The thermal forces +originating from those optothermal phenomena are employed to control the +translational movements of the targets and drive their stable rotation. + +Fig. 1 Overview of optothermal rotation. Orbiting, in-plane and out-of-plane spinning of +different colloids and living objects can stem from a variety of optothermal phenomena + +Thermo- +Thermo- +osmosis +electro- +electricity +kinetics +convection +Thermo- +Thermo- +Thermo- +phoresis +capillarity +Thermo- +Single-cell +Micro/nano +mechanics +-motors +- +3D bio-imagingincluding thermophoresis, thermoelectricity, thermo-electrokinetics, thermo-osmosis, +thermal convection, and thermo-capillarity. Optothermal rotation is promising in opening +new paths toward single-cell mechanics, 3D bio-imaging, and micro/nanomotors. +2.1. Thermophoresis +Thermophoresis (also known as thermodiffusion or the Soret effect) is defined as +the directed migration of tiny particles (e.g., ions, molecules, and colloidal particles) +along a temperature gradient in liquids.64-67 The temperature gradient functions as a +general force that directs the suspended particles to a cold or warm region at a certain +velocity given by +𝒖 = −𝐷𝑇∇𝑇 (1) +where DT is the thermophoretic mobility and ∇t denotes the temperature gradient. in a +nonuniform temperature field, the movement of the particles is also affected by Brownian +diffusion which competes with thermodiffusion. Thus, Soret coefficient (ST = DT /D) is +proposed to describe the thermophoretic migration in a general manner. With ST > 0 (OR +ST < 0), the suspended particles show thermophobicity (or thermophilicity) and move +toward the cold (or hot) region. In addition, a large ST in magnitude means that the +thermophoresis dominates over the Brownian motion, indicating that the particle can have +more directional motion under temperature gradient fields. +2.2. Thermoelectricity +Liquid thermoelectricity describes the generation of an electric field, i.e., thermoelectric +field, from the charge separation of ions in electrolyte solutions under a temperature field.38 +In a thermal equilibrium state, cations couple with anions due to electrostatic interactions. +While in a non-equilibrium state, a thermal gradient drives cations and anions with +different speeds and directions, depending on the ions’ size and solvation energy.68 +Additionally, the diffusion of molecules with high polarity (e.g., water) under a certain +temperature can also lead to nanoscale separation of atoms with different +electronegativity.69 These spatially separated charges (ions and atoms with partial charges) +lead to a bulk thermoelectric field given by +𝑬𝐓𝐄 = ∫ e(𝑛+ − 𝑛−) dz +ε + (2) + +where n+ and n- are volumetric number densities of cations and anions, respectively. ε +denotes the solvent permittivity. Since most colloids have surface charges, a thermoelectric +field can be employed to regulate the translational and rotational motions of the colloids +through electrostatic interactions. +2.3 Thermo-electrokinetics +In analogy with optical electrokinetics, where the migration of colloids is induced by the +non-uniform electric field based on a photoconductive substrate,70, +71 thermo- +electrokinetics describes a similar electrokinetic (EK) migration that stems from the +electric fields from the temperature-responsive substrates.44, 72 Most colloids are negatively +charged due to their ionized acid groups on the surface.31 Similarly, substrates can carry +surface charges by coating a thin layer of acid groups (e.g., carboxylic acids). Particularly, +the decrease of temperature generally promotes the dissociation of the acid molecules (e.g., +COO- and H+) according to the Van’t Hoff equation73, 74. Therefore, the surface-charge +gradients of colloids and substrates can be tuned by the temperature field, which leads to +an EK torque that can power the rotation of the colloids. The EK force is given by +𝑭𝐄𝐊 = ∮ 𝜎𝑟𝑑𝐴𝑟∬ 𝑬∥𝑑𝑥𝑑𝑦 (𝟑) +where σT is the surface charge density of the colloidal particle, dAr is the differential area +element on the particle surface, and 𝑬∥ is the parallel component of the electric field. +2.4 Thermo-osmosis +Thermo-osmosis describes the flow parallel with a surface under a temperature gradient, +which is typically directed toward higher temperature regions. Specifically, excess +hydrostatic pressure can be induced by a temperature gradient within an electric double +layer near the surface. The gradient of the hydrostatic pressure is opposite to the +temperature gradient and leads to a creeping flow parallel to the surface. Thus, when a +colloidal particle stays in a temperature field, the force from the thermo-osmotic flow +drives the particle moving in the opposite direction of the flow at the velocity31 +𝒖 = − εζ2∇𝑇 +3η𝑇 (4) + +where η is the solvent viscosity, and ζ is the surface potential of the particle. Moreover, +when a substrate is heated, the thermo-osmotic flow in the vicinity of the substrate can also +be used to direct and rotate neighboring micro/nanoparticles.36, 75, 76 +2.5 Thermal convection +Thermal convection, known as natural convection, results from buoyance forces exerting +on fluids with heat-induced density variations.77 Like the violent vibration of atoms in +solids at high temperatures,78, 79 fluid molecules scatter and separate in the vicinity of a hot +spot, causing the fluid to be less dense. Due to buoyance forces, the less-dense fluid moves +upward, and the cooler fluid gets denser and sinks. Meanwhile, the surrounding fluid +moves toward the hot spot due to fluid continuity. Thermal convection is typically bulky +and not suitable for single-particle manipulation. Therefore, optical heating upon laser +illumination on light-absorbing nanostructures has been proposed to achieve localized +temperature gradient fields for accurate control of thermal convection flows and delicate +micro/nano-manipulation.80, 81 +2.6 Thermo-capillarity +Thermo-capillary flow is one type of Marangoni effect: mass transfer along a fluid-fluid +interface happens due to the gradient of surface tension. Surface tension can be tuned by +concentration, temperature, and electrical potential. The cool liquid that has a high surface +tension can pull the surrounding warmer liquid that has a lower surface tension. This leads +to a thermo-capillary flow, whose velocity is given by +𝒖 = d𝛾 +d𝑇 +∇𝑇 +η (𝟓) +where γ is the interfacial surface tension. Due to its capability of rapid mass transfer, the +optothermo-capillary flow at fluid-liquid interfaces near micro/nanostructures has been +utilized for particle manipulation, digital fabrication, sensing, and energy harvesting.82-87 +3. Optothermal rotation techniques +Different applications of optothermal rotation are demonstrated whereby rational +management of light, heat, and solutions in optothermal fluidic systems. This Feature +article focuses on the contributions of Zheng Research Group to the recent progress in + +optothermal rotation. According to the relationship between the rotation axis and substrate, +optothermal rotation can be classified into in-plane optothermal rotation and out-of-plane +optothermal rotation, which both lead to different types of applications and are discussed +separately in this section. +3.1. In-plane rotation +Opto-thermophoretic rotation techniques. Opto-thermophoresis has been widely +used in micro/nano-manipulation by tailoring temperature gradient fields and +managing ST of target objects.88 Duhr et al. demonstrated on-demand opto- +thermophoretic accumulation and depletion of DNA molecules by manipulating ST +with temperature and salt concentration.89 A similar dual manipulation of +synthesized nanoparticles was achieved by Weinert et al. using opto-thermophoretic +flows in the vicinity of microparticles with different ST.90 Later, Cichos et al +demonstrated the trapping and directed swimming of nanoparticles with +thermophoretic force fields generated from the optical heating of gold +micro/nanostructures.91, 92 More recently, we developed opto-thermophoretic +tweezers for dynamic and low-power manipulation of micro-/nano-objects including +lipid vesicles which are challenging to be trapped by optical tweezers.41, 48 Despite +the tremendous progress in opto-thermophoretic manipulation, opto-thermophoretic +rotation is still challenging. +To fill this gap, we have developed opto-thermophoretic platforms for both rotational +and translational manipulation of synthesized and biological cells.49, 93 The working +principle of our platforms is illustrated in Fig. 2a. First, a real-time tailorable and +reconfigurable temperature field is established upon the illumination of a laser beam on a +uniform light-absorbing substrate (Fig. 2a(i)). Since the alignment of solvent molecules at +the liquid-particle interface depends on the surrounding temperature (Fig. 2a(ii)), a gradient +of interfacial entropy is then generated under the light-generated temperature field. +According to Anderson’s model, the entropy gradient causes an interfacial slip flow that +exerts a thermophoretic force on the particle’s surface.95 The thermophoretic velocity is +given by32 +𝒖 = −∇𝑇 +ε +2η𝑇 +2Λ𝑙 +2Λ𝑙 + Λ𝑝 +(1 + ∂lnε +∂ln𝑇)𝜁2 (6) + +where Λ𝑙 and Λp are the thermal conductivities of the liquid and the particle, respectively. +The permittivity term, ∂lnε/∂ln𝑇, is a function of interfacial entropy and the key for the +modulation of opto-thermophoretic movement. For instance, we achieved on-demand +opto-thermophoretic trapping of 1-μm polystyrene (PS) sphere in water as illustrated by +Fig. 2a(iii). + +Fig. 2 Opto-thermophoretic rotation techniques. (a) Working principle of opto- +thermophoretic manipulation of colloids. (b) Opto-thermophoretic rotation of a silver +nanowire by a thermoplasmonic with a 532-nm laser beam. (c) Opto-thermophoretic +rotation of two live yeast cells in water with two line-shaped laser beams at the wavelength +of 532 nm. The light-absorb substrate is the same as (b). Scale bars: (b) 5 µm and (c) 10 µm. +(a)-(b) Adapted with permission.49 Copyright 2018, American Chemical Society. (c) +Adapted with permission.93 Copyright 2017, American Chemical Society. +The rotation of micro/nanoscale colloids can be achieved on this platform through +geometric design and dynamic control of opto-thermophoretic potentials for torque +generation. The optical rotation of metallic anisotropic nanoparticles has been achieved by +exploiting optical aligning torque produced by laser beams with rotating linear +polarization.9 A similar opto-thermophoretic aligning torque can be achieved by reshaping +and steering the laser beams. Comprehensively, a line-shape beam spot can be obtained +through the digital micromirror device (DMD), which leads to a line-shape opto- +thermophoretic potential. This anisotropic potential can then capture anisotropic micro- +/nano-objects at the beam spot and rotate them until the status of minimum energy is +reached. By rotating the line-shape beam spot, a rotary anisotropic opto-thermophoretic + +Solventmolecules +Solvent +Bulkregion +Interface +Slipflow +Substrate +ili +Laserbeam +30 +126 +180.°potential well can be obtained which continuously rotates targets through the dynamic +minimum-energy status. As shown in Fig. 2b, a sliver (Ag) nanowire (12 µm × 100 nm) is +rotated in isopropyl alcohol (IPA) by a rotary one-dimensional (1D) beam spot,49 where +the offset between the position of the Ag nanowire and the minimum-energy position leads +to a trapping torque. In addition to synthesized particles, we have also achieved the rotation +of live biological cells.93 In water, parallel rotation of two yeast cells at an angular +resolution of one degree has been demonstrated (Fig. 2c). Moreover, our platform enables +the rotation of bacteria such as Escherichia coli. +Compared with other optical rotation techniques, opto-thermophoretic rotation has +shown its versatility in the dynamic rotation of low-dimensional objects of diverse sizes +and materials using a single low-power beam. In addition, the opto-thermophoretic torque +is not dictated by the refractive index contrast between the targets and surroundings, which +is superior to traditional optical tweezers. +Opto-thermoelectric rotation techniques. To date, thermoelectric fields have been +extensively applied to the transport of charged particles such as molecules, micelles, and +colloidal particles.34, 35, 66, 96 The thermoelectric transport of charged particles is controlled +by the electric force acting on the particles. Thus, the migration behaviors of the particles +can be predicted by their surface charges and the thermoelectric field generated from the +thermophoretic separation of ions.46 Nevertheless, the manipulation of colloids with +opposite surface charges in one solution is challenging due to the opposite signs of ST. +Lately, this challenge was addressed by us with the development of opto-thermoelectric +tweezers.42 By using a solution with ionic surfactants (e.g., cetyltrimethylammonium +chloride (CTAC)), different manipulations including trapping, pulling, assembling, and +printing of different micro-/nano-objects with arbitrary surface charges have been +achieved.40, 47, 55, 58, 97, 98 Specifically, the CTAC surfactant can be easily adsorbed on the +surface of most particles through electrostatic and hydrophobic interactions. This unifies +all suspended particles with positive surface charges. Meanwhile, a thermoelectric field +pointing to the hot region is built due to the thermophoretic separation of CTAC micelles +(CTA+) from anions (Cl-), which applies thermoelectric forces to the particles. 99 + +We have further extended the concept of opto-thermoelectric tweezers to an opto- +thermoelectric rotation.99 To achieve the rotation, a self-sustained thermoelectric field +should be established on target particles based on their non-uniform light-absorbing +structures (e.g., particles with half-coated metal). Due to the significant difference in light +absorption between the metal cap and the intrinsic part, a non-uniform temperature field is +established in the vicinity of the particle under illumination (Fig. 3b). Accordingly, a +thermoelectric field is produced from the separation between CTAC micelles and Cl- ions +around the particle, which then impose the thermoelectric force to the particle. The +interplay among the thermoelectric force, optical force, and Stokes drag force (Fig. 3c) +leads to a torque that drives the orbiting movements of the particle (Figs. 3d-3e). A high +rotation rate of 80 rpm can be obtained by simply increasing the optical power to 3.2 mW +for a higher thermoelectric field (Fig. 3f). + +Fig. 3 Opto-thermoelectric rotation technique. (a) 3D schematic illustration of an orbiting +metal-coated Janus particle driven by a laser beam. (b) Simulated temperature distribution + +(a) +(b) +CTACmicelle +X +Cl-ion ++ +T(K) +TE field +340 +PS +Au +330 +320 +2 +V +310 +1 μm +X +X +[(c) +Fo +FX +Fot +FY +X ++X +e) +t=0s +t=0.43s +100 +80 +RotationRate(rpm) +60 +t=0.81s +t=1.30s +40 ++X +20 +0 +3μm +1.5 +2.0 +2.5 +3.0 +Power (mW)of a 2.7-µm Au/PS Janus particle under the illumination of a 532-nm beam at a power of +2mW. The distribution of ions and established thermoelectric field are illustrated as well. +The orange and blue solid circles represent the CTAC micelles and Cl- ions, respectively. +The orange arrow denotes the direction of the thermoelectric field. (c) Schematics of the +thermoelectric force (purple), optical force (blue), and drag force (brown) acting on the +Janus particle. (d) Two-dimensional (2D) schematics of the orbital movements of the +particle powered by the beam. (e) Time-resolved dark-field images of the rotation of the +Janus particle. (f) The measured rotation rate as a function of the optical power of the beam. +Adapted with permission.99 Copyright 2020, Springer Nature. +Furthermore, opto-thermoelectric swimmers are developed on the same +platform, where rotary control with thermoelectric fields is utilized to achieve +delicate control of swimming directions. Specifically, if the laser beam is switched +for particle rotation to another loosely focused beam, the thermoelectric torque can +be turned into a thermoelectric force pointing from the PS hemisphere to the au +hemisphere to induce the translational motion of the particle. Through the on- +demand switch between two lasers, actively navigated swimming can be +demonstrated by alternately changing the propulsion and rotation states of the +particles with a feedback control system. With the capability to control particles over +all degrees of freedom in an automatic manner, opto-thermoelectric swimmers are +foreseen to open novel horizons in micro/nanorobotics. +Opto-thermocapillary rotation techniques. Marangoni flow has fuelled the development +of tiny motors ranging from nanometers to centimeters in the last two decades.100, 101 +Marangoni flow can be generated under external fields such as concentration and +temperature fields. Particularly, optical-heating-induced Marangoni flow, i.e., opto- +thermocapillary flow, has recently emerged as an effective solution for surfactant-free and +high-spatiotemporal-precision micro/nanomanipulation, including rotary control of micro- +/nano-objects.82, 102-104 As the thermocapillary flow typically occurs at the liquid-fluid +interface, the intrinsic Brownian motion will disturb the precise manipulation of objects, +especially for nano-objects below 100 nm. +Recently, we have achieved stable and fast in-plane rotation of sub-100 nm light- +absorbing particles via opto-thermocapillary forces (Fig. 4a).105 The key to opto- + +thermocapillary manipulation is the thin layer of polymers on the substrate, which can be +selectively turned into a quasi-liquid phase upon laser irradiation and lead to the +thermocapillary flow.106 In this work, a solid layer of CTAC is placed between a glass +substrate and 80-nm gold nanoparticles (AuNPs). Upon the illumination of a 660-nm laser +beam on the AuNP, a localized phase transition of CTAC is triggered by the optical heating +of the particle. As shown in Fig. 4b (top), a maximum temperature of 593.4 K is obtained +on the illuminated AuNP at the optical power of 10 mW. A very thin layer of CTAC in the +vicinity of the AuNP (<15 nm) is then heated above its phase transition temperature (505- +510 K). Therefore, a localized liquid-air interface is formed around the illuminated particle. +Owing to the multifaceted asymmetry of the AuNP, a temperature gradient exists at the +liquid/air interface contacting the AuNP. Meanwhile, thermocapillary stress was generated +on the particle’s surface because of the interfacial surface tension gradient. With dγ/dT ~- +0.097 mNm-1K-1. A considerable tangential opto-thermocapillary force can then be +obtained to trigger an in-plane rotation (Fig. 4c), while the synergy of capillary forces and +optical forces maintain the stable orbital movements (Fig. 4d). Figure 4e shows the real- +time rotation of an 80-nm AuNP under laser excitation, whose statistic displacement results +(Fig. 4f) indicate that the AuNP rotates stably in a circular orbit about the laser beam. + +This opto-thermocapillary rotation platform may open a novel path toward nanomachines. +Contrary to other existing light-powered machines, our nanomachines have suppressed +Brownian motion and could serve as fuel-free and gear-free engines to convert optical +energy into mechanical work for various nano-electro-mechanical systems. + +Fig. 4 Opto-thermocapillary rotation technique. (a) 3D schematic illustration of an orbiting +light-absorbing nanoparticle driven by a laser beam. (b) Top: Simulated 3D temperature +distribution of an 80-nm AuNP under the illumination of a 660-nm beam at a power of +10mW. Bottom: Simulated 2D temperature distribution of CTAC film near the AuNP. Scale +bar: 50 nm. (c) Simulated thermocapillary force (black squares) acting on the AuNP. The +tangential and radial components of the force are denoted by red triangles and green circles, +respectively. (d) Schematics of the thermocapillary force, optical force, and drag force +acting on the AuNP particle. (e) Time-resolved dark-field optical images of an orbiting 80 + +a) +(b) +T (K) +593.4 +Laser +Air +592.7 +FTC. +AuNP +T(K) +CTAC +Quasi-liquid +CTAC +550 +Glass +Air +500 +CTAC +450 + 400 +350 +300 +60 +IFTcl +40 +TC,r +TC,t +R +Force (fN) +Fo +20 +Fopt +FTc.r +0 +AuNP +Laser +FTC.t +-20- +-40 +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +Laser-particledistance(um) +t=3.91s +t=4.02s +0.6 +AuNP +t=os +0.3 +Laser +(uri) +0.0 +t=60s: +Laser +t=4.24s +t=4.37s +0.3 +0.6 +-0.6 +-0.3 +0.0 +0.3 +0.6 +X(μum)nm AuNP. Optical power: 6 mW. Scale bar: 1 μm. (f) Centroid tracking of the rotating +AuNP. Adapted with permission.105 Copyright 2022, American Chemical Society. +3.2. Out-of-plane rotation +Optothermal rotation techniques that enable the out-of-plane micro/nanorotation (i.e., +rotation of an object around an axis parallel to the substrate) are featured in this section. +Compared to in-plane rotation, out-of-plane rotation finds a variety of unique applications +in fields such as single-cell mechanics, 3D bio-imaging, and micro-/nano-surgery.16, 107-113 +However, out-of-plane rotation is known to be more challenging to accomplish especially +for highly symmetric or isotropic targets.114 In the past five years, a few groups have +proposed diverse strategies for out-of-plane rotation under broken symmetries generated +in light-driven temperature fields.44, 72, 75, 115-118 +Opto-thermo-electrokinetic rotation techniques. Historically, the main issue in +achieving light-driven out-of-plane rotation has centered in generating torque about an axis +perpendicular to the optical axis for rotation while simultaneously forming a trap well to +overcome Brownian motion for stable rotation. It is inherently challenging to produce +perpendicular optical torques despite the convenience of the generation of parallel optical +torques through different strategies.9 Light-driven out-of-plane rotation has been observed +occasionally,116 however, the maintenance of a stable rotation at a target location is +challenging. Moreover, it becomes more difficult to break symmetries for the rotation of +homogenous and perfectly symmetric spheres.12, 28 Recently, Xie et al. developed a dual- +beam optical tweezers system enabling the out-of-plane rotation of various cells for +microsurgeries.119-121 Nevertheless, its reliance on two high-power beams restricts its +application within the out-of-plane rotation of nano-objects, especially for ones of + +subwavelengths. Besides, the direct illumination of the two intense beams may also cause +photodamage to delicate cells. + +Fig. 5 Opto-thermo-electrokinetic rotation technique. (a) 3D schematic illustration of the +experimental setup. (b) Simulated 2D temperature distributions of the rotor, substrate, and +the surrounding liquid. Yellow arrows indicate discrete depletion forces on the rotor. (c) The +temperature effect on the dissociation of carboxylic function groups. (d) Schematics of +optothermal forces and torque on the rotor. The yellow, red, and black arrows denote the +depletion, thermoelectric, and thermo-electrokinetic forces, respectively. A net torque (black +curved arrow) can be generated on the particle at a certain position where a balance is reached +among the three forces. The temperature-dependent distributions of negative charges on the surface +of the particle and substrate are indicated by the “−” symbols. (e) Time-resolved fluorescent +images of out-of-plane rotary 2.8-µm PS particle. Optical power: 78.4 µW. Scale bar: 2 μm. + +PEG +Na +Rotor +separation +(d) +MEK. +TE +R-COO- +R-COOH +0 s +0.04 s +OWE +0.09s +0.14 s +3.0 +distance (um) +2.5 +2.0 +.5 +Critical PL +1.0 +0.5 +0.0(f) Left: Real-time dark-field optical images of a rotating 300-nm PS/Au Janus particle. +Right: Red-green-blue (RGB) intensity extracted from the dark-field optical images of the +Janus particle. The white dash rectangle in the left figure marks the selected area from which +the RGB intensity is recorded. (g) Rotation rate and laser-particle distance (DP-L) of a 2.8- +µm PS particle versus PEG concentration in 5% PBS solutions at an optical power of 196 +µW. The grey squares and black lines are experimentally measured (indicated as “EXP”) +and analytic (indicated as “ANA”) rotation rates, respectively. The grey region represents +the rotation rates calculated as the particle-substrate gap ranges from 30 to 32 nm. The +yellow diamonds and circles are measured and simulated (indicated as “SIM”) DP-L, +respectively. The dashed line marks the threshold PEG concentration (5%), where the +rotation rate reaches a maximum value. (h) The measured rotation rate of the PS particle in +5% PEG solutions with different PBS concentrations versus the optical power of the laser +beam. The dashed lines are linear fittings of the measured values. Adapted with +permission.72 Copyright 2022, American Association for the Advancement of Science. +To overcome these obstacles, we propose a universal rotation strategy based on opto- +thermo-electrokinetic coupling, which enables the out-of-plane rotation of particles with +diverse sizes, materials, and shapes using low-power and simple optics.72 The working +principle of opto-thermo-electrokinetic rotation is illustrated in Fig. 5a. The particles and +substrate are immersed in a solution containing polyethylene glycol (PEG) and phosphate- +buffered saline (PBS). A single laser beam is used to create a local temperature field +between the charged particle and a light-absorbing substrate with carboxylic acid– +terminated alkanethiol self-assembled monolayers. Under the temperature gradient, all the +ions and molecules diffuse from the hot to the cold region, leading to the charge separation +of Na+ and Cl- ions, and the concentration gradient of PEG molecules (Fig. 5b). Moreover, +the temperature gradient leads to the dissociation of acid functional groups on the +substrate‘s and particle’s surfaces (Fig. 5c),73, 74 which results in the gradients of surface +charge, respectively (Fig. 5d). Accordingly, the charge separation of Na+ and Cl- ions leads +a thermoelectric field, which exerts a repulsive thermoelectric force on the particle. +Meanwhile, the concentration gradient of the PEG molecule results in an attractive +depletion force to trap the particle around the laser beam.43, 122 More importantly, the +electrokinetic interaction between the surfaces of the particle and the substrate generates +an accumulative force with the line of action not passing through the particle’s centroid, + +which imposes a torque on the particle with the axis parallel to the substrate and finally +drive its out-of-plane rotation. Under the illumination of lower-power 532-nm beam, a 2.8- +µm PS microsphere in a 5% PEG/5% PBS solution rotates in an out-of-plane manner at the +rate of 32.0 rpm (Fig. 5e). The two bright spots are fluorescent nanobeads attached to the +particle for the observation of out-of-plane rotation. In contrast to other optical rotation +systems where the laser directly illuminates onto the object, the laser beam in our system +can be positioned away from the rotary objects to reduce the optical damage. +Our rotor platform displays general applicability to biological cells and synthetic +particles of diverse materials, sizes, and shapes. For example, the out-of-plane rotation of +a subwavelength PS/Au Janus has been achieved, whose rotation behavior is quantified by +the real-time RGB signals from the scattered light of the particle (Fig. 5f). We have also +explored the dependence of the rotation rate and DP-L on PEG and PBS concentrations, and +optical power. In detail, the rotation rate decreases with PEG concentration while DP-L +shows an opposite trend (Fig. 5g). In addition, the rotation rate gradually decreases as PBS +concentration increases from 5% to 15%. Since thermo-electrokinetic interactions increase +with temperature gradient, a higher rotation rate can be obtained using a beam with a higher +power (Fig. 5h). +In short, our platform shows advantages compared to conventional optical rotation +platforms in terms of simplicity, universality, and biocompatibility. The rotation of +homogenous and perfectly symmetric spheres is accomplished here, which has not been +achievable until now in any other light-driven rotor platforms because of their restricted +demands on the asymmetric shape12 or birefringence28 of particles. Furthermore, out-of- +plane rotation of single biological cells, which has only been implemented via a high-power +dual-beam system in the past, is accomplished here using a low-power laser beam. With +its simple optical setup, wide applicability, and low-power operation, our rotor platform +becomes promising in various scientific research and applications. +Opto-thermo-osmotic rotation techniques. Thermo-osmotic flow, generated on the +surfaces of the particles123 or substrates75 in nonuniform temperature fields, can be utilized +for the manipulation of particles resting on substrates. The flow velocity is given by124 +𝑣to = − 1 +η ∫ +𝑑𝑧𝑧ℎ(𝑧) ∇𝑇 +𝑇 +∞ +0 + (7) + + +where h is the excess specific enthalpy in the boundary layer, which is positive for common +solid substrates (e.g., glass) and generates flow towards the cold region. In contrast, some +synthetic membranes show negative enthalpies that drive the flow toward the hot region. +Therefore, thermo-osmotic flows can be modulated on demand by surface chemistry for +particle rotation.36 Lou et al have developed a model that predicts the opto-thermo-osmotic +rotation of different colloidal particles on different substrates.75 Later, Heidari et al. +experimentally observed an orientation change of a Janus particle induced by thermo- +osmotic flows from a glass slide in experiments.116 +More recently, we report a new strategy for the out-of-plane rotation of single cells +using opto-thermo-osmotic flows generated from light-absorbing substrates.115 The +experimental setup is shown in Fig. 6a. Two laser beams (785 nm and 532 nm) are focused +at the same position on the substrate for simultaneous trapping and rotation of a single cell. +The substrate is carefully designed by depositing a large amount of sub-100nm AuNPs on +a glass slide to achieve localized temperature fields upon laser irradiation as well as obtain +low surface charge for powerful thermo-osmotic flows. Meanwhile, the substrate has been +designed to have high transparency for a 785-nm laser beam for a considerable trapping +force. By rationally selecting an optical power intensity for the 785-nm beam, a single cell +can first be stably trapped by the optical gradient force (Fig. 6b). Then, upon the +illumination of the 532-nm beam, a nonuniform temperature field is established for the +generation of the thermo-osmotic flow, which powers the out-of-plane rotation of the cell +(Fig. 6c). The cross-section of the flow fields is shown in Fig. 6d. At the optical powers of +0.2 mW/µm2 (532 nm) and 1 mW/µm2 (785 nm), a rotation rate of ~1 Hz is obtained. At +higher optical powers, a higher rate can be obtained due to enhanced thermo-osmotic flows +(Fig. 6e). With its precise control of single-cell rotation, the opto-thermo-osmotic platform + +may provide an effective solution to the measurement of the ligand-receptor binding +kinetics at the single-cell level (see the inset of Fig. 6a). + +Fig. 6 Opto-thermo-osmotic rotation techniques. (a) Experimental setup. HWP: half-wave +plate. ODF: optical density filter. FL: focused lens. Inset: the receptors of the rotary cell +interacting with the ligand molecules immobilized on the substrate. BS: beam splitter. BE: +beam expander. (b) Schematics and time-resolved optical images showing the trapping of a +yeast cell (S. cerevisiae) by a 785-nm beam. (c) Schematics and time-resolved optical +images showing the stable rotation of the yeast cell with the secondary beam (532 nm). Scale +bars: 5 µm. (d) Cross-sectional view of the simulated flow velocity in the vicinity of the cell. +Scale bar: 1 μm. (e) The dependence of the rotational rate of the trapped cell on the optical + +Light +Inlet +Outlet +Substrate +F +Objective +100X +Camera +V +BS +JFL +532nm +BS +BE +HWP +ODF +785nm +BE +HWP +ODF +Trapping +t=os +t=0.6s +t=1.2s +国 +国 +C) +Rotation +t=1.5s +=21S +t=2.7s +d) +(e) +v(μm/s) +35 +Rotation +30 +2 +25 +(ZH): +20 +4 +15 +10 +5 +Substrate +0 +Laserbeamcenter +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Power(mW/um)power intensity of the 532-nm laser. The optical power intensity of the 785-nm beam is +fixed at 1 mW/μm2. Adapted with permission.115 Copyright 2022, Springer Nature. +Optothermal convective rotation techniques. Providing delicate control of the +temperature field,81 light-generated thermal convection has been extensively exploited for +the manipulation of arbitrary micro-/nano-objects, such as DNA,125, +126 silver +nanoparticles,127 and living cells.128 Recently, the optothermal convective flow has also +been exploited for the rotational manipulation of synthesized particles and biological cells. +For instance, Kumar et al. produced optothermal convective torques by heating a gold +substrate with a laser beam for the out-of-plane rotation of hexagonal-shaped particles and +single cells.118 Dai et al. proposed a novel strategy for the out-of-plane rotational +manipulation of hydrogel microstructures in additional to in-plane translational by the +synergy of optothermal convection and opto-thermocapillary flows near a surface bubble117 +Later, they also demonstrated in-plane rotation and versatile assembly of micro-objects in +different shapes based on the same convection and capillary flows stemming from surface +bubbles.129 +3.3 Potential applications +Featuring simple optics, low power, and wide applicability to different objects, the +optothermal rotation leads to a wide range of applications including single-cell mechanics, +3D bio-imaging, and micro/nanomotors. +Single-cell mechanics. Rotation of single cells is important for the identification of cellular +phenotypes, cell-cell communication, and, particularly, single-cell mechanics.107, 130 +Among different measurements of single-cell mechanics,131 cell adhesion has attracted a +lot of attention as it is highly involved in many biological processes such as viral infections. +However, the existing single-cell adhesion kinetics measurements can only give qualitative +results since they are usually performed under conditions far away from the in situ +physiological environments.132 For instance, in the study of SARS-CoV-2’s entry into host +cells, the difference in measured affinities of ACE2: S1 (the key parameter for infection) +can be as large as one order of magnitude among different methods. For instance, the +affinity measured by surface plasmon resonance assay is ~15nM133 while that by the atomic +force microscopy assay is ~120 nM134. In comparison, our optothermal rotation techniques +show the following striking advantages: 1. Our measurement was conducted in a situation + +close to the real cell adhesion process, which can hardly be achieved by previous methods +(Fig. 7a). 2. The measurement can be performed for diverse cells directly in complex +clinical samples such as human urine without any pre-treatment (Fig. 7b). 3. Both shear +affinity and tensile affinity can be measured in our platform (Fig. 7c). +3D bio-imaging. Enabling 3D interrogation of biological samples, contact-free rotation +techniques pave new ways to non-invasive 3D bio-imaging with high resolution in all +dimensions. By taking sequential images in hundreds of 2D focal planes, volumetric +imaging shows high lateral resolution. However, this type of imaging technique commonly +is insufficient to resolve sub-cellular events due to its low axial resolution (> 1 µm) caused +by the missing cone along the z-axis.135 Although advanced imaging systems such as total +internal reflection fluorescence (TIRF) and selective plane illumination microscopy +(SPIM) can further improve high lateral resolution, they still suffer from relatively low +axial resolution and limited optical overage. In contrast, out-of-plane rotation of single cells +combined with TIRF or SPIM is promising to construct 3D images of various biological +samples with high resolution in all dimensions. Whyte et al. realized out-of-plane rotation +of red blood cells and cancer cells with a fiber-based dual-beam laser trapping.19 Then, +they evolved it into an optofluidic device that enables single-cell rotation around an axis +perpendicular to the imaging plane for 3D single-cell tomography.136 However, all these +platforms show limited throughput and are not suitable to be combined with TIRF due to +the large gaps between the rotary cells and the substrates. As our optothermal rotation +techniques can rotate the living cells near the substrates (Fig. 7d), our platform displays a +high possibility to be integrated with TIRF and SPIM for high-resolution 3D reconstruction +of various living cells. Moreover, by using digital micromirror devices or spatial light +modulators, parallel rotations of multiple cells can be achieved by splitting the incident +laser into multiple laser beams, which significantly improves the throughput of the 3D +imaging.93 + + +Fig. 7 Biological applications of optothermal rotation techniques. (a) Left: Schematic +illustration of the adhesion process of a cell. With a flow of biofluids (e.g., blood, urine), +the cell will first attach to the endothelium cells as a substrate at an inclined angle. During +the rolling, the receptors on the cell and the ligands on the substrate experiences shear forces (Ft, +F’t). The curved arrow indicates the torque on the cell generated by the fluidic flow. Right: on the +optothermal rotation platform, the cell suspended in liquid is first trapped on the +functionalized substrate, mimicking the cell pre-attachment. Then, the measurement of the +cell’s shear adhesion kinetics can be obtained through the analysis of the light-driven out- +of-plane rotation behavior of the cell. (b) Optical images of bacteria and cells trapped and +rotated in clinical samples during the rolling adhesion measurement. Others: urinary +organisms that we can hardly identify. Scale bars: 5 μm. (c) Qa (τa≥t), the fraction of chitin’s +transient adhesion with lifetime τa≥t, versus t for urinary yeast cells. The dissociate constants of the +adhesion event (koff-1=6.79 s-1 and k off-1=1.56 s-1) were extracted by fitting the experimental data +with double exponential decay curves. (d) Time-resolved optical images of a rolling yeast cell. (a)- +(c) Adapted with permission.115 Copyright 2022, Springer Nature. (d) Adapted with +permission.137 Copyright 2020, Institute of Electrical and Electronics Engineers. +Micro/Nanomotors. Micro/nanomotors are micro/nanoscale devices that enable the +conversion of energy into mechanical work. Optical, electric, chemical, and magnetic +energies have been used to power micro/nanomotors for the applications of environmental +remediation, cargo delivery, and micro/nanorobotics.37, 138-141 Among them, motors driven + +Withflow +SCRAFA +Flow +E.coli +Candida +10° +Koff-1 +Koff-2 +6.79 +1.56 +10-1 +Q.(tzt) +Neutrophil +Others +102 +Urinaryyeast cells +103 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +t (s) +Os +S +S +S +5Sby optical heating (i.e., optothermal motors) show unique advantages such as high general +applicability, excellent temporal and spatial control with localized fields, and so on.72, 105, +142, 143 However, most existing optothermal motors require immersion in a liquid +environment, limiting their range of applications. For instance, liquid-state motors cannot +be utilized in devices, such as hard drives, that need to work in a dry environment. +Moreover, the precise control of a micro/nanomotor in liquid could be challenging due to +strong Brownian motion. In comparison, operating in a solid environment, our opto- +thermo-capillary nanomotors circumvent those problems and shed light on the +underdeveloped field of solid-state micro/nanomotors.105 +4. Summary and outlook +Diverse optothermal rotation techniques are developed based on different working +mechanisms, which can be implemented on different substrates (thermally +responsive or not), in different media (e.g., surfactant solutions, biological media, or +solid polymers), and to different target objects (e.g., colloidal particles or living +cells). They can either have in-plane or out-of-plane rotation frequency ranging from +Hz to kHz, which leads to various applications. We conclude this Feature Article by +proposing a general guideline for selecting suitable optothermal rotation techniques +in terms of different target objects, followed by a discussion about the existing +challenges, and future opportunities in this field. +4.1 General guideline +For the optimum performance, we provide a general guideline for the choices of +proper optothermal rotation techniques for different target objects, including +nanoparticles, microparticles, and biological cells. +Nanoparticles. The primary challenge in controlling the rotation of nanoparticles is +their prominent Brownian motion. To counter this intrinsic effect, opto- +thermoelectric, solid-state opto-thermocapillary, and opto-thermo-electrokinetic +rotation techniques can provide strong trap wells to suppress the Brownian motion +for the precise and stable rotation of nanoparticles. Specifically, opto-thermoelectric +and solid-state opto-thermocapillary rotations are designed for highly anisotropic + +and light-absorbing nanoparticles, in which nonuniform temperature fields can be +established under uniform illumination to generate rotation torques. In comparison, +opto-thermo-electrokinetic rotation is more universal for the rotation of different +types of nanoparticles regardless of their shapes, components, and light absorption. +However, high surface charges of nanoparticles are typically required. +Microparticles. Due to the less prominent Brownian motion, optothermal rotation +of microparticles can be achieved by all the means discussed in this article. We +should note that for opto-thermophoretic rotation, asymmetry in either shape or light +absorptivity of microparticles is required to produce torques for the rotation. +Furthermore, for microparticles with a size larger than 10 µm, optothermal +convective rotation can be the most suitable candidate as the convective flow offers +larger forces and torques than other optothermal rotation techniques. +Biological cells. When considering the biocompatibility of different optothermal +rotation techniques, we should point out that the solution components should be +biologically safe, and the operational optical power should be low to reduce +photodamage. Therefore, opto-thermophoretic, opto-thermo-electrokinetic, and +opto-thermo-osmotic rotation techniques are ideal candidates for the rotation of +biological cells at the single-cell level. The biological media typically contain lots +of ions that make the suspended living cells cold-seeking under a temperature field. +Accordingly, opto-thermophoretic rotation techniques are more suitable for robust +cells that can survive in less-salty environments (such as yeast cells). In contrast, +since the opto-thermo-electrokinetic and opto-thermo-osmotic rotation techniques +can offer extra forces to counteract the thermophoretic forces, they can function in +various biological medias for the stable rotation of living cells (even for some +delicate mammalian cells such as leukocytes).44 +4.2 Challenges and opportunities +The main challenges of the current optothermal rotation techniques are the potential +thermal damage and limited rotation rates. Though the optothermal rotation can +minimize phototoxicity due to the low optical power, thermal stress from optical +heating could lead to the photothermal degradation of delicate objects.144 Especially + +for optothermal convective and opto-thermo-capillary rotation techniques, higher +optical powers are typically required compared to the other optothermal rotation +techniques. In addition, it could be challenging to obtain an ultrahigh rotation rate +(e.g., GHz) on an optothermal rotation platform. Intuitively, a larger rotation rate +can be obtained from the higher optothermal torques powered by the higher optical +power. However, the increasing temperature might reach the boiling point of the +solution, where vapor bubbles can form and introduce strong Marangoni convection +to suppress the optothermal torques. One possible solution to these two issues is to +add a chip-size heat sink to lower the environmental temperature.145 Within a lower +environmental temperature, the target temperature difference for the generation of +optothermal torques can be obtained with a lower peak temperature, which can +reduce the thermal damage and avoid the formation of vapor bubbles40, 93 +Optothermal rotation of objects down to a few nanometers or even atomic scale +can be attractive to the study of virology and particle physics. Multiple optothermal +effects have proven to be able to direct the migration of the particles with these small +sizes.37 To implement the rotation of these extremely tiny objects, one promising +strategy is to use the femtosecond laser in optothermal rotation techniques. +Specifically, a femtosecond laser pulse can generate a highly localized and steeper +temperature field. Under a higher temperature gradient, a more robust trap well and +a larger optothermal torque can be produced, which is promising to achieve stable +rotary control of nano-/atomic-scale items. Besides, due to the high energy density +of a femtosecond laser beam, it can directly heat the liquid through nonlinear +interactions to establish temperature fields. Thus, all existing optothermal rotation +techniques can work without the requirement of light-absorbing substrates or +objects, which further broadens the applicability of optothermal rotations. +We believe that, once the above-mentioned bottlenecks are overcome, more +applications can be implemented by optothermal rotation. For instance, with the +capability to noninvasively rotate smaller objects such as viruses, optothermal +rotation can be used in single-virus mechanics. Specifically, the rolling adhesion of +SARS-CoV-2 (~0.1 µm) can be measured to help better understand the viral +infection +process. +Another +promising +application +is +cell +classification. + +Distinguishing different types of cells of high similarity is extremely challenging for +existing optical imaging techniques. Optothermal rotation of cells integrated into +standard optical microscopy will provide a collection of multiangle images at a high +efficiency and, in combination with machine-learning-enhanced image analysis, +could effectively reduce the quantity of the training samples and improve +classification accuracy. +Author Contributions +H.D. and Y.Z. conceived the idea, designed the frame, and wrote the manuscript. +Z.C. and C.P. assisted with the manuscript revision. Y.Z. supervised the project. +Conflicts of interest +The authors declare no competing interests. +Acknowledgments +H.D., Z.C., and Y.Z. acknowledge the financial support of the National Institute of +General Medical Sciences of the National Institutes of Health (R01GM146962) and +the National Science Foundation (NSF-ECCS-2001650). C.P. additionally +acknowledges the financial support of UT Austin Undergraduate Research +Fellowship. +Notes and references + +1. +D. G. Grier, Nature, 2003, 424, 810-816. +2. +A. Ashkin, Nature, 1987, 330, 608-609. +3. +P. H. Jones, O. M. Maragò and G. Volpe, Optical Tweezers: Principles and Applications, +Cambridge University Press, Cambridge, 2015. +4. +C. J. Bustamante, Y. R. Chemla, S. Liu and M. D. Wang, Nature Reviews Methods Primers, 2021, +1, 25. +5. +L.-M. Zhou, Y. Shi, X. Zhu, G. Hu, G. Cao, J. Hu and C.-W. Qiu, ACS Nano, 2022, 16, 13264- +13278. + +6. +A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, Opt. Lett., 1986, 11, 288-290. +7. +A. 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Eng., 2016, 100, 170-178. + + diff --git a/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/load_file.txt b/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..746f4e4e170b011a18368d0339ec89daf4d59d04 --- /dev/null +++ b/UtE3T4oBgHgl3EQfEgmt/content/tmp_files/load_file.txt @@ -0,0 +1,2000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf,len=1999 +page_content='Optothermal rotation of micro-/nano-objects in liquids Authors: Hongru Ding,1 Zhihan Chen,2 Carolina Ponce,1 and Yuebing Zheng1,2,* Affiliations: 1Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2Materials Science & Engineering Program and Texas Materials Institute, The University of Texas at Austin, Austin, TX 78712, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Correspondence to: zheng@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Abstract Controllable rotation of micro-/nano-objects provides tremendous opportunities for cellular biology, three-dimensional (3D) imaging, and micro/nanorobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Among different rotation techniques, optical rotation is particularly attractive due to its contactless and fuel-free operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' However, optical rotation precision is typically impaired by the intrinsic optical heating of the target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optothermal rotation, which harnesses light-modulated thermal effects, features simpler optics, lower operational power, and higher applicability to various objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In this Feature Article, we discuss the recent progress of optothermal rotation with a focus on work from our research group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We categorize the various rotation techniques based on distinct physical mechanisms, including thermophoresis, thermoelectricity, thermo-electrokinetics, thermo- osmosis, thermal convection, and thermo-capillarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Benefiting from the different rotation modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', in-plane and out-of-plane rotation), diverse applications in single-cell mechanics, 3D bio- imaging, and micro/nanomotors are demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We conclude the article with our perspectives on the operating guidelines, existing challenges, and future directions of optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Introduction Awarded the Nobel Prize in Physics 2018, optical tweezers have proved to be an effective instrument for non-contact spatial manipulation of micro/nanoparticles as well as living biological samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='1-8 On the basis of optical tweezers, various optical rotation platforms have been developed for exquisite control of micro/nanoscale targets over their rotational degrees of freedom using light (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', laser beams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='9-11 The light-powered rotation, termed optical rotation, holds significant application potential in nano-torque sensing,12-14 nanosurgery,15, 16 micro/nanofluidic systems,17, 18 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' To achieve optical rotation, dynamic intensity profile,19, 20 nonlinear polarization,12, 21, 22 optical angular momenta,23, 24 or radiation pressure25 is typically utilized to produce asymmetric light-matter interactions for the generation of optical torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition, sophisticated designs are also required on the rotating targets, which typically possess on-demand geometry, optical birefringence, or specific compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='26-29 The rigorous requirements for light beams and properties of particles largely restrict the broader applications of optical rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, the optical rotation of nanoparticles is extremely challenging as optical forces and torques decrease substantially with particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 Though high-power light beams can be used for rotation at the nanoscale level, the resultant optical heating will lead to strong Brownian motions that significantly reduce the frequency stability of the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='30 To overcome the bottlenecks of complex optics, limited applicability, and heat-induced instability of optical rotation, heat-mediated optical rotation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', optothermal rotation, has been developed for rotary control of micro-/nano-objects by leveraging optical heating to generate thermal forces and torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Over the past two decades, researchers have established multiple theories to describe the translational migration of colloids and living objects under temperature gradient fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='31-36 Consequently, different optothermal manipulation techniques have been developed,37-52 leading to applications in nanofabrication,53-58 chemical and biological sensing,59-61 and cargo delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='62, 63 Based on this, optothermal rotation has been further developed recently for precise and versatile rotations of colloidal particles and living cells with striking advantages such as low operation power, high applicability, and long working distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Through the synergy of diverse light-induced thermal forces and/or optical forces, researchers, including us, manage to rotate objects of diverse sizes, materials, and shapes with low-power laser beams and simple optics for the growing demands in sophisticated biological measurements, imaging, and the development of nano-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This Feature Article focuses on the development of the optothermal rotation of micro/nanoscale objects in liquids (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We first introduce the fundamentals of different thermal forces that can drive stable optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Then, different approaches to achieving in-plane and out-of-plane rotation in respect of the substrates are reviewed based on their working mechanisms and various potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Finally, we provide our perspectives on existing challenges and future directions of optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fundamental mechanisms Different physical mechanisms for optothermal rotation in liquids are introduced in this section, including thermophoresis, thermoelectricity, thermo-electrokinetics, thermo-osmosis, thermal convection, and thermo-capillarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The thermal forces originating from those optothermal phenomena are employed to control the translational movements of the targets and drive their stable rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 1 Overview of optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Orbiting, in-plane and out-of-plane spinning of different colloids and living objects can stem from a variety of optothermal phenomena Thermo- Thermo- osmosis electro- electricity kinetics convection Thermo- Thermo- Thermo- phoresis capillarity Thermo- Single-cell Micro/nano mechanics motors 3D bio-imagingincluding thermophoresis, thermoelectricity, thermo-electrokinetics, thermo-osmosis, thermal convection, and thermo-capillarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optothermal rotation is promising in opening new paths toward single-cell mechanics, 3D bio-imaging, and micro/nanomotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thermophoresis Thermophoresis (also known as thermodiffusion or the Soret effect) is defined as the directed migration of tiny particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', ions, molecules, and colloidal particles) along a temperature gradient in liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='64-67 The temperature gradient functions as a general force that directs the suspended particles to a cold or warm region at a certain velocity given by 𝒖 = −𝐷𝑇∇𝑇 (1) where DT is the thermophoretic mobility and ∇t denotes the temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' in a nonuniform temperature field, the movement of the particles is also affected by Brownian diffusion which competes with thermodiffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thus, Soret coefficient (ST = DT /D) is proposed to describe the thermophoretic migration in a general manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With ST > 0 (OR ST < 0), the suspended particles show thermophobicity (or thermophilicity) and move toward the cold (or hot) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition, a large ST in magnitude means that the thermophoresis dominates over the Brownian motion, indicating that the particle can have more directional motion under temperature gradient fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thermoelectricity Liquid thermoelectricity describes the generation of an electric field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', thermoelectric field, from the charge separation of ions in electrolyte solutions under a temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='38 In a thermal equilibrium state, cations couple with anions due to electrostatic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' While in a non-equilibrium state, a thermal gradient drives cations and anions with different speeds and directions, depending on the ions’ size and solvation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='68 Additionally, the diffusion of molecules with high polarity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', water) under a certain temperature can also lead to nanoscale separation of atoms with different electronegativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='69 These spatially separated charges (ions and atoms with partial charges) lead to a bulk thermoelectric field given by 𝑬𝐓𝐄 = ∫ e(𝑛+ − 𝑛−) dz ε (2) where n+ and n- are volumetric number densities of cations and anions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' ε denotes the solvent permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Since most colloids have surface charges, a thermoelectric field can be employed to regulate the translational and rotational motions of the colloids through electrostatic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 Thermo-electrokinetics In analogy with optical electrokinetics, where the migration of colloids is induced by the non-uniform electric field based on a photoconductive substrate,70, 71 thermo- electrokinetics describes a similar electrokinetic (EK) migration that stems from the electric fields from the temperature-responsive substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='44, 72 Most colloids are negatively charged due to their ionized acid groups on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='31 Similarly, substrates can carry surface charges by coating a thin layer of acid groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', carboxylic acids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Particularly, the decrease of temperature generally promotes the dissociation of the acid molecules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', COO- and H+) according to the Van’t Hoff equation73, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Therefore, the surface-charge gradients of colloids and substrates can be tuned by the temperature field, which leads to an EK torque that can power the rotation of the colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The EK force is given by 𝑭𝐄𝐊 = ∮ 𝜎𝑟𝑑𝐴𝑟∬ 𝑬∥𝑑𝑥𝑑𝑦 (𝟑) where σT is the surface charge density of the colloidal particle, dAr is the differential area element on the particle surface, and 𝑬∥ is the parallel component of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='4 Thermo-osmosis Thermo-osmosis describes the flow parallel with a surface under a temperature gradient, which is typically directed toward higher temperature regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Specifically, excess hydrostatic pressure can be induced by a temperature gradient within an electric double layer near the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The gradient of the hydrostatic pressure is opposite to the temperature gradient and leads to a creeping flow parallel to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thus, when a colloidal particle stays in a temperature field, the force from the thermo-osmotic flow drives the particle moving in the opposite direction of the flow at the velocity31 𝒖 = − εζ2∇𝑇 3η𝑇 (4) where η is the solvent viscosity, and ζ is the surface potential of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, when a substrate is heated, the thermo-osmotic flow in the vicinity of the substrate can also be used to direct and rotate neighboring micro/nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='36, 75, 76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 Thermal convection Thermal convection, known as natural convection, results from buoyance forces exerting on fluids with heat-induced density variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='77 Like the violent vibration of atoms in solids at high temperatures,78, 79 fluid molecules scatter and separate in the vicinity of a hot spot, causing the fluid to be less dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Due to buoyance forces, the less-dense fluid moves upward, and the cooler fluid gets denser and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Meanwhile, the surrounding fluid moves toward the hot spot due to fluid continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thermal convection is typically bulky and not suitable for single-particle manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Therefore, optical heating upon laser illumination on light-absorbing nanostructures has been proposed to achieve localized temperature gradient fields for accurate control of thermal convection flows and delicate micro/nano-manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='80, 81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 Thermo-capillarity Thermo-capillary flow is one type of Marangoni effect: mass transfer along a fluid-fluid interface happens due to the gradient of surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Surface tension can be tuned by concentration, temperature, and electrical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The cool liquid that has a high surface tension can pull the surrounding warmer liquid that has a lower surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This leads to a thermo-capillary flow, whose velocity is given by 𝒖 = d𝛾 d𝑇 ∇𝑇 η (𝟓) where γ is the interfacial surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Due to its capability of rapid mass transfer, the optothermo-capillary flow at fluid-liquid interfaces near micro/nanostructures has been utilized for particle manipulation, digital fabrication, sensing, and energy harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='82-87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optothermal rotation techniques Different applications of optothermal rotation are demonstrated whereby rational management of light, heat, and solutions in optothermal fluidic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This Feature article focuses on the contributions of Zheng Research Group to the recent progress in optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' According to the relationship between the rotation axis and substrate, optothermal rotation can be classified into in-plane optothermal rotation and out-of-plane optothermal rotation, which both lead to different types of applications and are discussed separately in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In-plane rotation Opto-thermophoretic rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Opto-thermophoresis has been widely used in micro/nano-manipulation by tailoring temperature gradient fields and managing ST of target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='88 Duhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' demonstrated on-demand opto- thermophoretic accumulation and depletion of DNA molecules by manipulating ST with temperature and salt concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='89 A similar dual manipulation of synthesized nanoparticles was achieved by Weinert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' using opto-thermophoretic flows in the vicinity of microparticles with different ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='90 Later, Cichos et al demonstrated the trapping and directed swimming of nanoparticles with thermophoretic force fields generated from the optical heating of gold micro/nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='91, 92 More recently, we developed opto-thermophoretic tweezers for dynamic and low-power manipulation of micro-/nano-objects including lipid vesicles which are challenging to be trapped by optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='41, 48 Despite the tremendous progress in opto-thermophoretic manipulation, opto-thermophoretic rotation is still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' To fill this gap, we have developed opto-thermophoretic platforms for both rotational and translational manipulation of synthesized and biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='49, 93 The working principle of our platforms is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' First, a real-time tailorable and reconfigurable temperature field is established upon the illumination of a laser beam on a uniform light-absorbing substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2a(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Since the alignment of solvent molecules at the liquid-particle interface depends on the surrounding temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2a(ii)), a gradient of interfacial entropy is then generated under the light-generated temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' According to Anderson’s model, the entropy gradient causes an interfacial slip flow that exerts a thermophoretic force on the particle’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='95 The thermophoretic velocity is given by32 𝒖 = −∇𝑇 ε 2η𝑇 2Λ𝑙 2Λ𝑙 + Λ𝑝 (1 + ∂lnε ∂ln𝑇)𝜁2 (6) where Λ𝑙 and Λp are the thermal conductivities of the liquid and the particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The permittivity term, ∂lnε/∂ln𝑇, is a function of interfacial entropy and the key for the modulation of opto-thermophoretic movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For instance, we achieved on-demand opto-thermophoretic trapping of 1-μm polystyrene (PS) sphere in water as illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2a(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2 Opto-thermophoretic rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) Working principle of opto- thermophoretic manipulation of colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Opto-thermophoretic rotation of a silver nanowire by a thermoplasmonic with a 532-nm laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Opto-thermophoretic rotation of two live yeast cells in water with two line-shaped laser beams at the wavelength of 532 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The light-absorb substrate is the same as (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bars: (b) 5 µm and (c) 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a)-(b) Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='49 Copyright 2018, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='93 Copyright 2017, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The rotation of micro/nanoscale colloids can be achieved on this platform through geometric design and dynamic control of opto-thermophoretic potentials for torque generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The optical rotation of metallic anisotropic nanoparticles has been achieved by exploiting optical aligning torque produced by laser beams with rotating linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='9 A similar opto-thermophoretic aligning torque can be achieved by reshaping and steering the laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Comprehensively, a line-shape beam spot can be obtained through the digital micromirror device (DMD), which leads to a line-shape opto- thermophoretic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This anisotropic potential can then capture anisotropic micro- /nano-objects at the beam spot and rotate them until the status of minimum energy is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' By rotating the line-shape beam spot, a rotary anisotropic opto-thermophoretic Solventmolecules Solvent Bulkregion Interface Slipflow Substrate ili Laserbeam 30 126 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='°potential well can be obtained which continuously rotates targets through the dynamic minimum-energy status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2b, a sliver (Ag) nanowire (12 µm × 100 nm) is rotated in isopropyl alcohol (IPA) by a rotary one-dimensional (1D) beam spot,49 where the offset between the position of the Ag nanowire and the minimum-energy position leads to a trapping torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition to synthesized particles, we have also achieved the rotation of live biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='93 In water, parallel rotation of two yeast cells at an angular resolution of one degree has been demonstrated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, our platform enables the rotation of bacteria such as Escherichia coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Compared with other optical rotation techniques, opto-thermophoretic rotation has shown its versatility in the dynamic rotation of low-dimensional objects of diverse sizes and materials using a single low-power beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition, the opto-thermophoretic torque is not dictated by the refractive index contrast between the targets and surroundings, which is superior to traditional optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Opto-thermoelectric rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' To date, thermoelectric fields have been extensively applied to the transport of charged particles such as molecules, micelles, and colloidal particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='34, 35, 66, 96 The thermoelectric transport of charged particles is controlled by the electric force acting on the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thus, the migration behaviors of the particles can be predicted by their surface charges and the thermoelectric field generated from the thermophoretic separation of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='46 Nevertheless, the manipulation of colloids with opposite surface charges in one solution is challenging due to the opposite signs of ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Lately, this challenge was addressed by us with the development of opto-thermoelectric tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='42 By using a solution with ionic surfactants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', cetyltrimethylammonium chloride (CTAC)), different manipulations including trapping, pulling, assembling, and printing of different micro-/nano-objects with arbitrary surface charges have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='40, 47, 55, 58, 97, 98 Specifically, the CTAC surfactant can be easily adsorbed on the surface of most particles through electrostatic and hydrophobic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This unifies all suspended particles with positive surface charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Meanwhile, a thermoelectric field pointing to the hot region is built due to the thermophoretic separation of CTAC micelles (CTA+) from anions (Cl-), which applies thermoelectric forces to the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 99 We have further extended the concept of opto-thermoelectric tweezers to an opto- thermoelectric rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='99 To achieve the rotation, a self-sustained thermoelectric field should be established on target particles based on their non-uniform light-absorbing structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', particles with half-coated metal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Due to the significant difference in light absorption between the metal cap and the intrinsic part, a non-uniform temperature field is established in the vicinity of the particle under illumination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Accordingly, a thermoelectric field is produced from the separation between CTAC micelles and Cl- ions around the particle, which then impose the thermoelectric force to the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The interplay among the thermoelectric force, optical force, and Stokes drag force (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3c) leads to a torque that drives the orbiting movements of the particle (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3d-3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' A high rotation rate of 80 rpm can be obtained by simply increasing the optical power to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 mW for a higher thermoelectric field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3 Opto-thermoelectric rotation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) 3D schematic illustration of an orbiting metal-coated Janus particle driven by a laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Simulated temperature distribution (a) (b) CTACmicelle X Cl-ion + T(K) TE field 340 PS Au 330 320 2 V 310 1 μm X X [(c) Fo FX Fot FY X +X e) t=0s t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='43s 100 80 RotationRate(rpm) 60 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='81s t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='30s 40 +X 20 0 3μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 Power (mW)of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='7-µm Au/PS Janus particle under the illumination of a 532-nm beam at a power of 2mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The distribution of ions and established thermoelectric field are illustrated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The orange and blue solid circles represent the CTAC micelles and Cl- ions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The orange arrow denotes the direction of the thermoelectric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Schematics of the thermoelectric force (purple), optical force (blue), and drag force (brown) acting on the Janus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Two-dimensional (2D) schematics of the orbital movements of the particle powered by the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (e) Time-resolved dark-field images of the rotation of the Janus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (f) The measured rotation rate as a function of the optical power of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='99 Copyright 2020, Springer Nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Furthermore, opto-thermoelectric swimmers are developed on the same platform, where rotary control with thermoelectric fields is utilized to achieve delicate control of swimming directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Specifically, if the laser beam is switched for particle rotation to another loosely focused beam, the thermoelectric torque can be turned into a thermoelectric force pointing from the PS hemisphere to the au hemisphere to induce the translational motion of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Through the on- demand switch between two lasers, actively navigated swimming can be demonstrated by alternately changing the propulsion and rotation states of the particles with a feedback control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With the capability to control particles over all degrees of freedom in an automatic manner, opto-thermoelectric swimmers are foreseen to open novel horizons in micro/nanorobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Opto-thermocapillary rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Marangoni flow has fuelled the development of tiny motors ranging from nanometers to centimeters in the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='100, 101 Marangoni flow can be generated under external fields such as concentration and temperature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Particularly, optical-heating-induced Marangoni flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', opto- thermocapillary flow, has recently emerged as an effective solution for surfactant-free and high-spatiotemporal-precision micro/nanomanipulation, including rotary control of micro- /nano-objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='82, 102-104 As the thermocapillary flow typically occurs at the liquid-fluid interface, the intrinsic Brownian motion will disturb the precise manipulation of objects, especially for nano-objects below 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Recently, we have achieved stable and fast in-plane rotation of sub-100 nm light- absorbing particles via opto-thermocapillary forces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='105 The key to opto- thermocapillary manipulation is the thin layer of polymers on the substrate, which can be selectively turned into a quasi-liquid phase upon laser irradiation and lead to the thermocapillary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='106 In this work, a solid layer of CTAC is placed between a glass substrate and 80-nm gold nanoparticles (AuNPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Upon the illumination of a 660-nm laser beam on the AuNP, a localized phase transition of CTAC is triggered by the optical heating of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4b (top), a maximum temperature of 593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='4 K is obtained on the illuminated AuNP at the optical power of 10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' A very thin layer of CTAC in the vicinity of the AuNP (<15 nm) is then heated above its phase transition temperature (505- 510 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Therefore, a localized liquid-air interface is formed around the illuminated particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Owing to the multifaceted asymmetry of the AuNP, a temperature gradient exists at the liquid/air interface contacting the AuNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Meanwhile, thermocapillary stress was generated on the particle’s surface because of the interfacial surface tension gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With dγ/dT ~- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='097 mNm-1K-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' A considerable tangential opto-thermocapillary force can then be obtained to trigger an in-plane rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4c), while the synergy of capillary forces and optical forces maintain the stable orbital movements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Figure 4e shows the real- time rotation of an 80-nm AuNP under laser excitation, whose statistic displacement results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4f) indicate that the AuNP rotates stably in a circular orbit about the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' This opto-thermocapillary rotation platform may open a novel path toward nanomachines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Contrary to other existing light-powered machines, our nanomachines have suppressed Brownian motion and could serve as fuel-free and gear-free engines to convert optical energy into mechanical work for various nano-electro-mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4 Opto-thermocapillary rotation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) 3D schematic illustration of an orbiting light-absorbing nanoparticle driven by a laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Top: Simulated 3D temperature distribution of an 80-nm AuNP under the illumination of a 660-nm beam at a power of 10mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Bottom: Simulated 2D temperature distribution of CTAC film near the AuNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bar: 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Simulated thermocapillary force (black squares) acting on the AuNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The tangential and radial components of the force are denoted by red triangles and green circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Schematics of the thermocapillary force, optical force, and drag force acting on the AuNP particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (e) Time-resolved dark-field optical images of an orbiting 80 a) (b) T (K) 593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='4 Laser Air 592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='7 FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' AuNP T(K) CTAC Quasi-liquid CTAC 550 Glass Air 500 CTAC 450 400 350 300 60 IFTcl 40 TC,r TC,t R Force (fN) Fo 20 Fopt FTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='r 0 AuNP Laser FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='t 20- 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 Laser-particledistance(um) t=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='91s t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='02s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 AuNP t=os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 Laser (uri) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 t=60s: Laser t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='24s t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='37s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 X(μum)nm AuNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optical power: 6 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bar: 1 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (f) Centroid tracking of the rotating AuNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='105 Copyright 2022, American Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Out-of-plane rotation Optothermal rotation techniques that enable the out-of-plane micro/nanorotation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', rotation of an object around an axis parallel to the substrate) are featured in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Compared to in-plane rotation, out-of-plane rotation finds a variety of unique applications in fields such as single-cell mechanics, 3D bio-imaging, and micro-/nano-surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='16, 107-113 However, out-of-plane rotation is known to be more challenging to accomplish especially for highly symmetric or isotropic targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='114 In the past five years, a few groups have proposed diverse strategies for out-of-plane rotation under broken symmetries generated in light-driven temperature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='44, 72, 75, 115-118 Opto-thermo-electrokinetic rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Historically, the main issue in achieving light-driven out-of-plane rotation has centered in generating torque about an axis perpendicular to the optical axis for rotation while simultaneously forming a trap well to overcome Brownian motion for stable rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' It is inherently challenging to produce perpendicular optical torques despite the convenience of the generation of parallel optical torques through different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='9 Light-driven out-of-plane rotation has been observed occasionally,116 however, the maintenance of a stable rotation at a target location is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, it becomes more difficult to break symmetries for the rotation of homogenous and perfectly symmetric spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='12, 28 Recently, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' developed a dual- beam optical tweezers system enabling the out-of-plane rotation of various cells for microsurgeries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='119-121 Nevertheless, its reliance on two high-power beams restricts its application within the out-of-plane rotation of nano-objects, especially for ones of subwavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Besides, the direct illumination of the two intense beams may also cause photodamage to delicate cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5 Opto-thermo-electrokinetic rotation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) 3D schematic illustration of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Simulated 2D temperature distributions of the rotor, substrate, and the surrounding liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Yellow arrows indicate discrete depletion forces on the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) The temperature effect on the dissociation of carboxylic function groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Schematics of optothermal forces and torque on the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The yellow, red, and black arrows denote the depletion, thermoelectric, and thermo-electrokinetic forces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' A net torque (black curved arrow) can be generated on the particle at a certain position where a balance is reached among the three forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The temperature-dependent distributions of negative charges on the surface of the particle and substrate are indicated by the “−” symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (e) Time-resolved fluorescent images of out-of-plane rotary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='8-µm PS particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optical power: 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='4 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bar: 2 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' PEG Na Rotor separation (d) MEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' TE R-COO- R-COOH 0 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='04 s OWE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='09s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='14 s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 distance (um) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 Critical PL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0(f) Left: Real-time dark-field optical images of a rotating 300-nm PS/Au Janus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Right: Red-green-blue (RGB) intensity extracted from the dark-field optical images of the Janus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The white dash rectangle in the left figure marks the selected area from which the RGB intensity is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (g) Rotation rate and laser-particle distance (DP-L) of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='8- µm PS particle versus PEG concentration in 5% PBS solutions at an optical power of 196 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The grey squares and black lines are experimentally measured (indicated as “EXP”) and analytic (indicated as “ANA”) rotation rates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The grey region represents the rotation rates calculated as the particle-substrate gap ranges from 30 to 32 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The yellow diamonds and circles are measured and simulated (indicated as “SIM”) DP-L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The dashed line marks the threshold PEG concentration (5%), where the rotation rate reaches a maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (h) The measured rotation rate of the PS particle in 5% PEG solutions with different PBS concentrations versus the optical power of the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The dashed lines are linear fittings of the measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='72 Copyright 2022, American Association for the Advancement of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' To overcome these obstacles, we propose a universal rotation strategy based on opto- thermo-electrokinetic coupling, which enables the out-of-plane rotation of particles with diverse sizes, materials, and shapes using low-power and simple optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='72 The working principle of opto-thermo-electrokinetic rotation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The particles and substrate are immersed in a solution containing polyethylene glycol (PEG) and phosphate- buffered saline (PBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' A single laser beam is used to create a local temperature field between the charged particle and a light-absorbing substrate with carboxylic acid– terminated alkanethiol self-assembled monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Under the temperature gradient, all the ions and molecules diffuse from the hot to the cold region, leading to the charge separation of Na+ and Cl- ions, and the concentration gradient of PEG molecules (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, the temperature gradient leads to the dissociation of acid functional groups on the substrate‘s and particle’s surfaces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5c),73, 74 which results in the gradients of surface charge, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Accordingly, the charge separation of Na+ and Cl- ions leads a thermoelectric field, which exerts a repulsive thermoelectric force on the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Meanwhile, the concentration gradient of the PEG molecule results in an attractive depletion force to trap the particle around the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='43, 122 More importantly, the electrokinetic interaction between the surfaces of the particle and the substrate generates an accumulative force with the line of action not passing through the particle’s centroid, which imposes a torque on the particle with the axis parallel to the substrate and finally drive its out-of-plane rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Under the illumination of lower-power 532-nm beam, a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='8- µm PS microsphere in a 5% PEG/5% PBS solution rotates in an out-of-plane manner at the rate of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 rpm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The two bright spots are fluorescent nanobeads attached to the particle for the observation of out-of-plane rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In contrast to other optical rotation systems where the laser directly illuminates onto the object, the laser beam in our system can be positioned away from the rotary objects to reduce the optical damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Our rotor platform displays general applicability to biological cells and synthetic particles of diverse materials, sizes, and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For example, the out-of-plane rotation of a subwavelength PS/Au Janus has been achieved, whose rotation behavior is quantified by the real-time RGB signals from the scattered light of the particle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We have also explored the dependence of the rotation rate and DP-L on PEG and PBS concentrations, and optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In detail, the rotation rate decreases with PEG concentration while DP-L shows an opposite trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition, the rotation rate gradually decreases as PBS concentration increases from 5% to 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Since thermo-electrokinetic interactions increase with temperature gradient, a higher rotation rate can be obtained using a beam with a higher power (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 5h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In short, our platform shows advantages compared to conventional optical rotation platforms in terms of simplicity, universality, and biocompatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The rotation of homogenous and perfectly symmetric spheres is accomplished here, which has not been achievable until now in any other light-driven rotor platforms because of their restricted demands on the asymmetric shape12 or birefringence28 of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Furthermore, out-of- plane rotation of single biological cells, which has only been implemented via a high-power dual-beam system in the past, is accomplished here using a low-power laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With its simple optical setup, wide applicability, and low-power operation, our rotor platform becomes promising in various scientific research and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Opto-thermo-osmotic rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thermo-osmotic flow, generated on the surfaces of the particles123 or substrates75 in nonuniform temperature fields, can be utilized for the manipulation of particles resting on substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The flow velocity is given by124 𝑣to = − 1 η ∫ 𝑑𝑧𝑧ℎ(𝑧) ∇𝑇 𝑇 ∞ 0 (7) where h is the excess specific enthalpy in the boundary layer, which is positive for common solid substrates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', glass) and generates flow towards the cold region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In contrast, some synthetic membranes show negative enthalpies that drive the flow toward the hot region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Therefore, thermo-osmotic flows can be modulated on demand by surface chemistry for particle rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='36 Lou et al have developed a model that predicts the opto-thermo-osmotic rotation of different colloidal particles on different substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='75 Later, Heidari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' experimentally observed an orientation change of a Janus particle induced by thermo- osmotic flows from a glass slide in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='116 More recently, we report a new strategy for the out-of-plane rotation of single cells using opto-thermo-osmotic flows generated from light-absorbing substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='115 The experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Two laser beams (785 nm and 532 nm) are focused at the same position on the substrate for simultaneous trapping and rotation of a single cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The substrate is carefully designed by depositing a large amount of sub-100nm AuNPs on a glass slide to achieve localized temperature fields upon laser irradiation as well as obtain low surface charge for powerful thermo-osmotic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Meanwhile, the substrate has been designed to have high transparency for a 785-nm laser beam for a considerable trapping force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' By rationally selecting an optical power intensity for the 785-nm beam, a single cell can first be stably trapped by the optical gradient force (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Then, upon the illumination of the 532-nm beam, a nonuniform temperature field is established for the generation of the thermo-osmotic flow, which powers the out-of-plane rotation of the cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The cross-section of the flow fields is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' At the optical powers of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 mW/µm2 (532 nm) and 1 mW/µm2 (785 nm), a rotation rate of ~1 Hz is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' At higher optical powers, a higher rate can be obtained due to enhanced thermo-osmotic flows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With its precise control of single-cell rotation, the opto-thermo-osmotic platform may provide an effective solution to the measurement of the ligand-receptor binding kinetics at the single-cell level (see the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 6 Opto-thermo-osmotic rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' HWP: half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' ODF: optical density filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' FL: focused lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Inset: the receptors of the rotary cell interacting with the ligand molecules immobilized on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' BS: beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' BE: beam expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Schematics and time-resolved optical images showing the trapping of a yeast cell (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' cerevisiae) by a 785-nm beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Schematics and time-resolved optical images showing the stable rotation of the yeast cell with the secondary beam (532 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bars: 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Cross-sectional view of the simulated flow velocity in the vicinity of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bar: 1 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (e) The dependence of the rotational rate of the trapped cell on the optical Light Inlet Outlet Substrate F Objective 100X Camera V BS JFL 532nm BS BE HWP ODF 785nm BE HWP ODF Trapping t=os t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6s t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2s 国 国 C) Rotation t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5s =21S t=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='7s d) (e) v(μm/s) 35 Rotation 30 2 25 (ZH): 20 4 15 10 5 Substrate 0 Laserbeamcenter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 Power(mW/um)power intensity of the 532-nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The optical power intensity of the 785-nm beam is fixed at 1 mW/μm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='115 Copyright 2022, Springer Nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optothermal convective rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Providing delicate control of the temperature field,81 light-generated thermal convection has been extensively exploited for the manipulation of arbitrary micro-/nano-objects, such as DNA,125, 126 silver nanoparticles,127 and living cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='128 Recently, the optothermal convective flow has also been exploited for the rotational manipulation of synthesized particles and biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For instance, Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' produced optothermal convective torques by heating a gold substrate with a laser beam for the out-of-plane rotation of hexagonal-shaped particles and single cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='118 Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' proposed a novel strategy for the out-of-plane rotational manipulation of hydrogel microstructures in additional to in-plane translational by the synergy of optothermal convection and opto-thermocapillary flows near a surface bubble117 Later, they also demonstrated in-plane rotation and versatile assembly of micro-objects in different shapes based on the same convection and capillary flows stemming from surface bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='129 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='3 Potential applications Featuring simple optics, low power, and wide applicability to different objects, the optothermal rotation leads to a wide range of applications including single-cell mechanics, 3D bio-imaging, and micro/nanomotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Single-cell mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Rotation of single cells is important for the identification of cellular phenotypes, cell-cell communication, and, particularly, single-cell mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='107, 130 Among different measurements of single-cell mechanics,131 cell adhesion has attracted a lot of attention as it is highly involved in many biological processes such as viral infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' However, the existing single-cell adhesion kinetics measurements can only give qualitative results since they are usually performed under conditions far away from the in situ physiological environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='132 For instance, in the study of SARS-CoV-2’s entry into host cells, the difference in measured affinities of ACE2: S1 (the key parameter for infection) can be as large as one order of magnitude among different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For instance, the affinity measured by surface plasmon resonance assay is ~15nM133 while that by the atomic force microscopy assay is ~120 nM134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In comparison, our optothermal rotation techniques show the following striking advantages: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Our measurement was conducted in a situation close to the real cell adhesion process, which can hardly be achieved by previous methods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The measurement can be performed for diverse cells directly in complex clinical samples such as human urine without any pre-treatment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Both shear affinity and tensile affinity can be measured in our platform (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 3D bio-imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Enabling 3D interrogation of biological samples, contact-free rotation techniques pave new ways to non-invasive 3D bio-imaging with high resolution in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' By taking sequential images in hundreds of 2D focal planes, volumetric imaging shows high lateral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' However, this type of imaging technique commonly is insufficient to resolve sub-cellular events due to its low axial resolution (> 1 µm) caused by the missing cone along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='135 Although advanced imaging systems such as total internal reflection fluorescence (TIRF) and selective plane illumination microscopy (SPIM) can further improve high lateral resolution, they still suffer from relatively low axial resolution and limited optical overage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In contrast, out-of-plane rotation of single cells combined with TIRF or SPIM is promising to construct 3D images of various biological samples with high resolution in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Whyte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' realized out-of-plane rotation of red blood cells and cancer cells with a fiber-based dual-beam laser trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='19 Then, they evolved it into an optofluidic device that enables single-cell rotation around an axis perpendicular to the imaging plane for 3D single-cell tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='136 However, all these platforms show limited throughput and are not suitable to be combined with TIRF due to the large gaps between the rotary cells and the substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' As our optothermal rotation techniques can rotate the living cells near the substrates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 7d), our platform displays a high possibility to be integrated with TIRF and SPIM for high-resolution 3D reconstruction of various living cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, by using digital micromirror devices or spatial light modulators, parallel rotations of multiple cells can be achieved by splitting the incident laser into multiple laser beams, which significantly improves the throughput of the 3D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='93 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 7 Biological applications of optothermal rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a) Left: Schematic illustration of the adhesion process of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' With a flow of biofluids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', blood, urine), the cell will first attach to the endothelium cells as a substrate at an inclined angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' During the rolling, the receptors on the cell and the ligands on the substrate experiences shear forces (Ft, F’t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The curved arrow indicates the torque on the cell generated by the fluidic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Right: on the optothermal rotation platform, the cell suspended in liquid is first trapped on the functionalized substrate, mimicking the cell pre-attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Then, the measurement of the cell’s shear adhesion kinetics can be obtained through the analysis of the light-driven out- of-plane rotation behavior of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (b) Optical images of bacteria and cells trapped and rotated in clinical samples during the rolling adhesion measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Others: urinary organisms that we can hardly identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Scale bars: 5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (c) Qa (τa≥t), the fraction of chitin’s transient adhesion with lifetime τa≥t, versus t for urinary yeast cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The dissociate constants of the adhesion event (koff-1=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='79 s-1 and k off-1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='56 s-1) were extracted by fitting the experimental data with double exponential decay curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Time-resolved optical images of a rolling yeast cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (a)- (c) Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='115 Copyright 2022, Springer Nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (d) Adapted with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='137 Copyright 2020, Institute of Electrical and Electronics Engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Micro/Nanomotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Micro/nanomotors are micro/nanoscale devices that enable the conversion of energy into mechanical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optical, electric, chemical, and magnetic energies have been used to power micro/nanomotors for the applications of environmental remediation, cargo delivery, and micro/nanorobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='37, 138-141 Among them, motors driven Withflow SCRAFA Flow E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='coli Candida 10° Koff-1 Koff-2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='56 10-1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' (tzt) Neutrophil Others 102 Urinaryyeast cells 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='5 t (s) Os S S S 5Sby optical heating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', optothermal motors) show unique advantages such as high general applicability, excellent temporal and spatial control with localized fields, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='72, 105, 142, 143 However, most existing optothermal motors require immersion in a liquid environment, limiting their range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For instance, liquid-state motors cannot be utilized in devices, such as hard drives, that need to work in a dry environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Moreover, the precise control of a micro/nanomotor in liquid could be challenging due to strong Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In comparison, operating in a solid environment, our opto- thermo-capillary nanomotors circumvent those problems and shed light on the underdeveloped field of solid-state micro/nanomotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='105 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Summary and outlook Diverse optothermal rotation techniques are developed based on different working mechanisms, which can be implemented on different substrates (thermally responsive or not), in different media (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', surfactant solutions, biological media, or solid polymers), and to different target objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', colloidal particles or living cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' They can either have in-plane or out-of-plane rotation frequency ranging from Hz to kHz, which leads to various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We conclude this Feature Article by proposing a general guideline for selecting suitable optothermal rotation techniques in terms of different target objects, followed by a discussion about the existing challenges, and future opportunities in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='1 General guideline For the optimum performance, we provide a general guideline for the choices of proper optothermal rotation techniques for different target objects, including nanoparticles, microparticles, and biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The primary challenge in controlling the rotation of nanoparticles is their prominent Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' To counter this intrinsic effect, opto- thermoelectric, solid-state opto-thermocapillary, and opto-thermo-electrokinetic rotation techniques can provide strong trap wells to suppress the Brownian motion for the precise and stable rotation of nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Specifically, opto-thermoelectric and solid-state opto-thermocapillary rotations are designed for highly anisotropic and light-absorbing nanoparticles, in which nonuniform temperature fields can be established under uniform illumination to generate rotation torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In comparison, opto-thermo-electrokinetic rotation is more universal for the rotation of different types of nanoparticles regardless of their shapes, components, and light absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' However, high surface charges of nanoparticles are typically required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Microparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Due to the less prominent Brownian motion, optothermal rotation of microparticles can be achieved by all the means discussed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We should note that for opto-thermophoretic rotation, asymmetry in either shape or light absorptivity of microparticles is required to produce torques for the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Furthermore, for microparticles with a size larger than 10 µm, optothermal convective rotation can be the most suitable candidate as the convective flow offers larger forces and torques than other optothermal rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' When considering the biocompatibility of different optothermal rotation techniques, we should point out that the solution components should be biologically safe, and the operational optical power should be low to reduce photodamage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Therefore, opto-thermophoretic, opto-thermo-electrokinetic, and opto-thermo-osmotic rotation techniques are ideal candidates for the rotation of biological cells at the single-cell level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' The biological media typically contain lots of ions that make the suspended living cells cold-seeking under a temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Accordingly, opto-thermophoretic rotation techniques are more suitable for robust cells that can survive in less-salty environments (such as yeast cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In contrast, since the opto-thermo-electrokinetic and opto-thermo-osmotic rotation techniques can offer extra forces to counteract the thermophoretic forces, they can function in various biological medias for the stable rotation of living cells (even for some delicate mammalian cells such as leukocytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='2 Challenges and opportunities The main challenges of the current optothermal rotation techniques are the potential thermal damage and limited rotation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Though the optothermal rotation can minimize phototoxicity due to the low optical power, thermal stress from optical heating could lead to the photothermal degradation of delicate objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='144 Especially for optothermal convective and opto-thermo-capillary rotation techniques, higher optical powers are typically required compared to the other optothermal rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' In addition, it could be challenging to obtain an ultrahigh rotation rate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', GHz) on an optothermal rotation platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Intuitively, a larger rotation rate can be obtained from the higher optothermal torques powered by the higher optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' However, the increasing temperature might reach the boiling point of the solution, where vapor bubbles can form and introduce strong Marangoni convection to suppress the optothermal torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' One possible solution to these two issues is to add a chip-size heat sink to lower the environmental temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='145 Within a lower environmental temperature, the target temperature difference for the generation of optothermal torques can be obtained with a lower peak temperature, which can reduce the thermal damage and avoid the formation of vapor bubbles40, 93 Optothermal rotation of objects down to a few nanometers or even atomic scale can be attractive to the study of virology and particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Multiple optothermal effects have proven to be able to direct the migration of the particles with these small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='37 To implement the rotation of these extremely tiny objects, one promising strategy is to use the femtosecond laser in optothermal rotation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Specifically, a femtosecond laser pulse can generate a highly localized and steeper temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Under a higher temperature gradient, a more robust trap well and a larger optothermal torque can be produced, which is promising to achieve stable rotary control of nano-/atomic-scale items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Besides, due to the high energy density of a femtosecond laser beam, it can directly heat the liquid through nonlinear interactions to establish temperature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Thus, all existing optothermal rotation techniques can work without the requirement of light-absorbing substrates or objects, which further broadens the applicability of optothermal rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' We believe that, once the above-mentioned bottlenecks are overcome, more applications can be implemented by optothermal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' For instance, with the capability to noninvasively rotate smaller objects such as viruses, optothermal rotation can be used in single-virus mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Specifically, the rolling adhesion of SARS-CoV-2 (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='1 µm) can be measured to help better understand the viral infection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Another promising application is cell classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Distinguishing different types of cells of high similarity is extremely challenging for existing optical imaging techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Optothermal rotation of cells integrated into standard optical microscopy will provide a collection of multiangle images at a high efficiency and, in combination with machine-learning-enhanced image analysis, could effectively reduce the quantity of the training samples and improve classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Author Contributions H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' conceived the idea, designed the frame, and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' assisted with the manuscript revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Conflicts of interest The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Acknowledgments H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=', and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' acknowledge the financial support of the National Institute of General Medical Sciences of the National Institutes of Health (R01GM146962) and the National Science Foundation (NSF-ECCS-2001650).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' additionally acknowledges the financial support of UT Austin Undergraduate Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Notes and references 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Grier, Nature, 2003, 424, 810-816.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Wei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Xie and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Zheng, Lab Chip, 2017, 17, 3061-3070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} +page_content=' Lin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfEgmt/content/2301.04297v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..b6ed5e1fe1b4a00a61c24ea58f32b7a0f0c79c8d --- /dev/null +++ b/XdFOT4oBgHgl3EQf8zQL/content/tmp_files/2301.12967v1.pdf.txt @@ -0,0 +1,2961 @@ +Hierarchical learning, forecasting coherent spatio-temporal individual and aggregated +building loads +Julien Leprincea,b,∗, Henrik Madsenb, Jan Kloppenborg Møllerb, Wim Zeilera +aTechnical University of Eindhoven, 5 Groene Loper, Eindhoven 5600 MB, the Netherlands +bTechnical University of Denmark, Building 303B Matematiktorvet, Lyngby 2800, Denmark +Abstract +Optimal decision-making compels us to anticipate the future at different horizons. However, in many domains connecting together +predictions from multiple time horizons and abstractions levels across their organization becomes all the more important, else +decision-makers would be planning using separate and possibly conflicting views of the future. This notably applies to smart grid +operation. To optimally manage energy flows in such systems, accurate and coherent predictions must be made across varying +aggregation levels and horizons. Such hierarchical structures are said to be coherent when values at different scales are equal when +brought to the same level, else would need to be reconciled. With this work, we propose a novel multi-dimensional hierarchical +forecasting method built upon structurally-informed machine-learning regressors and established hierarchical reconciliation taxon- +omy. A generic formulation of multi-dimensional hierarchies, reconciling spatial and temporal hierarchies under a common frame is +initially defined. Next, a coherency-informed hierarchical learner is developed built upon a custom loss function leveraging optimal +reconciliation methods. Coherency of the produced hierarchical forecasts is then secured using similar reconciliation technics. The +outcome is a unified and coherent forecast across all examined dimensions, granting decision-makers a common view of the future +serving aligned decision-making. The method is evaluated on two different case studies to predict building electrical loads across +spatial, temporal, and spatio-temporal hierarchies, benchmarked against base and multi-task regressors. Although the regressor +natively profits from computationally efficient learning thanks to the unification of independent forecasts into a global multi-task +model, results displayed disparate performances, demonstrating the value of hierarchical-coherent learning in only one setting. Yet, +supported by a comprehensive result analysis, existing obstacles were clearly delineated, presenting distinct pathways for future +work. Particularly, investigating reduced number of model weights and varying native multi-output regressors can tackle the two +preeminent challenges that are the curse of dimensionality and coherency-learning complications from scaled trees respectively. +Overall, the paper expands and unites traditionally disjointed hierarchical forecasting methods providing a fertile route toward a +novel generation of forecasting regressors. +Keywords: Hierarchical forecasting, Coherency, Spatio-temporal dimensions, Deep learning, Smart building +1. Introduction +A better anticipation of the future supports better decision- +making. This is true across all sectors. Yet, more accurate fore- +casts alone often do not suffice. When dealing with different +abstraction levels across a system or organization, it is com- +monly more important to obtain coherent predictions across all +considered layers and horizons, not to result in unaligned deci- +sions or possibly even conflicting ones [1]. This obstacle arises +in multiple domains, including tourism [2, 3], retail [4], stock +management [5] and smart grid management [6], which show- +cases this matter quite adequately. +Traditionally, smart grid operators focused on forecasting the +system’s total demand. However, with the increasing adoption +of smart meters at grid edges and substations, the focus is shift- +∗Corresponding author +Email addresses: j.j.leprince@tue.nl (Julien Leprince ), +hmad@dtu.dk (Henrik Madsen), jkmo@dtu.dk (Jan Kloppenborg Møller), +w.zeiler@tue.nl (Wim Zeiler) +ing. Grid management now benefits from high-frequency mea- +surements available at multiple levels of aggregation allowing +accurate forecast estimations across both spatial and temporal +scales, i.e., from sub-meters to regional-level, with per seconds +to monthly aggregated information [6, 7, 8]. Yet, the pluralities +and independence of models and their consequent forecasts in- +evitably produce inconsistencies across aggregation levels, i.e., +lower-level predictions might not sum up to higher-level ones +and vice-versa [9]. The consequent challenge decision-makers +are now faced with is to obtain coherent predictions across the +different horizons and scales of the system. Hierarchical struc- +tures (or trees) are said to be coherent when their values at the +disaggregate and aggregate scales are equal when brought to +the same level [2]. Should forecasts not be coherent, decision- +making units would be planning using diverging views of the +future. Optimal decision-making consequently requires fore- +casts to be coherent across all considered dimensional hierar- +chies. +Preprint submitted to Applied Energy +January 31, 2023 +arXiv:2301.12967v1 [cs.LG] 30 Jan 2023 + +1.1. Hierarchical forecasting +Enforcing coherency in hierarchical structures is a concept +that dates back to 1942 [10] and was first defined in 1988 as rec- +onciliation [11]. It leverages linear balancing equations from +covariance compositions inherent to hierarchical structures to +optimally re-adjust coherency mismatches. +Hyndman et al. +[12] later reformulated the approach with a unifying statistical +method, independent of prediction models, along with notations +more appropriate to hierarchical forecasting. +Hierarchical forecasting can thus be defined as the process in +which coherent predictions need to be made within a fixed hi- +erarchical structure. Commonly, forecasts are first estimated +separately considering each series of the hierarchy in a dis- +jointed manner. These forecasts are designated as independent +base forecasts [3]. Generating base forecasts for each series +implies that specialized models can be developed for each part +of the hierarchy, incorporating node-specific available informa- +tion [2]. Base forecasts are then linearly combined (reconciled) +leveraging available information across the hierarchy to ensure +coherency; a process employed by all hierarchical forecasting +approaches as of to date [1, 3, 5, 6, 11, 12, 13, 14, 15, 16, 17, +18, 19, 20, 21]. +1.1.1. Reconciliation approaches +Predominant reconciliation techniques comprise traditional +bottom-up and top-down approaches, trace minimization, op- +timal combinations, and recently developed machine-learning +methods. +Bottom-up hierarchical forecasting consists in generating +base forecasts at the very bottom level of the hierarchy and +enforce coherency through their direct aggregation across the +tree [22]. The greatest advantage of this approach is that it +can draw information from the most disaggregated levels of the +tree, consequently avoiding any information loss from aggrega- +tion [3]. However, series located at tree leaves tend to possess +low signal-to-noise ratios making them more difficult to pre- +dict. This is particularly true when dealing with smart-meter +electrical demands which are notoriously volatile. Consump- +tion peaks are indeed driven by often highly stochastic occu- +pant behaviors that are close to intractable, consequently mak- +ing bottom-up aggregation unlikely to provide accurate fore- +casts across the upper levels of the tree [12]. +Top-down hierarchical forecasting on the other hand only +generates forecasts for the top level of the hierarchy (tree-root) +and proceeds to disaggregate and distribute it down the hierar- +chy from either historical [23] or forecasted [3] proportions of +the data. The approach commonly favors higher aggregation +levels of the tree with more accurate predictions and is notably +valuable for low-count data. However, aggregation is not with- +out a large loss of information as temporal dynamics and other +individual series characteristics cannot be exploited [3]. Addi- +tionally, as the success of this approach depends solely on one +top-level model, it possesses a higher degree of risk from model +misspecifications or inaccuracies [24]. Given both bottom-up +and top-down approaches inadequate to profit from the rich- +ness of information across a given hierarchy, optimal combina- +tion techniques emerged. Linearly reconciling base forecasts +towards coherency, these approaches allowed interactions be- +tween different levels of the hierarchy, leveraging in particular +correlations and covariances present in such structures [12]. +However, estimating the covariance structure of a hierarchy +from base forecasts is challenging. Indeed, Wickramasuriya et +al. [13] declared that the covariance matrix of the coherency +errors is ”impossible to estimate in practice due to identifiabil- +ity conditions” such that even with high-frequency data avail- +able, assumptions on its form must be made [25]. The ordi- +nary least-square (OLS) estimator was particularly developed +by Hyndman et al. +[12] and Athanasopoulos et al. +[3] to +avoid this problem. +Their approach demonstrated improved +results compared to other commonly adopted techniques. A +weighted least squares (WLS) approach, considering variances +from the variance-covariance matrix diagonal but ignoring the +off-diagonal covariance elements, was put forward by Hynd- +man et al. [26]. Wickramasuriya et al. [13] later provided the +theoretical justification for estimating variances from base fore- +cast error variances. They proposed a generalized least-squares +(GLS) estimator and found the incorporation of correlation in- +formation into the reconciliation process to benefit forecasting +accuracy, with resulting reconciled forecasts guaranteed to be, +in mean or in sample, at least, as good as their base forecasts, +given a particular covariance structure. +Finally, in recent years, machine learning approaches have +made their way into hierarchical forecasting. Relying on power- +ful statistical regressors and the availability of larger and richer +data sets, machine learning emerges as an appealing and suit- +able tool for estimating the persistently challenging covariance +matrix. Spiliotis et al. [5] put forward such an approach em- +ploying a bottom-up method to reconcile predictions from Ran- +dom Forest and XGBoost regressors. Taking as input the base +forecasts of all the series of the hierarchy, the reconciled tree +is then obtained from bottom-up aggregation. It allows non- +linear combinations of the base forecasts, extending conven- +tional linear approaches thanks to its machine-learning nature. +Sagheer et al. [27] proceeded to obtain coherent hierarchies +from deep long-short term memory (DLSTM) recurrent neu- +ral networks by applying transfer learning across their hierar- +chies in a bottom-up fashion. They evaluated their approach on +national-scale Brazilian electrical power production as well as +Australian domestic tourism data. In another work, Mancuso et +al. [28] proposed a method to unify the two prevailing processes +that are forecasting and reconciliation. By including hierarchi- +cal information in the forecasting process through a customized +loss function, they allow the network to train towards reconciled +forecasts using a top-down disaggregation process. +None of these approaches, however, include the general +formulation of hierarchical forecasting within their learning +framework. This limits their reconciliation approaches to en- +compass solely traditional approaches, i.e., bottom-up or top- +down, which, as has been mentioned, only exploit a fraction of +the available information of hierarchical structures. +1.1.2. Dimensional considerations +While numerous works have first approached the reconcili- +ation of hierarchical structures from a spatial (cross-sectional) +2 + +dimensional frame perspective [3, 5, 6, 11, 12, 13, 14, 15, 16], +temporal hierarchies have also been the center of recent atten- +tion within the field [1, 17, 18, 19, 20, 21]. +Athanasopoulos et al. [17] first introduced the notion of tem- +poral hierarchies with forecasting reconciliation performed in +the temporal dimension. Quite similarly to spatial reconcilia- +tion, base forecasts are independently produced across a defined +set of temporal aggregation levels, e.g., weekly, daily, quarter- +daily, hourly to per-minute or seconds granularities. This allows +models to capture temporal-specific characteristics of the times +series across the hierarchical-structure, e.g., trends or seasonal- +ity possessing particular time-frames. Base temporal-forecasts +are then reconciled across all forecasting horizons and temporal +tree-structure, allowing aligned decisions across multiple plan- +ning horizons [2]. Nystrup et al. [1] notably proposed temporal +estimators accounting for autocorrelation structures to reconcile +electric grid load forecasts. It was found that auto- and cross- +covariances significantly improved forecast accuracy uniformly +across all temporal aggregation levels. +It thus becomes clear that both spatial and temporal hier- +archical forecasts produce substantial empirical accuracy im- +provements. By dealing with parameter estimation errors and +model misspecifications, forecast combinations have demon- +strated significant error variance reduction across numerous +works [29, 24, 30]. +Exploiting both available hierarchi- +cal dimensions to further improve prediction accuracies con- +sequently emerges as not only appealing but quite evident. +Kourentzes and Athanasopoulos [2] notably advanced a frame- +work to produce spatial- and temporal-coherent forecasts (des- +ignated as cross-temporal), supporting all hierarchical lev- +els with short- to long-term forecasts. +Their work demon- +strated empirical evidence that leveraging both dimensions in +reconciliation offered improved accuracies compared to uni- +dimensional reconciliation, i.e., spatial or temporal. A find- +ing certainly due to the complete information exposure the ap- +proach provides. Spiliotis et al. [9] later proposed a cross- +aggregation process to iteratively generate coherency across +spatial hierarchies from multiple temporal aggregations applied +to electricity consumption forecasting. Punia et al. [31] intro- +duced a similar framework leveraging deep learning algorithms +applied to supply chain base forecasts. Their approach, how- +ever, produced coherency solely from bottom-up approaches. +While the advantage of multi-dimensional hierarchical fore- +cast has become evident, there exists, as of today, no generic +formulation of these approaches. Indeed, while Spiliotis et al. +[9] stated that it is possible, in principle, to design a summing +matrix S that accounts for both considered dimensions of rec- +onciliation, a theoretical formulation of S and its subsequent +reconciliation approaches was not put forward. +Indeed, the +design of a reconciliation estimator that fully captures scaling +issues and cross-sectional interdependencies is not straightfor- +ward. Yet, this deprives multi-dimensional reconciliations of +exploiting custom dimensional considerations. The principal +counterargument to undertaking such formulations is grounded +on the fact that multi-dimensional hierarchies generate increas- +ingly large tree structures that could soon become intractable to +estimate. Recently, however, the work of Nystrup et al. [20] +proposed a dimensionality reduction technique to counter this +problem. +Using eigendecomposition when reconciling fore- +casts, maximum information can be extracted from the error +structure using available data. They find that uniformly im- +proved predictions can be obtained across all aggregation lev- +els, with the estimator achieving state-of-the-art accuracy all +the while being applicable to hierarchies of all sizes. +1.2. Motivation +This comprehensive state-of-the-art overview underlines the +following shortcomings; +(i) Base forecasts are typically produced separately, consid- +ering each series of the hierarchy in a disjointed man- +ner. While this procedure allows the independence and +hierarchically-tailored design of these models, it is inher- +ently deprived from the benefits of data information (learn- +ing) transfer across models. +(ii) Machine-learning reconciliation approaches have exhib- +ited clear forecast improvements potential. Yet, developed +approaches have, so far, not proceeded to put forward a +unified method for machine-learning based hierarchical- +forecasting. +This limits considered reconciliations ap- +proaches to the more information-limited bottom-up and +top-down approaches [5, 31]. Embedding advanced rec- +onciliation techniques, e.g., optimal combinations, in the +learning process of machine learning regressors is, as of +today, still missing. +(iii) Although advantages of leveraging multi-dimensional hi- +erarchies in forecasting has become evident, a generic for- +mulation of such hierarchical-combinations is still needed. +Existing tools have demonstrated effective dimensionality +reductions of large hierarchies [20], presenting promising +solutions to the problem of dimension intractability. +This study proposes a response to this appeal and puts for- +ward a generic multi-dimensional formulation for hierarchical +forecasting with machine-learning. We put together a unified +and adaptable forecasting and reconciliation method founded +on native multi-task machine-learning regressors while fram- +ing multi-dimensional hierarchical-forecasting approaches in a +generic way. Contributions of this work can be summarized as +five-fold; +1. We develop a unified machine-learning-based hierarchical +forecasting approach. This grants (i) a unique forecast- +ing model the benefit of a complete information overview +across its hierarchy, while (ii) including coherency con- +straints within its learning process as well as (iii) be- +ing adaptable to either independent or combined forecast- +ing and reconciliation processes. It establishes a unified +method generating accurate and coherent forecasts at all +levels of the hierarchy thanks to a custom hierarchical +loss function leveraging coherency information from es- +tablished field-taxonomy. +3 + +2. To best exploit available information embedded within +multi-dimensional data, we formulate a generic multi- +dimensional extension of conventional hierarchical fore- +casting methods. +In particular, we address the prob- +lem of diverging reconciliation considerations in a multi- +dimensional setting with uni-dimensional couplings of +the covariance estimator. This allows the unification of +multi-hierarchical structures under a common frame, fuel- +ing both traditional and machine-learning approaches with +ever-richer and transferable (learning) information. +3. In the interest of addressing the dimensional tractability +of our approach, we put forward dimensionality reduc- +tion prospects and illustrate them both theoretically and +in practice with an applied demonstration. +4. Our study considers two substantial smart-meter data sets +including an established open source, i.e., the Building +Data Genome project 2 (BDG2) [32]. +This allows the +grounding of our approach thanks to a first-of-its-kind per- +formance benchmark in the field of electric-meter hierar- +chical predictions, which we render fully replicable. +5. To +best +serve +knowledge +dissemination +and +re- +search reproducibility, +we open-source all our de- +veloped +code +under +the +public +GitHub +repository +https://github.com/JulienLeprince/hierarchicallearning. +The greatest advantage of this approach is granting ac- +cess to the regressor a complete information overview of the +considered (multi-dimensional) hierarchy. This permits both +a cross-dimensional, data-rich learning process as well as a +hierarchically-informed training for hierarchical forecasting. +The outcome is a unified and coherent forecast across all ex- +amined dimensions, granting decision-makers a common view +of the future serving aligned and better decisions. +The rest of this paper is organized as follows: Section 2 de- +tails traditional hierarchical forecasting prospects and extends +them to multi-dimensional frames, while Section 3 presents the +hierarchical learning methods put forward with developed cus- +tom loss functions. Section 4 introduces the implementation +specifics of our applied method from two different case studies, +and Section 5 reports the performance results of the method. +Highlighted findings are analyzed and detailed under the dis- +cussion Section 6, followed by Section 7 which concludes the +paper and reveals future work outlooks. +2. Hierarchical forecasting +In this section, we present the foundations of hierarchical +forecasting as defined by Athanasopoulos et al. [3] and Wick- +ramasuriya et al. [13] and extend them to multi-dimensional +frames with a generic formulation. We discuss dimensionality +tractability limitations and offer dimensionality reduction con- +siderations to address them. +21 +31 +33 +22 +34 +35 +11 +36 +k1 = m +k-levels +k2 = 3 +kK = 1 +m = 6 +32 +Figure 1: A two-level hierarchical tree diagram. +2.1. Hierarchical structures +Let us refer to the simple hierarchy of Fig. 1 to demon- +strate the methodology. Every element (node) of the hierar- +chy (tree) can be labeled as yk j, where the subscripts k and j +stand for the aggregation-level and node observations respec- +tively. We define k1 as the most aggregate level of the hierarchy +(tree root), i.e., node y11, and kK as the most disaggregate level +(tree leaves), i.e., nodes yK j where j ∈ [1 : m] and K = 3. In +such a setting, two important components must be considered; +the number of nodes in the bottom level of the hierarchical tree, +which is denoted as m, and the total number of nodes on the +tree n. Here n = 9 and m = 6. +Stacking +all +tree +elements +in +a +n-dimensional +vec- +tor +y += +(y11, y21, y22, y31, y32, y33, y34, y35, y36)T, +and +bottom-level +observations +in +an +m-dimensional +vector +b = (y31, y32, y33, y34, y35, y36)T, we can write +y = S b, +(1) +where S is the summation matrix, here expressed as +S = +�������������� +1 +1 +1 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +1 +1 +1 +Im +�������������� +, +(2) +which is of dimension n×m, and Im is an identity matrix of size +m. S maps the hierarchical structure of the tree, where from the +tree leaves b the complete hierarchy y can be reproduced. No- +tice how S captures the coherency requirements within the hi- +erarchy, integrated here as the linear summations of the bottom- +level observations. +2.1.1. Uni-dimensional +Hierarchical structures encompassed within hierarchical +forecasting have, as of today, treated either one of the two fol- +lowing dimensional frames, namely, temporal T or spatial S +(sectional). +4 + +We define spatial dimensional perspectives as a unique inter- +element dimension, which places itself in opposition to the +previously-defined cross-sectional dimensions [2, 9, 17, 33], +which aggregated elements from very different entities together, +e.g., stock management, resulting in considerable heterogeneity +within ”one” (but in fact, multi-) dimension. It is our proposal +to re-frame these cross-sectional considerations into separate +dimensions to allow clear delineations of multi-dimensional +frames, as we later detail in Sec. 2.3. +Although structures of any shape or form can be designed in +both dimensions, it is common for temporal hierarchies to adopt +symmetrical structures, with k-level values being homogeneous +across the trees’ aggregation levels. +Taking the exemplified +symmetrical hierarchy of Fig. 1, one could consider removing +nodes y32 and y33; resulting in a hierarchy where m = 4, n = 7 +and node y21 being consequently removed as a redundant ele- +ment of y31. This would result in a non-symmetrical tree which, +in the temporal domain, implies non-equally spaced measure- +ment points (or sampling rate) across the considered aggrega- +tion level and the ones above it. +Typically, for symmetrical trees, there are k ∈ {k1, ..., kK} ag- +gregation levels, where k is a factor of m, with k1 = m, kK = 1, +and m/k is the number of observations at aggregation level k. +The summation matrix of temporal hierarchies can therefore be +expressed as [1] +S T = +������������ +Im/k1 ⊗ 1k1 +... +Im/kK ⊗ 1kK +������������ +, +(3) +where ⊗ is the Kronecker product and 1k is a k-vector of ones. +To generically define the formulation of the summation ma- +trix of any uni-dimensional hierarchy H, however, one needs +to consider the eventuality of non-homogeneous k-level values +across aggregation levels as well as uneven tree-depths. To this +end, we define +sij = +������� +1, +if yi +is ancestor of +yK j, +0, +if yi +is not ancestor of +yK j, +(4) +where sij is a matrix element of the summation matrix S H given +a fixed hierarchical structure H and yi here refers to the i-th +element of y. The subscripts i and j go from 1 to n − m and m +respectively. They refer to the considered tree node element i +and tree leaf element j. This sets the matrix element of a given +node i to either 1 or 0 if it is an ancestor of the leaf element +j, or, in other words, whether it is a result of the aggregation of +the corresponding tree-leaf element yK j or not respectively. The +summation matrix can then be expressed as +S H = +������������������������������ +s11 +. . . +s1 j +. . . +s1m +... +... +... +si1 +. . . +sij +. . . +sim +... +... +... +s(n−m)1 +. . . +s(n−m)j +. . . +s(n−m)m +Im +������������������������������ +. +(5) +This enables the formulation of any hierarchical structure to +a summation matrix, e.g., from event-based or equally spaced +time-series measurements for temporal hierarchies T , to non- +symmetrical or homogeneous aggregation structures for spatial +hierarchies S. +2.1.2. Multi-dimensional +Multi-dimensional hierarchies are the product of two uni- +dimensional structures and can be obtained from function com- +position of separate hierarchical structures over another one. +Fig. 2 illustrates the derivation of a spatio-temporal ST hi- +erarchy from two disjointed spatial S and temporal T structure +compositions, i.e., SoT and T oS. The resulting tree structures +demonstrate fundamental equivalences, with all tree nodes pos- +sessing identical bonds linking one element to the other, and +consequently producing a unique hierarchical structure ST . +The formulation of the multi-dimensional summation matrix +in a generic way, can thus be expressed as a Kronecker product, +where +S ST ≡ +������� +S SoT = +S S ⊗ S T , +S T oS = +S T ⊗ S S, +(6) +from which the resulting spatio-temporal summation matrix +S ST is of dimension nSnT × mSmT , which, in the example of +Fig. 2, yields 3· 7 × 2 · 4 = 21 × 8. The equivalence of SoT and +T oS is attained via varying orderings of the nSnT -dimensional +vector yST . These are derived from alternative transpose defi- +nitions of the observation matrix YSoT such that +YSoT = YT +T oS = +������������ +y11 +. . . +y1nT +... +... +... +ynS1 +. . . +ynSnT +������������ +, +(7) +where uni-dimensional vectors yS and yT are stacked together +to form an observation matrix YSoT of dimension (nS, nT ). The +yST equivalent vectors can then obtained with +yST ≡ +������� +ySoT = +vec(YT +SoT ), +yT oS = +vec(YT +T oS). +(8) +In +the +exemplified +structures +of +Fig. +2, +we +obtain +ySoT += +(yA1, yB1, yC1, ..., yA7, yB7, yC7)T +and +yT oS += +(yA1, ..., yA7, yB1, ..., yB7, yC1, ..., yC7)T. +With structural combinations of two disjointed dimensional +hierarchies producing a unique bi-dimensional structure, it con- +sequently follows that multi-dimensional combinations can be +exploited in a similar manner. By chaining function composi- +tions of considered singular dimensions over summation matri- +ces and y vectors, any combination of dimensional frames can +be considered. +2.1.3. Dimensionality reduction +Multi-dimensional trees, however, introduce a key limitation: +the dimensional explosion of hierarchical structures from func- +tion composition. With the multiplication of dimensions from +summation matrices, what was then considered a tractability +5 + +4 +A +B +C +2 +4 +5 +3 +6 +7 +1 +2 +4 +5 +3 +6 +7 +1 +2 +4 +5 +3 +6 +7 +1 +B2 +B4 +B5 +B3 +B6 +B7 +B1 +C2 +C4 +C5 +C3 +C6 +C7 +C1 +A2 +A4 +A5 +A3 +A6 +A7 +A1 +2 +5 +3 +6 +7 +1 +A +B +C +A +B +C +A +B +C +A +B +C +A +B +C +A +B +C +A +B +C +A4 +B4 +C4 +A5 +B5 +C5 +A2 +B2 +C2 +A6 +B6 +C6 +A7 +B7 +C7 +A3 +B3 +C3 +A1 +B1 +C1 +2 +4 +5 ++ +3 +6 +7 +1 +A +B +C +Temporal +hierarchy +T +Spatial +hierarchy +S +SoT +ToS +ST +Figure 2: Schematic of spatio-temporal ST hierarchical structure conception from either SoT or T oS structure composition, both producing an equivalent ST +tree structures. Highlighted nodes (in grey) reveal opportunities for dimensionality reduction by dropping nodes of little dimensional interest, i.e., high temporal +granularity in high spatial aggregation levels, and low temporal frequencies in high spatial granularities. +A +B +C +Spatial +hierarchy +S +1 +2 +4 +Temporal +hierarchy +T ++ +A +B +C +1 +2 +4 +A +B +C +A +B +C +A +B +C +Topological covariance matrix +3 +3 +B +C +A +1 +2 +4 +3 +1 +2 +4 +3 +1 +2 +4 +3 +Summation matrix and y vector +S +T +S +T +ToS +SoT +ST +Figure 3: Exemplified illustrations of hierarchical derivations of summation matrix, y vector and topological covariance matrix from spatio-temporal SoT or T oS +function composition. +6 + +shortcoming has now become an inevitable obstacle needing +overcoming. +However, multi-dimensional hierarchies bring with them +a consequential consideration: multi-dimensional aggregation +levels. Indeed, such trees encompass more than former uni- +dimensional high- or low-aggregation levels, they consist of +deep structures where multi-dimensional aggregation combi- +nations demand investigation. Spatio-temporal hierarchies, for +example, display dissimilar insights from high-temporal-low- +spatial aggregation levels, low-temporal-low-spatial or high- +temporal-high-spatial ones. +It thus comes to light that, given a defined insight-driven ap- +plication, subsets of certain multi-dimensional aggregation re- +gions can be of limited use. High-frequency forecasts at very +aggregate geographical levels might be of great value to grid +operators contemplating frequency control in power systems, +but not so much when forecasting tourism flows for instance +[2]. +Considering the end-goal application of optimal smart- +grid control from electric load forecasting of grid edges (smart- +building meter), low temporal frequencies and low spatial ag- +gregations would be of little interest. Indeed, frequency control +focuses on rather high-frequency samplings at medium-high +spatial aggregation levels. However, should the end-goal appli- +cation be optimal cooperative control of smart-building neigh- +borhoods, then low temporal frequencies and high spatial ag- +gregations would become the dimensional frame of lesser con- +cern. Fig. 2 highlights these bi-dimensional nodes over the +hierarchical structure conception, i.e., in grey, revealing the po- +tential of dimensionality reduction within multi-dimensional hi- +erarchies. +Therefore, while using spatio-temporal coherent forecasts of- +fer benefits to decision-making, not all outputs from these hier- +archies are effectively useful, opening the door to dimensional- +ity reduction. +2.2. Reconciliation methods +Traditionally, forecast reconciliation starts by generating an +initial forecast of the tree independently for each node, referred +to as base forecasts ˆy. +This set of hierarchical forecasts is +stacked in the same manner as the y vector. Because of the in- +dependence of the base forecasts, in most cases, they do not ex- +hibit coherency properties throughout their hierarchical struc- +tures. By introducing a matrix +G = �0m × (n−m) | Im +�, +(9) +of order m × n that extracts the m bottom-level forecasts, the +reconciliation constraint is formulated as +˜y = SG˜y. +(10) +Reconciliation is necessary when base forecasts ˆy do not satisfy +this constraint [1]. In such situations, Eq. (10) becomes ˜y = +SGˆy, where G maps the base forecasts into the reconciled tree- +leaves and S sums these up to a set of coherent forecasts ˜y. +SG can thus be thought of as a reconciliation matrix taking the +incoherent base forecasts as input and reconciling them to ˜y. A +major drawback of traditional approaches is that G, as defined +in Eq. (9), only considers information from a single level. +2.2.1. Optimal reconciliation +To include the exploitation of all aggregation levels in an op- +timal manner, Hyndman et al. [12] and later, Van Erven and +Cugliari [14] and Athanasopoulos et al. [17] formulated the +reconciliation problem, as linear regression models. Exploit- +ing either spatial or temporal hierarchical structures, reconciled +forecasts are found employing the generalized least-squares es- +timate: +minimize +�˜y − ˆy�TΣ−1�˜y − ˆy�, +subject to +˜y = SG˜y, +(11) +where ˜y ∈ Rn is the decision variable of the optimization prob- +lem and S ∈ Rn×m and G ∈ Rm×n are constant matrices defined +by the structure of the hierarchy. The parameter Σ ∈ Rn×n is +the positive definite covariance matrix of the coherency errors +ε = ˜y − ˆy, which are assumed to be multivariate Gaussian and +unbiased, i.e., with zero mean. +If Σ were known, the solution to (11) would be given by the +generalized least-squares (GLS) estimator +˜y = S �S TΣ−1S �−1S TΣ−1ˆy, +(12) +which has been employed in close to all notable hierarchical +forecasting works over the last years [1, 2, 3, 6, 9, 12, 13, 17]. +The precision matrix Σ−1 is used to scale discrepancies from the +base forecasts, hence, is often referred to as a weight matrix. +The recurrent challenge in estimating Σ−1 stems from its di- +mension n × n which can potentially become very large. +2.3. Multi-dimensional reconciliation +Traditional uni-dimensional estimators can be coupled to- +gether topologically to form multi-dimensional ones in a similar +manner to the summation matrix, with +Σ† +ST ≡ +������� +Σ† +SoT = +Σ† +S ⊗ Σ† +T , +Σ† +T oS = +Σ† +T ⊗ Σ† +S, +(13) +where Σ† refers to the topological covariance matrix of a given +covariance matrix Σ. This allows uni-dimensional estimators +ΣS and ΣT to incorporate dimension-specific topological con- +siderations and produce a suitable multi-dimensional estimator +ΣST . +The topological covariance matrix is characterized by ele- +ments of either 0 or 1 that indicate the mapping form assump- +tion of the considered covariance matrix. Once the topological +covariance matrix is identified, we simply populate it with the +scaling parameters dictated by the reconciliation approach con- +sidered to obtain the covariance matrix. Figure 3 exemplifies +the identification of multidimensional topological covariance +matrices from both SoT and T oS dimensional-derivations. +To address dimensional considerations in traditional esti- +mators of the covariance matrix applied in reconciliation, we +present four state-of-the-art estimators, namely, identity, struc- +tural, variance, and covariance scaling with shrinkage, while +detailing dimensional deliberations individually. +7 + +2 +4 +5 +3 +6 +7 +1 +Considered +hierarchy +H +Summation matrix +and y vector +Topological +covariance matrices +diagonal +k-level +full +Covariance matrices +identity - id +structural - str +variance - hvar +variance - svar +covariance - kcov +covariance - cov +Figure 4: Example illustration of the covariance matrices considered in this work along with their associated topological covariance matrices. +8 + +2.3.1. Identity +A simplifying assumption proposed by Hyndman et al. [12] +puts the following identity approximation forward +Σid = In. +(14) +This simplistic approach has been shown to work well in prac- +tice [3] and allows to bypass the estimation of the covariance +matrix. It ignores scale differences (captured by the variances) +and interrelations (captured by the covariances) information of +the observations within the hierarchical structure, which makes +it independent of dimensional frame considerations. +Deep neural networks can be expected to build upon such +simple relationships and approximate the more complex depen- +dencies of the hierarchy thanks to its automated feature selec- +tion, as we later detail in Section 3. We refer to this approach +with the subscript id. +2.3.2. Structural scaling +Structural scaling was proposed by Athanasopoulos et al. +[17] as a solution to cases where forecast errors are not avail- +able for some aggregation levels. It assumes the variance of +each bottom-level base forecast error σ2 +K is equal and that these +are uncorrelated between nodes. Therefore, higher-level error +variances are the sum of the error variances of tree leaves se- +ries connected to them. By introducing a diagonal matrix Λstr +with each element containing the number of forecast errors con- +tributing to that aggregation level, they define +Σstr = σ2 +KΛstr, +(15) +Λstr = diag(S 1m), +(16) +where 1 ∈ Rm is a column vector. The hierarchy illustrated +in Fig. 1, for instance, gives Λstr = diag(6, 3, 3, 1, 1, 1, 1, 1, 1). +The estimator is independent from the considered forecasting +method, since no estimation of the variance of the forecast er- +rors is needed, making it computationally efficient [1]. +If considering a temporal dimensional frame, the estimator +depends only on the seasonal period m of the tree leaves. While +with spatial perspectives, the estimator can suffer from hetero- +geneity within aggregation levels, e.g., residential and commer- +cial buildings typically have heterogeneous electricity demand +patterns and scale. Hence, assuming a common forecasting er- +ror variance across all leaves-series is not suitable [2]. We refer +to this approach as str. +2.3.3. Variance scaling +Another estimator proposed by Athanasopoulos et al. [17], +referred to as variance scaling, scales the base forecasts using +the variance of the residuals. It includes separate variance es- +timates for each aggregation level and assumes either homo- +geneous or non-homogeneous error variance within, but not +across, a level. Given the hierarchy presented in Fig. 1, this +gives +Σsvar = Λsvar = diag(σ2 +k1, σ2 +k2, σ2 +k2, σ2 +k3, . . . , σ2 +k3), +(17) +Σhvar = Λhvar = diag(σ2 +11, σ2 +21, σ2 +22, σ2 +31, . . . , σ2 +37), +(18) +for homogeneous and heterogeneous variances respectively. By +definition, this scaling ignores correlations across and within +aggregation levels and can be considered as an alternative +weighted least-squares estimator. +Similarly to structural scaling, spatial and temporal dimen- +sional scaling can differ due to their intrinsic heterogeneous and +homogeneous error variances respectively; we refer to these es- +timators as hvar and svar. It follows, that Σsvar and Σhvar are +appropriate to temporal T and spatial S dimensional scalings +respectively. +2.3.4. Covariance scaling +To exploit important information about a time series at differ- +ent frequencies (temporal dimension) or inter-scale differences +(spatial dimension), Nystrup et al. [1] argue that potential in- +formation in the autocorrelation structure should be included. +They consequently proposed a covariance scaling for temporal +hierarchies estimating the full covariance matrix within each +aggregation level, while ignoring correlations between them. +Following these footsteps, we explore both full and k-level, +or so-called block, covariance estimates such that, for the hier- +archy illustrated in Fig. 1, the estimator is either +Σcov = Λ1/2 +hvarRΛ1/2 +hvar, or +(19) +Σkcov = Λ1/2 +hvarRkΛ1/2 +hvar, +(20) +where R and Rk refer to the full and k-level empirical cross- +correlation matrix respectively, +R = +������������ +1 +. . . +ρ11,36 +... +... +... +ρ11,36 +. . . +1 +������������ +, +(21) +Rk = +���������������������������� +1 +0 +0 +0 +. . . +0 +0 +1 +ρ21,22 +0 +. . . +0 +0 +ρ22,21 +1 +0 +. . . +0 +0 +0 +0 +1 +. . . +ρ31,36 +... +... +... +... +... +... +0 +0 +0 +ρ31,36 +. . . +1 +���������������������������� +. +(22) +With increasing difficulties in estimating the full covari- +ance matrix from high-dimensional hierarchies, even with high- +frequency data available, special forms are commonly assumed. +To alleviate this burden, Ledoit and Wolf [34] proposed a Stein- +type shrinkage estimator of the sample covariance matrix. Fol- +lowing these footsteps, Nystrup et al. [1] considered a shrink- +age estimator of the cross-correlation rather than the cross- +covariance matrix to avoid problems with heteroscedasticity. +Their estimator is based on decomposing the cross-covariance +matrix into two diagonal (heterogeneous) variance matrices +Λ1/2 +hvar and a shrunk cross-correlation matrix Rsrk. +The estimator is defined as +Σsrk = Λ1/2 +hvarRsrkΛ1/2 +hvar, +(23) +Rsrk = (1 − λ)R + λIn, +(24) +9 + +where 0 ≤ λ ≤ 1 is a regularization parameter to control the +degree of shrinkage towards the identity matrix. +When λ = 1, shrinkage scaling is equivalent to scaling by the +diagonal variance matrix Λhvar. When λ = 0, it is equivalent to +scaling by the sample covariance matrix. A closed-form solu- +tion for the optimal value of λ was derived by Ledoit and Wolf +[34] by minimizing the mean squared error. This shrinkage es- +timator is ideal for a small number of data points with a large +number of parameters. With an assumed constant variance, the +optimal shrinkage parameter is expressed by, +λ = Σi�jVar(σij) +Σi�jσ2 +ij +, +(25) +where σij is the i jth element of the covariance matrix from the +base forecast errors. The variance of the estimated covariance, +Var(σi j), is computed as depicted in Appendix A of Sch¨afer and +Strimmer [35]. +Therefore, in contrast to the preceding variance and struc- +tural scaling estimators, this allows strong interrelations be- +tween time series in the hierarchy to be captured, while shrink- +age alleviates the complexities of the estimation of Σsrk due to +its size. +We refer to the shrunken estimators of Eqs. (19) and (20) as +cov and kcov respectively. +It should be noted that a variety of other well-performing es- +timators remain, including, but not limited to, Markov [1] or +spectral scaling [20] supported by alternative inverse covari- +ance shrinkage GLASSO method [1]. In the intent of limiting +the scope of this work to the evaluation of a novel hierarchical +regressor, however, the afore-presented prevailing covariance +approximation methods are favored. Figure 4 provides a visual +illustration of the encompassed techniques along with their as- +sociated topological covariance matrices. +2.4. Evaluation method +The accuracy evaluation of hierarchical forecasting perfor- +mances requires the consideration of an important principle that +common forecasting methods are exempt from, i.e., the struc- +tural scale differences inherent to hierarchical structures. In- +deed, by its nature, hierarchical forecasting creates outputs of +increasing orders of magnitudes, typically characterized by the +aggregation levels of the tree, i.e., k-levels. It consequently be- +comes crucial to take these hierarchically-impended scale dif- +ferences into account when undertaking the accuracy perfor- +mance evaluation of hierarchical forecasts, else these would +consistently produce poorer performances for the top levels of +the aggregation, where predicted values possess larger magni- +tudes. +This is commonly done by treating each aggregation level of +the tree separately first, then evaluating the relative per-level +performance of the reconciliation phase over the base fore- +cast, allowing the removal of scale differences between aggre- +gation levels. However, a relative performance evaluation does +not allow the comparison of approaches across case studies +nor the distinctive performances of forecasting and reconcili- +ation phases, which is why we propose to complement relative +performance evaluations with measures based on structurally- +scaled errors to provide an evaluation method more suited to +the evaluation of hierarchical learning regressors. +2.4.1. Relative measures +The prevailing approach employed to evaluate hierarchical +forecasting accuracy consists in scaling the accuracy perfor- +mance of the reconciliation phase over a reference base fore- +cast. +This can be done by exploiting either Relative Mean +Squared Error (RelMSE) [2] or Relative Root Mean Squared +Error (RRMSE) [1, 17, 19]. Both depict the improvement of a +given reconciliation approach compared to base forecast. We +favor RelMSE over RRMSE to align with the commonly em- +ployed Mean Squared Error (MSE) loss function of machine +learning models. The RelMSE can be expressed as +RelMSEk = +MSEk +MSEbase +k +− 1, +(26) +where the RelMSEk is computed for each aggregation level k +and k ∈ {1, 2, ..., K}. A negative entry describes a percentage +improvement of the reconciled forecast over the base forecast. +The MSEk is computed as the average error of all prediction +steps of a given aggregation level k from +MSEk j = 1 +h +h +� +t=1 +e2 +k j,t, +(27) +MSEk = 1 +Nk +Nk +� +j=1 +MSEk j, +(28) +where ek j,t = yk j,t − ˆyk j,t is the forecast error at a starting ref- +erence time t ∈ Rh of an node k j with k being the aggregation +level of the hierarchy possessing Nk elements and j the node ob- +servation. The starting reference time t points to the very first +time step considered in the hierarchy and is employed to anchor +the nomenclature of temporal as well as spatio-temporal hierar- +chies, which usually encase time frames of [t, t+m], in similar +notations as spatial ones. +2.4.2. Measures based on scaled errors +An alternative way of removing the inherent structural- +scale differences present in hierarchical structures is producing +structurally-scaled errors. This can be achieved by dividing the +error vector et = yt − ˆyt by the structural vector κstr, where +each element contains the number of nodes contributing to the +forecasted error of that aggregation level, such that +κstr = S 1m, +(29) +estr +t += et ⊘ κstr, +(30) +where ⊘ is a Hadamard division and estr +t +is the structurally +scaled error vector at a time step t. The hierarchy illustrated +in Fig. 1, for instance, gives κstr = (6, 3, 3, 1, 1, 1, 1, 1, 1). +Structurally-scaled errors can then be employed in any +given evaluation metric. +We consequently define the Mean +10 + +Structurally-Scaled Square Error (MS3E) as +MS3Ek j = 1 +h +h +� +t=1 +estr +k j,t +2, +(31) +which can be averaged either per aggregation level or over the +entire hierarchy. +3. Hierarchical learning +While traditional hierarchical forecasting approaches have +treated forecasting and reconciliation phases separately, we pro- +pose to unify these steps under a singular machine-learning +method. To introduce our approach in a step-wise manner, let +us first provide a comprehensive overview of the diverse ways +machine learning may be employed within the frame of hier- +archical forecasting, supported by the illustrative schemes pro- +vided in Fig. 5. +We start by detailing the forecasting phase composing hier- +archical forecasting with machine learning and continue with +the description of the reconciliation step and its subsequent ap- +proaches. +3.1. Hierarchical forecasting with machine learning +Machine learning regressors employed for hierarchical fore- +casting can here be employed in one of three ways. +3.1.1. Independent forecasting +First, with independent models each forecasting a unique +node of the hierarchy, see (1a) of Fig. 5. The models leverage +data information uniquely related to the considered node and +do not exchange information with one another. They produce +typical independent (base) forecasts of the hierarchy. Notable +variations of this process involve either; (i) exploiting transfer +learning across the models to allow exchange of information +throughout the hierarchy. The work of Sagheer et al. [27] pre- +cisely employed such a scheme using a top-down approach to +determine coefficients of the lower-level models as proportions +of the learnt top-level one. This process secures the coherency +of the forecasted tree all the while providing privacy protec- +tion of data from one site to another, as transferred model co- +efficients retrieve data sharing dependence. Or (ii) by employ- +ing a unique single-output model for the forecasting of each +node of the tree, namely a multivariate learner. This allows +one model to gather more information as it learns from a much +larger database than independent models. However, the dis- +advantage of this approach originates from the generalization +intention of the learnt model applied to what could be, very dif- +ferent forecasted processes, e.g., heterogeneous buildings. This +is why this approach works best when considering processes +exhibiting similar characteristics, e.g., time series range, and +typical patterns, which are generally obtained through a prior +clustering phase [36]. In addition, because the approach relies +on the formulation of a unique model, any miss-specification +could drastically impact the performance of the forecast, con- +sequently making its design a key consideration for scientists. +The loss function of independent forecasting regressors are +typically designed around a given error metric, e.g. +mean +squared error, describing the differences between forecasted +and true values. Typically +Lb�Y, �Y|Θ� = 1 +h +h +� +t=1 +�yt − ˆyt +�2, +(32) +where Lb denotes the mean square loss function between the +predicted independent base forecast set �Y subject to a set of +parameters Θ and a set of observed values Y. +3.1.2. Multi-task forecasting +Second, by taking the concept of multivariate regressors even +further, a multi-task regressor can be contemplated, see (1b) of +Fig. 5. The regressor now produces a hierarchical, dependent, +forecast of the tree as a single vector output. The model notably +accepts features from the bottom layer of the tree for spatial +hierarchies, as aggregate levels would provide redundant infor- +mation already present in the tree leaves. However, temporal +hierarchies typically benefit from the inclusion of aggregate- +level features, allowing them to exploit important information +about the time series at different frequencies. Requirements for +coherency are, however, not included with such a scheme. +The loss function of multi-task learners is similar to single- +task ones other than considering vector rather than point errors, +i.e., +Lh�Y, �Y|Θ� = 1 +h +h +� +t=1 +� +yt − ˆyt +�2. +(33) +3.1.3. Hierarchical forecasting +This takes us to our third and last approach, crystallizing the +intention and concepts behind the contributions of our work, +namely, hierarchical forecasting, see (1c) of Fig. +5. +This +technique builds on the aforementioned multi-task forecasting +model while extending it with the inclusion of a coherency- +informed learning process thanks to a custom loss function em- +ploying established coherency taxonomy from the literature. +The coherency loss function is formulated as the difference be- +tween the predicted values ˆy and its reconciled counterpart ˜y, +following the reconciliation constraint of Eq. (12). The co- +herency loss function Lc can consequently be expressed as +Lc�Y, �Y|Θ� = 1 +h +h +� +t=1 +� +ˆyt − S �S TΣ−1S �−1S TΣ−1ˆyt +�2. +(34) +To combine both accuracy and coherency in the learning pro- +cess of the regressor, the coherency loss is added to the hi- +erarchical loss function defined in Eq. +(33) forming the +hierarchical-coherent loss function Lhc, +Lhc�Y, �Y|Θ� = αLh +t + (1 − α)Lc +t , +(35) +where α ∈ [0, 1] weights the hierarchical loss against the co- +herency loss. This avoids the over-adjustment of weights during +the training of the regressor due to the addition of the coherency +11 + +1 + x1 + x2 + x3 +x +ˆ +ỹ +y +1 +2 +3 +independent +(base) forecast +coherent +forecast +Forecasting +2 +Reconciliation +Considered +hierarchy +H +input +features +Traditional Hierarchical Forecasting Method +R +BB + + x2 + x3 +x +ˆ +ỹ + yh +hierarchical +forecast +coherent +forecast +Hierarchical +Forecasting +Hard-constrained +Reconciliation +input +features +Coherency +loss function +Hierarchical Learning Method +BB +x +ˆ +y +Independent +Forecasting +BB +BB +(1a) +x +ˆ +y +Multi-task +Forecasting +BB +(1b) +yh +x +ˆ +Hierarchical +Forecasting +BB +Coherency +loss function +(1c) +yh +y +ˆ +Soft-constrained +Reconciliation +BB +Coherency +loss function +ˆ +(2a) +R +y +Hard-constrained +Reconciliation +ˆ +(2b) +ỹ +Figure 5: Hierarchical forecasting methods with highlighted traditional approach steps (top left) and proposed hierarchical learning method (top right). Forecasting +possibilities employing machine learning (here illustrated with the acronym BB, standing for Black Box) are illustrated (1a-c) along with two reconciliation +approaches (2a-b). Forecasting methods encompass independent forecasting (1a), multi-task forecasting (1b), and our proposed hierarchical learning method +(1c), working as a combined forecasting and reconciliation learner. Reconciliation approaches presented cover our machine learning method employed as a soft- +constrained coherency enforcement over the base forecast (2a) and the traditional hard-constrained coherency enforcing method (2b). +loss to the loss function. We typically set α to 0.75 for hierar- +chical forecasting to favor accuracy learning of produced pre- +dictions over coherency, yet this parameter should commonly +be tuned by hyper-parameter optimization in the validation pro- +cess of the model development, see Sec. 4.3.4 for implementa- +tion details. +The method regroups numerous key advantages of ma- +chine learning-based forecasts. +With large and rich multi- +dimensional data to learn from the regressor effectively makes +use of all the information provided by the most detailed layer +of the hierarchy, i.e., the tree leaves, all the while incorporating +hierarchical structure information as a soft-constrained learning +mechanism. Loss function augmentation via regularization and +penalty methods has grown to become the most popular way +of introducing constraints in deep learning [37, 38, 39]. Al- +though the approach comes at the price of sacrificing hard con- +straints, it has been shown that soft-constrained penalty meth- +ods perform well in practice and often exceed hard constraint +methods [40, 41]. In addition, machine learning approaches +are powerful at capturing non-linear relationships in the tar- +geted predicted values. In particular, deep-learning methods +are known for effective and automatic feature extraction from +the data, thus reducing the need for guesswork and heuristics, +which could provide a much-needed solution to the problem of +non-identifiability of the covariance matrix. +Its disadvantages are similar to those of hierarchical forecast- +ing approaches. By relying on a unique model, architecture +considerations become paramount for the accurate performance +of the regressor and consequently require careful, tailored tun- +ing, e.g. with hyper-parameter grid-search. +3.2. Reconciliation with machine learning +While our proposed hierarchical learning approach (1c) blurs +the limit between the traditionally delineated forecasting and +reconciliation steps, it can also be employed as a classic recon- +ciliation step, see (2a) of Fig. 5. Proposed as a soft-constrained +coherency regressor, the machine learning model now takes +the entire base forecast ˆy as input and outputs a coherency- +informed forecast ˆyh. The weighting coefficient α presented in +Eq. (34) can here be set to 0.25 to favor coherent outputs for ex- +ample. The evaluation of such a scheme, lays, however, outside +the scope of this work, as our contribution targets hierarchi- +cal forecasting performance evaluation on varying dimensions. +This setup rather showcases the flexibility of our approach as +applicable to both the forecasting and reconciliation phases of +traditional hierarchical forecasting methods. +For hard-constrained reconciliation, optimal reconciliation is +considered, see (2b) of Fig. 5. It imposes coherency to its input +forecast and can be employed a posteriori to the hierarchical +learning step (1c) for eventual non-coherent outputs. In addi- +tion, as an established reconciliation method, it provides a good +benchmark to evaluate the performance of our proposed method +to both forecasting (1c) and reconciliation (2a). +4. Implementation +This section details the implementation-related details of our +study, namely, considered case studies, hierarchical structures, +and predictive-learning setup. +4.1. Case studies +Our study considers two large datasets of building smart- +meter measurements to demonstrate the usability and perfor- +mance of our method to real-life scenarios. +12 + +Fox_education_Willis +Fox_education_Otilia +Fox_utility_Marian +Fox_lodging_Helen +Fox_assembly_Johnnie +Fox_education_Yolande +Fox_assembly_Renna +Fox_lodging_Wallace +Fox_education_Elizabeth +Fox_education_Wendell +Fox_assembly_Boyce +Fox_education_Melinda +Fox_education_Andre +Fox_education_Gloria +Fox_office_Alice +Fox_education_Claude +Fox_education_Leota +Fox_office_Joy +Fox_assembly_Adrianne +Fox_public_Martin +Fox_education_Theodore +Fox_assembly_Bradley +Fox_education_Cynthia +Fox_lodging_Alana +Fox_education_Tonya +Fox_public_Lauren +Fox_public_Denny +Fox_education_Heriberto +Fox_lodging_Morris +Fox_health_Lorena +Fox_assembly_Leeanne +Fox_assembly_Jerrod +Fox_education_Miguelina +Fox_assembly_Tony +Fox_assembly_Gary +Fox_education_Long +Fox_education_Carleen +Fox_education_Louie +Fox_office_Essie +Fox_office_Molly +Fox_lodging_Jina +Fox_parking_Felipa +Fox_education_Dewayne +Fox_education_Shaun +Fox_office_Rowena +Fox_education_Kim +Fox_education_Jacqueline +Fox_parking_Lynelle +Fox_office_Bernard +Fox_education_Rosie +Fox_education_Lesley +Fox_education_Shawanda +Fox_assembly_Kathie +Fox_education_Henrietta +Fox_office_Easter +Fox_education_Virgil +Fox_warehouse_Pearl +Fox_assembly_Dixie +Fox_education_John +Fox_assembly_Christie +Fox_education_Claire +Fox_food_Scott +Fox_education_Marcelina +Fox_office_Juana +Fox_public_Rhonda +Fox_education_Sterling +Fox_religion_Maurice +Fox_lodging_Stephen +Fox_education_Melvin +Fox_warehouse_Lorretta +Fox_education_Virginia +Fox_lodging_Isabell +Fox_assembly_Audrey +Fox_assembly_Terrell +Fox_lodging_Stephan +Fox_education_Ollie +Fox_education_Delma +Fox_office_Israel +Fox_education_Elois +Fox_education_Suzan +Fox_office_Edythe +Fox_assembly_Cindy +Fox_assembly_Carlos +Fox_education_Kris +Fox_education_Geoffrey +Fox_education_Shirley +Fox_education_Gayla +Fox_office_Brandy +Fox_lodging_Winifred +Fox_office_Yong +Fox_lodging_Warren +Fox_public_Bart +Fox_food_Francesco +Fox_lodging_Angla +Fox_retail_Manie +Fox_education_Tamika +Fox_assembly_Sheldon +Fox_office_Vicki +Fox_education_Janina +Fox_office_Carson +Fox_education_Charles +Fox_education_Burton +Fox_assembly_Cecelia +Fox_office_Gaylord +Fox_office_Karima +Fox_office_Sheila +Fox_office_Margarita +Fox_assembly_Lakeisha +Fox_education_Stacia +Fox_lodging_Frances +Fox_assembly_Cathy +Fox_education_Ray +Fox_office_Clayton +Fox_office_Zachary +Fox_office_Demetrius +Fox_education_Rudolph +Fox_office_Berniece +Fox_education_Nilda +Fox_office_Thelma +Fox_parking_Tommie +Fox_office_Susanne +Fox_education_Ashli +Fox_education_Kendrick +Fox_education_Etta +Fox_education_Eddy +Fox_education_Elvira +Fox_education_Eldon +Fox_education_Lilly +Fox_education_Jaclyn +Fox_assembly_Emma +Fox_education_Dominique +Fox_education_Marlana +Fox_education_Leona +0.190242 +Figure 6: Hierarchical spatial tree structure of the Fox site from Case Study 2 +Table 1: Characteristics of assembled hierarchy per case study +Characteristics +Spatial +Temporal +Spatiotemporal +Case study 1 +n +[#] +383 +37 +14,171 +m +[#] +192 +24 +4,608 +horizon +[hours] +1 +24 +24 +Case study 2 +n +[#] +140 +37 +1,998 +m +[#] +133 +24 +1,200 +horizon +[hours] +1 +24 +24 +Case Study 1 considers a total of 225 homes located in +the Netherlands, a European region under the K¨oppen climate +classification index [42] Cfb which describes mild temperate, +fully humid and warm summer regions. Anonymized measure- +ments are gathered from smart-meters collected by energy dis- +tributor Eneco at resolutions of 10 seconds over a period of +3 years starting from January 1st 2019 to the 2nd of August +2021. Weather data is assembled from publicly available Royal +Netherlands Meteorological Institute (KNMI) weather stations +measurements [43], that are paired to each building thanks to +a geo-localization process using 4 (over the 6) ZIP code dig- +its; an aggregation level that allows no anonymized user to be +geographically isolated nor identified. +Case Study 2 employs the open data set from the Building +Data Genome project 2 (BDG2) [32]. This open set was se- +lected to allow reproducibility of our method while putting for- +ward a benchmark for hierarchical forecasting in the building +sector. +The BDG2 includes 3053 energy meters from 1636 +non-residential buildings grouped by site located in Europe +and, principally, North America. The set covers two full years +(2016–2017) at an hourly resolution with multi-meter building +measurements paired with site weather data. +4.2. Hierarchies +24H_0 +6H_0 +3H_0 +1H_0 +1H_1 +1H_2 +3H_1 +1H_3 +1H_4 +1H_5 +6H_1 +3H_2 +1H_6 +1H_7 +1H_8 +3H_3 +1H_9 +1H_10 +1H_11 +6H_2 +3H_4 +1H_12 +1H_13 +1H_14 +3H_5 +1H_15 +1H_16 +1H_17 +6H_4 +3H_6 +1H_18 +1H_19 +1H_20 +3H_7 +1H_21 +1H_22 +1H_23 +59.1932 +Figure 7: Hierarchical temporal tree structure for day-ahead forecasts +Spatial hierarchies are defined by hierarchically clustering +the prediction target time series, i.e., electricity demand. This +step is carried out employing the Ward variance minimization +algorithm [44]. The obtained hierarchy is reduced in size by +cutting the tree using a defined distance threshold over visual +inspection of the derived dendrogram. In this way, hierarchi- +cal structures located below the defined distance threshold will +be clustered together, effectively reducing the number of con- +nection nodes of the tree. Figure 6 illustrates the attained re- +duced tree of the Fox site of case study 2. Temporal hierar- +chies are considered with a horizon of one day (tree root sam- +pling frequency) while reaching down to granularities of hourly +sampling intervals (tree leaves). Aggregation levels encompass +sampling frequencies every 6 and 3 hours, resulting in a tree +with sampling frequencies of 1 day, 6 hours, 3 hours and 1 hour +per k-level, as illustrated in Fig. 7. Spatio-temporal trees are +then obtained as a result of the dimensional combination of spa- +tial and temporal hierarchies, as detailed under Sec. 2.1.2. To +limit the exponential explosion in tree size from dimensional +combination, spatial trees are limited to 50 leaves in case study +2. Table 1 details the different characteristics of the considered +hierarchies per case study. +4.3. Model learning setup +In both case studies, we proceed to resample the time-series +to hourly intervals. Time-series with no cumulative missing +values larger than 2 hours are considered and smaller gaps are +interpolated via a moving average using a window size of 8 +hours. +4.3.1. Feature engineering +Data sets are then treated per dimensional batches, namely, +per site, sub-site sample, or building for spatial, spatio- +temporal, and temporal dimensional hierarchies respectively. +Features are selected based on their Maximum Information Co- +efficient (MIC) [45] computed in relation to the learning target. +MIC is a powerful indicator that captured a wide range of as- +sociations both functional and not while providing a score that +roughly equals the coefficient of determination (R2) of the data +relative to the regression function. It ranges between values of +0 and 1, where 0 implies statistical independence and 1 a com- +pletely noiseless relationship. The advantage of using MIC for +feature engineering over the more commonly employed person +13 + +Σ1 +Σ2 +Σ3 +Σn−2 +n splits +scale fit & transform +train +test +1 +batch +2 +3 +n-2 +n-1 +coherency +loss Σ +Σid +Σ1 +Σ2 +Σn−3 +Σn−2 +scale transform +Σ estimation +Figure 8: Data partitioning, transformation and covariance matrix estimation +setup +correlation indicator [46] is that it captures non-linear relation- +ships present in the data, which deep-learning models are pop- +ularly capable of detecting. We retain features exhibiting MIC +values higher than 0.25, as electric loads can typically become +quite volatile and impede MIC values with noise. +Additionally, to feed the learner with the most relevant his- +torical information of the predicted target, we select the 3 top +auto-correlation values per temporal aggregation level above +0.25 as model input features. If no target auto-correlation value +is above 0.25, we consider the most recent historical informa- +tion, i.e., tk −1 where tk is the first k-level time-step value of the +predicted horizon. +Both MIC and autocorrelation selection thresholds are set- +tings that should typically be included in the hyper-parameter +optimization of the model validation phase. While evaluating +the performance of hierarchical regressors over three varying +dimensional considerations and two different case studies, this +work considers the tuning of these thresholds to lay outside of +its scope, as such computations rapidly become excessively bur- +densome. +4.3.2. Data partitioning and transformation +Training and testing sets are then defined employing Time- +SeriesSplit, a times-series cross-validator of the sklearn pack- +age [47], with equal test-size in a rolling window setup. We +next proceed to standard normalize the data per batch using +batch-specific available historical information such that each +batch-scaler is first fitted to the current, and past, training set. +The fitted-scaler is then employed to transform batch-specific +test sets, as depicted by Fig. 8. This process avoids data leak- +age situations, which refers to the inadvertent use of data from +test sets, or more generally data not available during inference +while training a model. This typically occurs when the data +is normalized prior to partitioning for cross-validation, i.e., by +performing smoothing or normalization over the whole series +before partitioning for training and testing [48]. +While it benefits the performance of (deep) neural networks +to normalize input features and predicted target, i.e., from un- +scaled yx to scaled yz, this shatters the hierarchical relationship +of the regressors’ outputs; thus, affecting the soundness of the +coherency loss-function. To integrate the coherency loss func- +tion in such a setting, hierarchical relationships of predicted val- +ues ˆyz are restored by reverse transformation prior to coherency +loss calculation. The obtained reversed-scaled prediction ˆyx is +reconciled to ˜yx following Eq. (12) and is finally re-scaled to +˜yz to calculate the coherency loss function against its original +predicted self ˆyz. +4.3.3. Coherency settings +The estimation of the covariance matrix is performed over +test sets. For the first batch training, we employ the identity +covariance estimate id as no forecasts are yet available. Each +batch training i then comes with a new covariance matrix esti- +mate Σi that is employed in the coherency loss function of the +next training set i + 1, see Fig. 8. This setup echoes the adap- +tive covariance matrix estimation proposed by [19] employed +for temporal hierarchies, anchored here quite organically in the +learning process of neural networks. +4.3.4. Designing hierarchical regressors +We select deep neural network regressors to best serve +the benchmarking of hierarchical-coherent forecasts. +Such +machine-learning regressors possess well-developed packages +supporting custom implementations that serve our approach +well. The regressor is structured as a series of sequential layers +decreasing proportionally in size, from initial input layer size +to the desired output dimension n. The optimal number of lay- +ers is selected from monitored test-set prediction performances +while step-wise increasing the network’s depths starting from +shallow 1-layer perceptrons. This allows the selected architec- +ture to serve an ”as simple as possible yet as complex as nec- +essary” design. Model hyper-parameters are later tuned over +a concise grid encompassing loss function parameter α, acti- +vation functions, and dropout fraction, further improving the +performance of the model. These tests resulted in the design of +a deep neural network of 3 layers, leveraging sigmoid activa- +tion functions and dropout ratios of 0.2 on all but the last layer +favoring a linear activation and no dropouts, and a retained α +coefficient value of 0.75 The presented models of Sec. 3 were +implemented in Python using the Keras package [49]. +5. Results +We describe the outcome of the implementation here over +spatial, temporal and spatio-temporal hierarchical structures per +case study. +In particular, we evaluate the accuracy and co- +herency of the forecasted building loads outlined in an anno- +tated heatmap and bar plot respectively, where the presented +coherency loss relates solely to the output of the forecasting +method, i.e., reconciliation referred to None, as reconciled fore- +casts all possess null coherency losses. Evaluated forecasting +methods cover the independent (base), multi-task, and hierar- +chical forecasting methods presented under Sec. 3.1. Hier- +archical forecasting and reconciliation methods each consider +the covariance approximations presented under Fig. 4, i.e., or- +dinary least square (id), structural (str), heterogeneous vari- +ance (hvar), homogeneous variance (svar), shrunken covari- +ance (cov) and shrunken k-level covariance (kcov). Necessary +14 + +0 +72000 +144000 + [kWh] +1.783e+05 +36.44 +12.87 +6.196 +13.12 +14.75 +35.68 +15.41 +None +4.122e+05 4.141e+05 4.045e+05 4.146e+05 4.042e+05 4.077e+05 4.105e+05 4.051e+05 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +5.703e+05 4.142e+05 4.046e+05 4.147e+05 4.042e+05 4.078e+05 4.105e+05 4.051e+05 +4.064e+05 4.141e+05 4.045e+05 4.147e+05 4.042e+05 4.078e+05 4.105e+05 4.051e+05 +4.277e+05 4.141e+05 4.046e+05 4.147e+05 4.042e+05 4.078e+05 4.105e+05 4.051e+05 +4.026e+05 4.141e+05 4.046e+05 4.147e+05 4.042e+05 4.078e+05 4.105e+05 4.051e+05 +4.181e+05 4.142e+05 4.046e+05 4.147e+05 4.042e+05 4.078e+05 4.106e+05 4.051e+05 +4.277e+05 4.141e+05 4.046e+05 4.147e+05 4.042e+05 4.078e+05 4.105e+05 4.051e+05 +405000 +410000 +415000 +420000 +425000 +430000 + MS3E [kWh] +Figure 9: Spatial hierarchy forecasting performance of case study 1. To allow +the differentiation of performances across the heatmap, extreme values were cut +off from the color map. +computational resources inherent to the forecasting methods are +also discussed. +5.1. Case study 1 +5.1.1. Spatial +Performances of spatial hierarchical forecasts are presented +under Fig. 9, where illustrated hierarchical losses showcase +svar as the best hierarchical forecast performer, with and with- +out reconciliation. The lowest hierarchical MS3E originates +from base forecast reconciled with id covariance matrix ap- +proximation, while hvar and kcov also notably perform quite +poorly for this forecasting method. Overall, the performance +of the forecasts seems to rely more on the selected forecasting +method rather than their reconciliation approaches. +Coherency losses seem in line with expected results; base +forecast is showcased as the most incoherent outcome, holding +coherency errors ranging up to 1.783e5 kWh, while multi-task +and hierarchical regressors score MS3Es of 36 kWh and 16 +kWh (on average) respectively. +5.1.2. Temporal +Temporal hierarchical forecasting performances, on the other +hand, portray a much different behavior. As illustrated by Fig. +10, it is here the base and multi-task regressors that possess the +lowest hierarchical losses, with 1.232e6 and 1.116e6 kWh re- +spectively. The poorer performer without reconciliation in this +setup is svar, with an MS3E of up to 3.23e6 kWh. Extreme +poor performances are noticeable for the cov and kcov reconcil- +iations of id and svar forecasting methods. Overall, the perfor- +mance of the forecasting methods here seems also more driven +by the considered forecasting method than reconciliation. +In terms of coherency, the base forecast surprisingly exhibits +the most coherent outputs with an MS3E of 7925 kWh. +It +0 +108000 +216000 + [kWh] +7925 +1.922e+05 +1.535e+05 +4.682e+04 +1.37e+05 +1.8e+05 +2.704e+05 +2.04e+05 +None +1.232e+06 1.166e+06 2.527e+06 1.974e+06 2.652e+06 +3.23e+06 +2.233e+06 2.532e+06 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +1.265e+06 1.318e+06 2.321e+06 1.755e+06 2.428e+06 3.041e+06 2.238e+06 2.347e+06 +1.227e+06 1.296e+06 2.301e+06 1.818e+06 2.456e+06 3.023e+06 2.198e+06 2.371e+06 +1.209e+06 +1.9e+06 +2.346e+06 1.825e+06 2.421e+06 2.968e+06 2.366e+06 2.393e+06 +1.21e+06 +1.535e+06 2.265e+06 1.804e+06 2.388e+06 3.009e+06 2.219e+06 2.315e+06 +1.318e+06 2.754e+06 6.478e+06 1.843e+06 2.778e+06 6.925e+06 2.562e+06 2.535e+06 +1.235e+06 1.911e+06 4.151e+06 1.861e+06 2.439e+06 4.091e+06 2.283e+06 2.295e+06 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 + MS3E [kWh] +1e6 +Figure 10: Temporal hierarchy forecasting performance of 40 buildings from +case study 1. To allow the differentiation of performances across the heatmap, +extreme values were cut off from the color map. +is followed by str, and all other hierarchical forecasts which +compare considerably worst featuring inconsistency errors of +4.682e4 kWh and 1.691e5 kWh (on average) respectively. +5.1.3. Spatio-temporal +Finally, spatio-temporal forecasting performances exposed in +Fig. 11 reveal contrasting outcomes compared to previous hi- +erarchies. First, all cov and kcov reconciliations here perform +extremely poorly, irrespective of the forecasting method em- +ployed, with hierarchical losses ranging between 1.57e6 and +2.62e6 kWh. +Similarly to the temporal hierarchy, base and +multi-task forecasts perform overall better than hierarchical +ones. The multi-task regressor without reconciliation is show- +cased as the best performer in this setup with an MS3E of +3.187e5 kWh. It can notably be observed here that all hier- +archical and multi-task forecast reconciliations do not improve +the accuracy of their original forecast. Additionally, exposed +performances here display a much stronger dependency on the +considered reconciliation approach than forecasting. +Concerning coherency losses, spatio-temporal hierarchies +produce two distinct performances; where base and multi-task +forecasts exhibit inconsistencies of 1 order of magnitude lower +than all hierarchical ones, i.e., 3.255e4 kWh against 1.878e5 +kWh on average. +5.2. Case study 2 +Concerning case study 2, the spatial hierarchical forecasting +performance presented under Fig. 12, depicts noticeable varia- +tions from case study 1. +5.2.1. Spatial +Here, the base case exhibits the most accurate forecast, al- +though at the cost of higher inconsistencies across the tree. +15 + +0 +86000 +172000 + [kWh] +3.462e+04 3.047e+04 +1.945e+05 2.163e+05 +1.699e+05 1.809e+05 1.783e+05 1.869e+05 +None +3.28e+05 +3.187e+05 3.905e+05 4.083e+05 3.954e+05 3.907e+05 3.925e+05 3.979e+05 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +3.467e+05 3.495e+05 6.276e+05 6.743e+05 5.983e+05 6.071e+05 +6.04e+05 +6.319e+05 +3.271e+05 3.451e+05 4.216e+05 +4.48e+05 +4.337e+05 4.247e+05 +4.26e+05 +4.318e+05 +3.255e+05 +4.59e+05 +5.232e+05 5.514e+05 5.338e+05 5.231e+05 5.261e+05 5.377e+05 +3.253e+05 3.442e+05 4.042e+05 4.208e+05 4.095e+05 4.063e+05 4.077e+05 4.104e+05 +1.57e+06 +2.352e+06 1.962e+06 1.869e+06 +2.12e+06 +1.947e+06 1.935e+06 2.034e+06 +2.62e+06 +2.021e+06 +2.2e+06 +2.061e+06 2.356e+06 2.275e+06 2.166e+06 2.278e+06 +350000 +400000 +450000 +500000 +550000 +600000 +650000 + MS3E [kWh] +Figure 11: Spatio-temporal hierarchy forecasting performance of 41 buildings +from case study 1. To allow the differentiation of performances across the +heatmap, extreme values were cut off from the color map. +0 +80 +160 +240 + [kWh] +244.8 +0.1558 +4.364 +0.3183 +11.98 +14.4 +65.02 +19.18 +None +615.1 +1179 +1231 +1285 +1285 +1188 +1717 +1663 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +834.6 +1179 +1233 +1277 +1288 +1190 +1721 +1664 +616.5 +1179 +1232 +1283 +1285 +1188 +1721 +1664 +636.4 +1180 +1230 +1279 +1288 +1191 +1727 +1663 +611.5 +1179 +1231 +1284 +1285 +1188 +1718 +1663 +577.1 +1180 +1235 +1286 +1290 +1195 +1654 +1677 +636.4 +1180 +1230 +1279 +1288 +1191 +1727 +1664 +600 +800 +1000 +1200 +1400 +1600 + MS3E [kWh] +Figure 12: Spatial hierarchy forecasting performance of the Fox site of case +study 2 +0 +24000 +48000 + [kWh] +180.4 +6.213e+04 +1.018e+04 +2127 +1.407e+04 1.099e+04 +2.416e+04 2.04e+04 +None +1593 +1164 +1.724e+04 +3.005e+04 +4.808e+04 +1.543e+04 +4.004e+04 +4.566e+04 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +1618 +6.327e+04 +3.506e+04 +4.163e+04 +5.97e+04 +3.115e+04 +4.481e+04 +5.86e+04 +1528 +3.626e+04 +3e+04 +3.438e+04 +5.637e+04 +2.953e+04 +4.319e+04 +5.717e+04 +1515 +3.217e+04 +3.753e+04 +3.541e+04 +6.09e+04 +3.739e+04 +4.445e+04 +6.193e+04 +1523 +2.81e+04 +3.885e+04 +3.655e+04 +5.865e+04 +3.294e+04 +4.319e+04 +5.644e+04 +1468 +3.974e+04 +3.397e+04 +3.469e+04 +5.751e+04 +3.424e+04 +4.442e+04 +5.609e+04 +1520 +3.362e+04 +3.514e+04 +3.466e+04 +5.814e+04 +3.563e+04 +4.372e+04 +5.854e+04 +10000 +20000 +30000 +40000 +50000 +60000 + MS3E [kWh] +Figure 13: Temporal hierarchy forecasting performance of 66 buildings from +the Fox site of case study 2 +Multi-task forecasts followed by structural, str, hierarchical +ones both produce the most coherent outcomes. Surprisingly, +while the multi-task forecast is trained without coherency in- +formation, its forecast displays the best coherency performance +in this scenario. +Overall, the best forecast accuracy is obtained from base +forecasting reconciled with the cov approximation, while the +worst performer for this scenario is the cov hierarchical fore- +casting with either kcov or hvar covariance approximations. It +displays hierarchical MS3Es ranging from 611.5 to 1.727e3 +kWh and coherency MS3Es varying between 0.156 and 245 +kWh. +5.2.2. Temporal +The averaged temporal hierarchical forecast performance of +66 buildings from the Fox site is exposed under Fig. 13. Fore- +casting performances are overall significantly worse than those +of spatial-hierarchies, with hierarchical MS3Es now ranging +between 1.164e3 and 6.327e4 kWh, while coherency losses +fluctuate from 180 to 6.213e4 kWh; an order of magnitude +about 3 times higher than temporal trees. +Here, the best- +performing forecast belongs to the multi-task forecast with no +reconciliation, which also displays the highest inconsistency +score. The lowest performing forecast interestingly resides with +the id reconciliation of that same forecast. +The most coherent forecast produced for temporal-trees pe- +culiarly originate from base forecasts, which neither share +information across the hierarchy, nor possess coherency- +knowledge. +Other hierarchical forecasts produce coherency +losses ranging between 2.120e3 and 2.416e4 kWh. +5.2.3. Spatio-temporal +Lastly, the forecast performance of spatio-temporal struc- +tures considering 50 buildings from the Fox site is presented un- +der Fig. 14. Similarly to the temporal-tree, hierarchical losses +16 + +0 +30000 +60000 + [kWh] +391.1 +7.603e+04 +2.567e+04 1.952e+04 +4.881e+04 +2.72e+04 +4.909e+04 4.764e+04 +None +1692 +1485 +1.335e+04 +4.077e+04 +1.939e+04 +1.576e+04 +1.957e+04 +1.869e+04 +base +multitask +hierarchical - id +hierarchical - str +hierarchical - hvar +hierarchical - svar +hierarchical - cov +hierarchical - kcov +Forecasting method +id +str +hvar +svar +cov +kcov +Reconciliation method +1846 +7.749e+04 +5.816e+04 +2.264e+05 +7.484e+04 +6.526e+04 +7.556e+04 +7.066e+04 +1644 +4.489e+04 +4.315e+04 +6.891e+04 +4.939e+04 +4.809e+04 +4.98e+04 +4.675e+04 +1637 +7.749e+04 +7.165e+04 +8.689e+04 +7.697e+04 +7.436e+04 +7.558e+04 +7.417e+04 +1622 +2.57e+04 +3.598e+04 +4.522e+04 +3.914e+04 +3.76e+04 +3.923e+04 +3.849e+04 +2024 +7.749e+04 +6.701e+06 +1.44e+06 +3.018e+07 +1.321e+09 +6.711e+05 +9.57e+09 +2119 +7.749e+04 +7.164e+04 +8.689e+04 +7.696e+04 +7.435e+04 +7.557e+04 +7.416e+04 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +80000 + MS3E [kWh] +Figure 14: Spatio-temporal hierarchy forecasting performance of 50 buildings +from the Fox site of case study 2. To allow the differentiation of performances +across the heatmap, extreme values were cut off from the color map. +Table 2: Averaged computing times (in seconds) of evaluated forecasting meth- +ods +tree size +forecasting method +n +base +multi-task +hierarchical +Case study 1 +14,171 +3.6 +392 +397.3 +383 +34.9 +90 +96.6 +37 +12.2 +12 +13 +Case study 2 +1,998 +2.5 +70 +77 +140 +22.5 +70 +80.6 +37 +2.2 +20 +19.3 +display much poorer performances compared to their spatial an- +tecedent, with MS3Es ranging between 1.485e3 and extreme +9.57e9 kWh values, while coherency losses vary from 391 to +7.603e4 kWh. +Mirroring the results from temporal-hierarchies, the forecast- +ing technique withholding the lowest hierarchical loss is the +multi-task learner without reconciliation which is also charac- +terized by the highest coherency loss. A series of extreme poor +performers are identified as a result of the cov reconciliation +over all hierarchical-learners. Contrary to temporal-tree, recon- +ciled forecasts performances here seem driven by the reconcil- +iation method rather than the considered forecasting technique. +Coherency scores display overall poor performances across +all hierarchical and multi-task learners with losses ranging 2 +orders of magnitude higher than the best case base regressor. +5.3. Computational prospects +Computational performances of forecasting approaches are +here considered, providing a complete overview of evaluated +methods. Table 2 presents the computation time of each fore- +casting method averaged over all training batches. Two antici- +pated findings can be noted from it. +First, the computing time is positively correlated to the size +of the hierarchy. One exception seems to deviate from that rule +in case study 2, between tree sizes of 1,998 and 140, which dis- +play relatively close computing times. Second, smaller regres- +sors, i.e, base, train faster than larger ones, namely, multi-task +and hierarchical. Both these observations can be explained by +the increasing number of weights to update in the larger regres- +sor. The more weights to update, the longer the training will +take. +Although independent regressors seem attractive due to their +noticeably faster computing times, it should be noted that the +displayed performances depict only the average computing time +of a unique independent regressor. Should such regressors not +be trained and tested in a distributed computational setup, then +these numbers would need to be multiplied by the hierarchy size +to obtain an appropriate estimation of the required computing +period. +6. Discussion +Although presented case studies bear varying results, these +also display a number of commonalities supporting interpreta- +tion and analysis, which are here discussed. +6.1. Hierarchical-coherency value +Unifying the forecast of hierarchical structures under one re- +gressor possesses attractive data-efficient prospects, i.e., cross- +tree information exchange combined with embedded-structural +learning provided from coherency loss. +However, produced +outcomes from hierarchical-coherent learners were only found +to bring added value in one setting, namely, the spatial hierar- +chy of case study 1. This can be explained by the similarities in +building loads of case study 1, which encompassed time series +of similar patterns and dynamics, all originating from residen- +tial constructions, while case study 2 included a broader collec- +tion of construction types covering offices, college classrooms, +lodging, warehouses, and parkings. Such profile diversities are +challenging to learn from limited measurements, particularly +for a large model involving considerable numbers of regression +weights. +It can consequently be found that while the results of the spa- +tial hierarchy of case study 1 are promising, these unveil, in fact, +important challenges hierarchical forecasting must face. +6.2. An efficient but arduous learning process +Although the outcome of hierarchical learning demonstrated +promising performances, identified in the spatial hierarchy of +case study 1, the resulting number of weights to update and pos- +sibly conflicting forecasted outputs can become burdensome, +i.e., as unveiled by the performance of the spatial hierarchy of +case study 2. Indeed, with hierarchical regressors growing in +size, their number of neuron connections increases by an expo- +nential factor of 2. This renders the learning process of these +models laborious as more data should support the learning of +larger number of weights. Additionally, multi-output regressors +are faced with the challenging task of predicting numerous out- +comes which might exhibit highly different, possibly antipodal, +dynamics. This also affects the learning process, which might +17 + +2017-02-03 +2017-02-13 +2017-02-23 +2017-03-05 +2017-03-15 +2017-03-25 +2017-04-04 +2017-04-14 +2017-04-24 +2017-05-04 +2017-05-14 +2017-05-24 +2017-06-03 +2017-06-13 +forecasted date +0 +100 +200 +300 +400 +500 +600 +electric loads [kWh] +Figure 15: Illustration of faulty coherent-learning from normalized trees. The +predicted (red) versus true (black) electric loads of the Fox assembly Lakeisha +temporal hierarchical tree showcase the mirrored top-level forecast predicted in +the negative domain. +struggle to identify these discrepancies from limited training +data. +6.3. Induced coherency over accuracy +Overall, temporal hierarchies of the considered case studies +were seen to perform significantly worse than spatial ones. This +significant change can be attributed to the combination of two +factors. First, the longer forecasting horizon of temporal trees +compared to spatial ones, i.e., 24 hours against 1, implies that +forecasts must rely on fewer data and less recent information +while dealing with higher uncertainties, thus negatively affect- +ing their performances. Secondly, building electrical loads are +endowed with a periodicity that falls precisely on the forecasted +horizon of 24 hours. This consequently leads to little variations +in the forecasted element of its hierarchy. And, while this char- +acteristic is desirable for ordinary forecasting, the addition of +the coherency-loss function, although weighted by the α coeffi- +cient - see Eq. (35), may push the regressor to produce constant +predictions, tailored more to coherency than accuracy, thus re- +sulting in unrealistic and inaccurate predictions. +6.4. Faulty coherent-learning from normalized trees +In some settings, hierarchical-coherent learning displayed +particularly poor performances from extreme hierarchical and +coherency errors, i.e., temporal and spatio-temporal hierar- +chies. Following further inspection, it was noticed that these +poor performers all withheld abnormal top-level forecasts +which mirrored their expected true values in the negative do- +main, as illustrated in Fig. 15. These undesirable, yet pecu- +liarly common, results can be traced back to the normaliza- +tion of the target hierarchical time series. Indeed, while neu- +ral networks benefit from normalized targets, serving fair and +balanced learning across the network’s weights, this also shat- +ters the coherency structure of the tree. The existing setup, de- +tailed in Sec. 4.3.2, proceeds to tackle this issue by reverse- +transforming these target values prior to the coherency con- +straint computation and re-scaling them for coherency loss cal- +culation. This ensures both loss functions, namely hierarchical +and coherency, see Eqs. (33) and (34) respectively, to oper- +ate on akin normalized time series. However, coherency learn- +ing can eventually produce adjustments larger than the origi- +nal normalization ranges, e.g., lowering the top-level forecast +ˆyz fully into the negative domain such that the reverse stan- +dard transformation ˆyx = ˆyz · u + s, where u and s refer to +the mean and standard deviation of the fitted time series respec- +tively, also produces a fully negative reverse-scaled ˆyx. This +evidently improper outcome consequently negatively impacts +both the learning and the forecasting performance of the regres- +sor and should be dealt with in future work. +7. Conclusion +Ensuring coherent previsions of the future is crucial to sup- +port better informed and aligned decision-making processes +across hierarchical structures. And while previous works have +attempted to exploit spatio-temporal hierarchical reconciliation +using disparate steps [2, 9, 31, 33], no common formulation of +multi-dimensional hierarchical structures had, to this date, been +proposed. Furthermore, traditional hierarchical forecasts use +disjointed forecasting and reconciliation processes that inher- +ently deprive forecasting algorithms of (i) the benefits of infor- +mation transfer across (hierarchical) models, as well as (ii) cap- +italizing on the coherency requirements of the produced fore- +cast. This paper proposes a solution to these shortcomings. +First, by formally defining multi-dimensional hierarchi- +cal structures, it extends conventional hierarchical forecasting +methods, allowing the exploitation of spatio-temporal struc- +tures unified under a common frame, i.e., a unique summation +and covariance matrix resulting from spatio-temporal function +composition. +Second, rather than considering reconciliation a posteriori +to forecasting, this work brings together independent forecast- +ing models into a unique machine-learning regressor embedded +with coherency information. This provides the regressor with +(i) a global overview of information across its hierarchy, permit- +ting a cross-dimensional and data-rich learning process, while +(ii) learning coherency-requirements as a soft constraint thanks +to a custom hierarchical-coherent loss function. The approach +can notably be tuned thanks to an adjustable α coefficient to ei- +ther consider multi-task, hierarchical or only reconciliation in +its learning process. Coherency of the produced hierarchical +forecasts can then be enforced as a hard constraint using es- +tablished reconciliation technics. The outcome is a unified and +coherent forecast across all examined dimensions, granting a +common view of the future serving aligned and better decision- +making. The approach provides a data-driven solution to as- +semble diverging parts of an organization and blend informa- +tion from varying sources, hierarchy levels, or scales [2]. +Third, we evaluated our approach on two different case stud- +ies, across all hierarchical dimensions, considering established +state-of-the-art reconciliation approaches. Results revealed spa- +tial hierarchies to perform best while temporal and spatiotem- +18 + +poral structures suffered from coinciding forecasted horizon +with the periodicity of electric loads from buildings. Although +the value potential of hierarchical-coherent learning was ob- +served in case study 1, the performances of the approach were +quite disparate in other settings. In this regard, a comprehen- +sive analysis was reported revealing important challenges the +approach faces. In particular, dealing with predicted outputs of +conflicting trends while fitting an exponentially large number +of weights to the model is a recurring fragility of the approach. +Additionally, correcting faulty coherency training from normal- +ized tree structures is another frailty future work may tackle. +Finally, to encourage knowledge dissemination we ren- +der our work fully replicable by open-sourcing developed +python implementations under the public GitHub repository +https://github.com/JulienLeprince/hierarchicallearning. +7.1. Outlooks and future work +This paper proposes a novel hierarchical learning method +yielding important implications for forecasting theory. Indeed, +by directly forecasting hierarchies this work opens the door to +leveraging multi-scale and multi-frequency measurement infor- +mation driving improved forecast accuracies. +It notably ex- +pands and unites traditionally disjointed methods together pro- +viding a path toward a novel generation of forecasting regres- +sors. +Meanwhile, numerous directions for future work can already +be distinguished, guiding attempts to tackle uncovered obsta- +cles. +As such, the curse of dimensionality endowed from +larger, unified hierarchical models can notably be undertaken +by investigating distributed and connected models working as +a hybrid solution between independent, but tractable regres- +sors and extensive hierarchical ones. In addition, varying na- +tive multi-output machine learning algorithms may be exam- +ined such as ensemble decision trees, Gaussian processes, K- +Neighbors, long short-term memory neural networks, and sup- +port vector machines. In particular, algorithms that deal with +different ranges of target values can naturally tackle issues with +coherency learning due to scaling. Finally, comparing hierar- +chical learning performances against established models, i.e., +grey- or white-box, that benefit from the inclusion of domain +expertise to tackle targeted behaviors, such as seasonality, ad- +vances another interesting endeavor for future work. +8. CRediT authorship contribution statement +Julien Leprince: Conceptualization, Methodology, Soft- +ware, Formal analysis, Investigation, Data curation, Writing - +original draft, Writing - review and editing, Visualization. Hen- +rik Madsen: Methodology, Supervision, Validation, Writing - +review and editing. Jan Kloppenborg Møller: Methodology, +Supervision, Validation, Writing - review and editing. Wim +Zeiler: Supervision, Funding acquisition. +All authors have read and agreed to the published version of +the manuscript. +9. 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Chollet, et al., Keras (2015). +URL https://github.com/fchollet/keras +20 + diff --git a/XdFOT4oBgHgl3EQf8zQL/content/tmp_files/load_file.txt b/XdFOT4oBgHgl3EQf8zQL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..291cef35c325023bde3633b55abd5fcec2f06408 --- /dev/null +++ b/XdFOT4oBgHgl3EQf8zQL/content/tmp_files/load_file.txt @@ -0,0 +1,1928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf,len=1927 +page_content='Hierarchical learning, forecasting coherent spatio-temporal individual and aggregated building loads Julien Leprincea,b,∗, Henrik Madsenb, Jan Kloppenborg Møllerb, Wim Zeilera aTechnical University of Eindhoven, 5 Groene Loper, Eindhoven 5600 MB, the Netherlands bTechnical University of Denmark, Building 303B Matematiktorvet, Lyngby 2800, Denmark Abstract Optimal decision-making compels us to anticipate the future at different horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, in many domains connecting together predictions from multiple time horizons and abstractions levels across their organization becomes all the more important, else decision-makers would be planning using separate and possibly conflicting views of the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This notably applies to smart grid operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To optimally manage energy flows in such systems, accurate and coherent predictions must be made across varying aggregation levels and horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Such hierarchical structures are said to be coherent when values at different scales are equal when brought to the same level, else would need to be reconciled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' With this work, we propose a novel multi-dimensional hierarchical forecasting method built upon structurally-informed machine-learning regressors and established hierarchical reconciliation taxon- omy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A generic formulation of multi-dimensional hierarchies, reconciling spatial and temporal hierarchies under a common frame is initially defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Next, a coherency-informed hierarchical learner is developed built upon a custom loss function leveraging optimal reconciliation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Coherency of the produced hierarchical forecasts is then secured using similar reconciliation technics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The outcome is a unified and coherent forecast across all examined dimensions, granting decision-makers a common view of the future serving aligned decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The method is evaluated on two different case studies to predict building electrical loads across spatial, temporal, and spatio-temporal hierarchies, benchmarked against base and multi-task regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Although the regressor natively profits from computationally efficient learning thanks to the unification of independent forecasts into a global multi-task model, results displayed disparate performances, demonstrating the value of hierarchical-coherent learning in only one setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Yet, supported by a comprehensive result analysis, existing obstacles were clearly delineated, presenting distinct pathways for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Particularly, investigating reduced number of model weights and varying native multi-output regressors can tackle the two preeminent challenges that are the curse of dimensionality and coherency-learning complications from scaled trees respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Overall, the paper expands and unites traditionally disjointed hierarchical forecasting methods providing a fertile route toward a novel generation of forecasting regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Keywords: Hierarchical forecasting, Coherency, Spatio-temporal dimensions, Deep learning, Smart building 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Introduction A better anticipation of the future supports better decision- making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This is true across all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Yet, more accurate fore- casts alone often do not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' When dealing with different abstraction levels across a system or organization, it is com- monly more important to obtain coherent predictions across all considered layers and horizons, not to result in unaligned deci- sions or possibly even conflicting ones [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This obstacle arises in multiple domains, including tourism [2, 3], retail [4], stock management [5] and smart grid management [6], which show- cases this matter quite adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Traditionally, smart grid operators focused on forecasting the system’s total demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, with the increasing adoption of smart meters at grid edges and substations, the focus is shift- ∗Corresponding author Email addresses: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='leprince@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='nl (Julien Leprince ), hmad@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='dk (Henrik Madsen), jkmo@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='dk (Jan Kloppenborg Møller), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='zeiler@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='nl (Wim Zeiler) ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Grid management now benefits from high-frequency mea- surements available at multiple levels of aggregation allowing accurate forecast estimations across both spatial and temporal scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', from sub-meters to regional-level, with per seconds to monthly aggregated information [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Yet, the pluralities and independence of models and their consequent forecasts in- evitably produce inconsistencies across aggregation levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', lower-level predictions might not sum up to higher-level ones and vice-versa [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The consequent challenge decision-makers are now faced with is to obtain coherent predictions across the different horizons and scales of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical struc- tures (or trees) are said to be coherent when their values at the disaggregate and aggregate scales are equal when brought to the same level [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Should forecasts not be coherent, decision- making units would be planning using diverging views of the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Optimal decision-making consequently requires fore- casts to be coherent across all considered dimensional hierar- chies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Preprint submitted to Applied Energy January 31, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='12967v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='LG] 30 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical forecasting Enforcing coherency in hierarchical structures is a concept that dates back to 1942 [10] and was first defined in 1988 as rec- onciliation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It leverages linear balancing equations from covariance compositions inherent to hierarchical structures to optimally re-adjust coherency mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hyndman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [12] later reformulated the approach with a unifying statistical method, independent of prediction models, along with notations more appropriate to hierarchical forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical forecasting can thus be defined as the process in which coherent predictions need to be made within a fixed hi- erarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Commonly, forecasts are first estimated separately considering each series of the hierarchy in a dis- jointed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' These forecasts are designated as independent base forecasts [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Generating base forecasts for each series implies that specialized models can be developed for each part of the hierarchy, incorporating node-specific available informa- tion [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Base forecasts are then linearly combined (reconciled) leveraging available information across the hierarchy to ensure coherency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' a process employed by all hierarchical forecasting approaches as of to date [1, 3, 5, 6, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Reconciliation approaches Predominant reconciliation techniques comprise traditional bottom-up and top-down approaches, trace minimization, op- timal combinations, and recently developed machine-learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Bottom-up hierarchical forecasting consists in generating base forecasts at the very bottom level of the hierarchy and enforce coherency through their direct aggregation across the tree [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The greatest advantage of this approach is that it can draw information from the most disaggregated levels of the tree, consequently avoiding any information loss from aggrega- tion [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, series located at tree leaves tend to possess low signal-to-noise ratios making them more difficult to pre- dict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This is particularly true when dealing with smart-meter electrical demands which are notoriously volatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Consump- tion peaks are indeed driven by often highly stochastic occu- pant behaviors that are close to intractable, consequently mak- ing bottom-up aggregation unlikely to provide accurate fore- casts across the upper levels of the tree [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Top-down hierarchical forecasting on the other hand only generates forecasts for the top level of the hierarchy (tree-root) and proceeds to disaggregate and distribute it down the hierar- chy from either historical [23] or forecasted [3] proportions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The approach commonly favors higher aggregation levels of the tree with more accurate predictions and is notably valuable for low-count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, aggregation is not with- out a large loss of information as temporal dynamics and other individual series characteristics cannot be exploited [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Addi- tionally, as the success of this approach depends solely on one top-level model, it possesses a higher degree of risk from model misspecifications or inaccuracies [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Given both bottom-up and top-down approaches inadequate to profit from the rich- ness of information across a given hierarchy, optimal combina- tion techniques emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Linearly reconciling base forecasts towards coherency, these approaches allowed interactions be- tween different levels of the hierarchy, leveraging in particular correlations and covariances present in such structures [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, estimating the covariance structure of a hierarchy from base forecasts is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, Wickramasuriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [13] declared that the covariance matrix of the coherency errors is ”impossible to estimate in practice due to identifiabil- ity conditions” such that even with high-frequency data avail- able, assumptions on its form must be made [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The ordi- nary least-square (OLS) estimator was particularly developed by Hyndman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [12] and Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [3] to avoid this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Their approach demonstrated improved results compared to other commonly adopted techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A weighted least squares (WLS) approach, considering variances from the variance-covariance matrix diagonal but ignoring the off-diagonal covariance elements, was put forward by Hynd- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Wickramasuriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [13] later provided the theoretical justification for estimating variances from base fore- cast error variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They proposed a generalized least-squares (GLS) estimator and found the incorporation of correlation in- formation into the reconciliation process to benefit forecasting accuracy, with resulting reconciled forecasts guaranteed to be, in mean or in sample, at least, as good as their base forecasts, given a particular covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Finally, in recent years, machine learning approaches have made their way into hierarchical forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Relying on power- ful statistical regressors and the availability of larger and richer data sets, machine learning emerges as an appealing and suit- able tool for estimating the persistently challenging covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spiliotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [5] put forward such an approach em- ploying a bottom-up method to reconcile predictions from Ran- dom Forest and XGBoost regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Taking as input the base forecasts of all the series of the hierarchy, the reconciled tree is then obtained from bottom-up aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It allows non- linear combinations of the base forecasts, extending conven- tional linear approaches thanks to its machine-learning nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Sagheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [27] proceeded to obtain coherent hierarchies from deep long-short term memory (DLSTM) recurrent neu- ral networks by applying transfer learning across their hierar- chies in a bottom-up fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They evaluated their approach on national-scale Brazilian electrical power production as well as Australian domestic tourism data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In another work, Mancuso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [28] proposed a method to unify the two prevailing processes that are forecasting and reconciliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By including hierarchi- cal information in the forecasting process through a customized loss function, they allow the network to train towards reconciled forecasts using a top-down disaggregation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' None of these approaches, however, include the general formulation of hierarchical forecasting within their learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This limits their reconciliation approaches to en- compass solely traditional approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', bottom-up or top- down, which, as has been mentioned, only exploit a fraction of the available information of hierarchical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Dimensional considerations While numerous works have first approached the reconcili- ation of hierarchical structures from a spatial (cross-sectional) 2 dimensional frame perspective [3, 5, 6, 11, 12, 13, 14, 15, 16], temporal hierarchies have also been the center of recent atten- tion within the field [1, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [17] first introduced the notion of tem- poral hierarchies with forecasting reconciliation performed in the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Quite similarly to spatial reconcilia- tion, base forecasts are independently produced across a defined set of temporal aggregation levels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', weekly, daily, quarter- daily, hourly to per-minute or seconds granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows models to capture temporal-specific characteristics of the times series across the hierarchical-structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', trends or seasonal- ity possessing particular time-frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Base temporal-forecasts are then reconciled across all forecasting horizons and temporal tree-structure, allowing aligned decisions across multiple plan- ning horizons [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Nystrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [1] notably proposed temporal estimators accounting for autocorrelation structures to reconcile electric grid load forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It was found that auto- and cross- covariances significantly improved forecast accuracy uniformly across all temporal aggregation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It thus becomes clear that both spatial and temporal hier- archical forecasts produce substantial empirical accuracy im- provements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By dealing with parameter estimation errors and model misspecifications, forecast combinations have demon- strated significant error variance reduction across numerous works [29, 24, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Exploiting both available hierarchi- cal dimensions to further improve prediction accuracies con- sequently emerges as not only appealing but quite evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Kourentzes and Athanasopoulos [2] notably advanced a frame- work to produce spatial- and temporal-coherent forecasts (des- ignated as cross-temporal), supporting all hierarchical lev- els with short- to long-term forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Their work demon- strated empirical evidence that leveraging both dimensions in reconciliation offered improved accuracies compared to uni- dimensional reconciliation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', spatial or temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A find- ing certainly due to the complete information exposure the ap- proach provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spiliotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [9] later proposed a cross- aggregation process to iteratively generate coherency across spatial hierarchies from multiple temporal aggregations applied to electricity consumption forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Punia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [31] intro- duced a similar framework leveraging deep learning algorithms applied to supply chain base forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Their approach, how- ever, produced coherency solely from bottom-up approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' While the advantage of multi-dimensional hierarchical fore- cast has become evident, there exists, as of today, no generic formulation of these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, while Spiliotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [9] stated that it is possible, in principle, to design a summing matrix S that accounts for both considered dimensions of rec- onciliation, a theoretical formulation of S and its subsequent reconciliation approaches was not put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, the design of a reconciliation estimator that fully captures scaling issues and cross-sectional interdependencies is not straightfor- ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Yet, this deprives multi-dimensional reconciliations of exploiting custom dimensional considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The principal counterargument to undertaking such formulations is grounded on the fact that multi-dimensional hierarchies generate increas- ingly large tree structures that could soon become intractable to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Recently, however, the work of Nystrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [20] proposed a dimensionality reduction technique to counter this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Using eigendecomposition when reconciling fore- casts, maximum information can be extracted from the error structure using available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They find that uniformly im- proved predictions can be obtained across all aggregation lev- els, with the estimator achieving state-of-the-art accuracy all the while being applicable to hierarchies of all sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Motivation This comprehensive state-of-the-art overview underlines the following shortcomings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (i) Base forecasts are typically produced separately, consid- ering each series of the hierarchy in a disjointed man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' While this procedure allows the independence and hierarchically-tailored design of these models, it is inher- ently deprived from the benefits of data information (learn- ing) transfer across models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (ii) Machine-learning reconciliation approaches have exhib- ited clear forecast improvements potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Yet, developed approaches have, so far, not proceeded to put forward a unified method for machine-learning based hierarchical- forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This limits considered reconciliations ap- proaches to the more information-limited bottom-up and top-down approaches [5, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Embedding advanced rec- onciliation techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', optimal combinations, in the learning process of machine learning regressors is, as of today, still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (iii) Although advantages of leveraging multi-dimensional hi- erarchies in forecasting has become evident, a generic for- mulation of such hierarchical-combinations is still needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Existing tools have demonstrated effective dimensionality reductions of large hierarchies [20], presenting promising solutions to the problem of dimension intractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This study proposes a response to this appeal and puts for- ward a generic multi-dimensional formulation for hierarchical forecasting with machine-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We put together a unified and adaptable forecasting and reconciliation method founded on native multi-task machine-learning regressors while fram- ing multi-dimensional hierarchical-forecasting approaches in a generic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Contributions of this work can be summarized as five-fold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We develop a unified machine-learning-based hierarchical forecasting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This grants (i) a unique forecast- ing model the benefit of a complete information overview across its hierarchy, while (ii) including coherency con- straints within its learning process as well as (iii) be- ing adaptable to either independent or combined forecast- ing and reconciliation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It establishes a unified method generating accurate and coherent forecasts at all levels of the hierarchy thanks to a custom hierarchical loss function leveraging coherency information from es- tablished field-taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To best exploit available information embedded within multi-dimensional data, we formulate a generic multi- dimensional extension of conventional hierarchical fore- casting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In particular, we address the prob- lem of diverging reconciliation considerations in a multi- dimensional setting with uni-dimensional couplings of the covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows the unification of multi-hierarchical structures under a common frame, fuel- ing both traditional and machine-learning approaches with ever-richer and transferable (learning) information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In the interest of addressing the dimensional tractability of our approach, we put forward dimensionality reduc- tion prospects and illustrate them both theoretically and in practice with an applied demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Our study considers two substantial smart-meter data sets including an established open source, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', the Building Data Genome project 2 (BDG2) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows the grounding of our approach thanks to a first-of-its-kind per- formance benchmark in the field of electric-meter hierar- chical predictions, which we render fully replicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To best serve knowledge dissemination and re- search reproducibility, we open-source all our de- veloped code under the public GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='com/JulienLeprince/hierarchicallearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The greatest advantage of this approach is granting ac- cess to the regressor a complete information overview of the considered (multi-dimensional) hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This permits both a cross-dimensional, data-rich learning process as well as a hierarchically-informed training for hierarchical forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The outcome is a unified and coherent forecast across all ex- amined dimensions, granting decision-makers a common view of the future serving aligned and better decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The rest of this paper is organized as follows: Section 2 de- tails traditional hierarchical forecasting prospects and extends them to multi-dimensional frames, while Section 3 presents the hierarchical learning methods put forward with developed cus- tom loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Section 4 introduces the implementation specifics of our applied method from two different case studies, and Section 5 reports the performance results of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Highlighted findings are analyzed and detailed under the dis- cussion Section 6, followed by Section 7 which concludes the paper and reveals future work outlooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical forecasting In this section, we present the foundations of hierarchical forecasting as defined by Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [3] and Wick- ramasuriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [13] and extend them to multi-dimensional frames with a generic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We discuss dimensionality tractability limitations and offer dimensionality reduction con- siderations to address them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 21 31 33 22 34 35 11 36 k1 = m k-levels k2 = 3 kK = 1 m = 6 32 Figure 1: A two-level hierarchical tree diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical structures Let us refer to the simple hierarchy of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1 to demon- strate the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Every element (node) of the hierar- chy (tree) can be labeled as yk j, where the subscripts k and j stand for the aggregation-level and node observations respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We define k1 as the most aggregate level of the hierarchy (tree root), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', node y11, and kK as the most disaggregate level (tree leaves), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', nodes yK j where j ∈ [1 : m] and K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In such a setting, two important components must be considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' the number of nodes in the bottom level of the hierarchical tree, which is denoted as m, and the total number of nodes on the tree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Here n = 9 and m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Stacking all tree elements in a n-dimensional vec- tor y = (y11, y21, y22, y31, y32, y33, y34, y35, y36)T, and bottom-level observations in an m-dimensional vector b = (y31, y32, y33, y34, y35, y36)T, we can write y = S b, (1) where S is the summation matrix, here expressed as S = �������������� 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 Im �������������� , (2) which is of dimension n×m, and Im is an identity matrix of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' S maps the hierarchical structure of the tree, where from the tree leaves b the complete hierarchy y can be reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' No- tice how S captures the coherency requirements within the hi- erarchy, integrated here as the linear summations of the bottom- level observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Uni-dimensional Hierarchical structures encompassed within hierarchical forecasting have, as of today, treated either one of the two fol- lowing dimensional frames, namely, temporal T or spatial S (sectional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4 We define spatial dimensional perspectives as a unique inter- element dimension, which places itself in opposition to the previously-defined cross-sectional dimensions [2, 9, 17, 33], which aggregated elements from very different entities together, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', stock management, resulting in considerable heterogeneity within ”one” (but in fact, multi-) dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It is our proposal to re-frame these cross-sectional considerations into separate dimensions to allow clear delineations of multi-dimensional frames, as we later detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Although structures of any shape or form can be designed in both dimensions, it is common for temporal hierarchies to adopt symmetrical structures, with k-level values being homogeneous across the trees’ aggregation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Taking the exemplified symmetrical hierarchy of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1, one could consider removing nodes y32 and y33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' resulting in a hierarchy where m = 4, n = 7 and node y21 being consequently removed as a redundant ele- ment of y31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This would result in a non-symmetrical tree which, in the temporal domain, implies non-equally spaced measure- ment points (or sampling rate) across the considered aggrega- tion level and the ones above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Typically, for symmetrical trees, there are k ∈ {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', kK} ag- gregation levels, where k is a factor of m, with k1 = m, kK = 1, and m/k is the number of observations at aggregation level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The summation matrix of temporal hierarchies can therefore be expressed as [1] S T = ������������ Im/k1 ⊗ 1k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Im/kK ⊗ 1kK ������������ , (3) where ⊗ is the Kronecker product and 1k is a k-vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To generically define the formulation of the summation ma- trix of any uni-dimensional hierarchy H, however, one needs to consider the eventuality of non-homogeneous k-level values across aggregation levels as well as uneven tree-depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To this end, we define sij = ������� 1, if yi is ancestor of yK j, 0, if yi is not ancestor of yK j, (4) where sij is a matrix element of the summation matrix S H given a fixed hierarchical structure H and yi here refers to the i-th element of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The subscripts i and j go from 1 to n − m and m respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They refer to the considered tree node element i and tree leaf element j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This sets the matrix element of a given node i to either 1 or 0 if it is an ancestor of the leaf element j, or, in other words, whether it is a result of the aggregation of the corresponding tree-leaf element yK j or not respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The summation matrix can then be expressed as S H = ������������������������������ s11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' s1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' s1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' si1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' sij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' sim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' s(n−m)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' s(n−m)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' s(n−m)m Im ������������������������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (5) This enables the formulation of any hierarchical structure to a summation matrix, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', from event-based or equally spaced time-series measurements for temporal hierarchies T , to non- symmetrical or homogeneous aggregation structures for spatial hierarchies S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Multi-dimensional Multi-dimensional hierarchies are the product of two uni- dimensional structures and can be obtained from function com- position of separate hierarchical structures over another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2 illustrates the derivation of a spatio-temporal ST hi- erarchy from two disjointed spatial S and temporal T structure compositions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', SoT and T oS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The resulting tree structures demonstrate fundamental equivalences, with all tree nodes pos- sessing identical bonds linking one element to the other, and consequently producing a unique hierarchical structure ST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The formulation of the multi-dimensional summation matrix in a generic way, can thus be expressed as a Kronecker product, where S ST ≡ ������� S SoT = S S ⊗ S T , S T oS = S T ⊗ S S, (6) from which the resulting spatio-temporal summation matrix S ST is of dimension nSnT × mSmT , which, in the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2, yields 3· 7 × 2 · 4 = 21 × 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The equivalence of SoT and T oS is attained via varying orderings of the nSnT -dimensional vector yST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' These are derived from alternative transpose defi- nitions of the observation matrix YSoT such that YSoT = YT T oS = ������������ y11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' y1nT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' ynS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' ynSnT ������������ , (7) where uni-dimensional vectors yS and yT are stacked together to form an observation matrix YSoT of dimension (nS, nT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The yST equivalent vectors can then obtained with yST ≡ ������� ySoT = vec(YT SoT ), yT oS = vec(YT T oS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (8) In the exemplified structures of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2, we obtain ySoT = (yA1, yB1, yC1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', yA7, yB7, yC7)T and yT oS = (yA1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', yA7, yB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', yB7, yC1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', yC7)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' With structural combinations of two disjointed dimensional hierarchies producing a unique bi-dimensional structure, it con- sequently follows that multi-dimensional combinations can be exploited in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By chaining function composi- tions of considered singular dimensions over summation matri- ces and y vectors, any combination of dimensional frames can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Dimensionality reduction Multi-dimensional trees, however, introduce a key limitation: the dimensional explosion of hierarchical structures from func- tion composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' With the multiplication of dimensions from summation matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' what was then considered a tractability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='B ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='hierarchy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='hierarchy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='SoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ToS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Figure 2: Schematic of spatio-temporal ST hierarchical structure conception from either SoT or T oS structure composition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' both producing an equivalent ST tree structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Highlighted nodes (in grey) reveal opportunities for dimensionality reduction by dropping nodes of little dimensional interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', high temporal granularity in high spatial aggregation levels, and low temporal frequencies in high spatial granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A B C Spatial hierarchy S 1 2 4 Temporal hierarchy T + A B C 1 2 4 A B C A B C A B C Topological covariance matrix 3 3 B C A 1 2 4 3 1 2 4 3 1 2 4 3 Summation matrix and y vector S T S T ToS SoT ST Figure 3: Exemplified illustrations of hierarchical derivations of summation matrix, y vector and topological covariance matrix from spatio-temporal SoT or T oS function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6 shortcoming has now become an inevitable obstacle needing overcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, multi-dimensional hierarchies bring with them a consequential consideration: multi-dimensional aggregation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, such trees encompass more than former uni- dimensional high- or low-aggregation levels, they consist of deep structures where multi-dimensional aggregation combi- nations demand investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spatio-temporal hierarchies, for example, display dissimilar insights from high-temporal-low- spatial aggregation levels, low-temporal-low-spatial or high- temporal-high-spatial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It thus comes to light that, given a defined insight-driven ap- plication, subsets of certain multi-dimensional aggregation re- gions can be of limited use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' High-frequency forecasts at very aggregate geographical levels might be of great value to grid operators contemplating frequency control in power systems, but not so much when forecasting tourism flows for instance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Considering the end-goal application of optimal smart- grid control from electric load forecasting of grid edges (smart- building meter), low temporal frequencies and low spatial ag- gregations would be of little interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, frequency control focuses on rather high-frequency samplings at medium-high spatial aggregation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, should the end-goal appli- cation be optimal cooperative control of smart-building neigh- borhoods, then low temporal frequencies and high spatial ag- gregations would become the dimensional frame of lesser con- cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2 highlights these bi-dimensional nodes over the hierarchical structure conception, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', in grey, revealing the po- tential of dimensionality reduction within multi-dimensional hi- erarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Therefore, while using spatio-temporal coherent forecasts of- fer benefits to decision-making, not all outputs from these hier- archies are effectively useful, opening the door to dimensional- ity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Reconciliation methods Traditionally, forecast reconciliation starts by generating an initial forecast of the tree independently for each node, referred to as base forecasts ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This set of hierarchical forecasts is stacked in the same manner as the y vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Because of the in- dependence of the base forecasts, in most cases, they do not ex- hibit coherency properties throughout their hierarchical struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By introducing a matrix G = �0m × (n−m) | Im �, (9) of order m × n that extracts the m bottom-level forecasts, the reconciliation constraint is formulated as ˜y = SG˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (10) Reconciliation is necessary when base forecasts ˆy do not satisfy this constraint [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In such situations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (10) becomes ˜y = SGˆy, where G maps the base forecasts into the reconciled tree- leaves and S sums these up to a set of coherent forecasts ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' SG can thus be thought of as a reconciliation matrix taking the incoherent base forecasts as input and reconciling them to ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A major drawback of traditional approaches is that G, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (9), only considers information from a single level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Optimal reconciliation To include the exploitation of all aggregation levels in an op- timal manner, Hyndman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [12] and later, Van Erven and Cugliari [14] and Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [17] formulated the reconciliation problem, as linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Exploit- ing either spatial or temporal hierarchical structures, reconciled forecasts are found employing the generalized least-squares es- timate: minimize �˜y − ˆy�TΣ−1�˜y − ˆy�, subject to ˜y = SG˜y, (11) where ˜y ∈ Rn is the decision variable of the optimization prob- lem and S ∈ Rn×m and G ∈ Rm×n are constant matrices defined by the structure of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The parameter Σ ∈ Rn×n is the positive definite covariance matrix of the coherency errors ε = ˜y − ˆy, which are assumed to be multivariate Gaussian and unbiased, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' If Σ were known, the solution to (11) would be given by the generalized least-squares (GLS) estimator ˜y = S �S TΣ−1S �−1S TΣ−1ˆy, (12) which has been employed in close to all notable hierarchical forecasting works over the last years [1, 2, 3, 6, 9, 12, 13, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The precision matrix Σ−1 is used to scale discrepancies from the base forecasts, hence, is often referred to as a weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The recurrent challenge in estimating Σ−1 stems from its di- mension n × n which can potentially become very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Multi-dimensional reconciliation Traditional uni-dimensional estimators can be coupled to- gether topologically to form multi-dimensional ones in a similar manner to the summation matrix, with Σ† ST ≡ ������� Σ† SoT = Σ† S ⊗ Σ† T , Σ† T oS = Σ† T ⊗ Σ† S, (13) where Σ† refers to the topological covariance matrix of a given covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows uni-dimensional estimators ΣS and ΣT to incorporate dimension-specific topological con- siderations and produce a suitable multi-dimensional estimator ΣST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The topological covariance matrix is characterized by ele- ments of either 0 or 1 that indicate the mapping form assump- tion of the considered covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Once the topological covariance matrix is identified, we simply populate it with the scaling parameters dictated by the reconciliation approach con- sidered to obtain the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Figure 3 exemplifies the identification of multidimensional topological covariance matrices from both SoT and T oS dimensional-derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To address dimensional considerations in traditional esti- mators of the covariance matrix applied in reconciliation, we present four state-of-the-art estimators, namely, identity, struc- tural, variance, and covariance scaling with shrinkage, while detailing dimensional deliberations individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 7 2 4 5 3 6 7 1 Considered hierarchy H Summation matrix and y vector Topological covariance matrices diagonal k-level full Covariance matrices identity - id structural - str variance - hvar variance - svar covariance - kcov covariance - cov Figure 4: Example illustration of the covariance matrices considered in this work along with their associated topological covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Identity A simplifying assumption proposed by Hyndman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [12] puts the following identity approximation forward Σid = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (14) This simplistic approach has been shown to work well in prac- tice [3] and allows to bypass the estimation of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It ignores scale differences (captured by the variances) and interrelations (captured by the covariances) information of the observations within the hierarchical structure, which makes it independent of dimensional frame considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Deep neural networks can be expected to build upon such simple relationships and approximate the more complex depen- dencies of the hierarchy thanks to its automated feature selec- tion, as we later detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We refer to this approach with the subscript id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Structural scaling Structural scaling was proposed by Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [17] as a solution to cases where forecast errors are not avail- able for some aggregation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It assumes the variance of each bottom-level base forecast error σ2 K is equal and that these are uncorrelated between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Therefore, higher-level error variances are the sum of the error variances of tree leaves se- ries connected to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By introducing a diagonal matrix Λstr with each element containing the number of forecast errors con- tributing to that aggregation level, they define Σstr = σ2 KΛstr, (15) Λstr = diag(S 1m), (16) where 1 ∈ Rm is a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The hierarchy illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1, for instance, gives Λstr = diag(6, 3, 3, 1, 1, 1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The estimator is independent from the considered forecasting method, since no estimation of the variance of the forecast er- rors is needed, making it computationally efficient [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' If considering a temporal dimensional frame, the estimator depends only on the seasonal period m of the tree leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' While with spatial perspectives, the estimator can suffer from hetero- geneity within aggregation levels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', residential and commer- cial buildings typically have heterogeneous electricity demand patterns and scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hence, assuming a common forecasting er- ror variance across all leaves-series is not suitable [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We refer to this approach as str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Variance scaling Another estimator proposed by Athanasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [17], referred to as variance scaling, scales the base forecasts using the variance of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It includes separate variance es- timates for each aggregation level and assumes either homo- geneous or non-homogeneous error variance within, but not across, a level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Given the hierarchy presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1, this gives Σsvar = Λsvar = diag(σ2 k1, σ2 k2, σ2 k2, σ2 k3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' , σ2 k3), (17) Σhvar = Λhvar = diag(σ2 11, σ2 21, σ2 22, σ2 31, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' , σ2 37), (18) for homogeneous and heterogeneous variances respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By definition, this scaling ignores correlations across and within aggregation levels and can be considered as an alternative weighted least-squares estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Similarly to structural scaling, spatial and temporal dimen- sional scaling can differ due to their intrinsic heterogeneous and homogeneous error variances respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' we refer to these es- timators as hvar and svar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It follows, that Σsvar and Σhvar are appropriate to temporal T and spatial S dimensional scalings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Covariance scaling To exploit important information about a time series at differ- ent frequencies (temporal dimension) or inter-scale differences (spatial dimension), Nystrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [1] argue that potential in- formation in the autocorrelation structure should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They consequently proposed a covariance scaling for temporal hierarchies estimating the full covariance matrix within each aggregation level, while ignoring correlations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Following these footsteps, we explore both full and k-level, or so-called block, covariance estimates such that, for the hier- archy illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1, the estimator is either Σcov = Λ1/2 hvarRΛ1/2 hvar, or (19) Σkcov = Λ1/2 hvarRkΛ1/2 hvar, (20) where R and Rk refer to the full and k-level empirical cross- correlation matrix respectively, R = ������������ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' ρ11,36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' ρ11,36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1 ������������ , (21) Rk = ���������������������������� 1 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0 0 1 ρ21,22 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0 0 ρ22,21 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0 0 0 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' ρ31,36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0 0 0 ρ31,36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1 ���������������������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (22) With increasing difficulties in estimating the full covari- ance matrix from high-dimensional hierarchies, even with high- frequency data available, special forms are commonly assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To alleviate this burden, Ledoit and Wolf [34] proposed a Stein- type shrinkage estimator of the sample covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Fol- lowing these footsteps, Nystrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [1] considered a shrink- age estimator of the cross-correlation rather than the cross- covariance matrix to avoid problems with heteroscedasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Their estimator is based on decomposing the cross-covariance matrix into two diagonal (heterogeneous) variance matrices Λ1/2 hvar and a shrunk cross-correlation matrix Rsrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The estimator is defined as Σsrk = Λ1/2 hvarRsrkΛ1/2 hvar, (23) Rsrk = (1 − λ)R + λIn, (24) 9 where 0 ≤ λ ≤ 1 is a regularization parameter to control the degree of shrinkage towards the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' When λ = 1, shrinkage scaling is equivalent to scaling by the diagonal variance matrix Λhvar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' When λ = 0, it is equivalent to scaling by the sample covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A closed-form solu- tion for the optimal value of λ was derived by Ledoit and Wolf [34] by minimizing the mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This shrinkage es- timator is ideal for a small number of data points with a large number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' With an assumed constant variance, the optimal shrinkage parameter is expressed by, λ = Σi�jVar(σij) Σi�jσ2 ij , (25) where σij is the i jth element of the covariance matrix from the base forecast errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The variance of the estimated covariance, Var(σi j), is computed as depicted in Appendix A of Sch¨afer and Strimmer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Therefore, in contrast to the preceding variance and struc- tural scaling estimators, this allows strong interrelations be- tween time series in the hierarchy to be captured, while shrink- age alleviates the complexities of the estimation of Σsrk due to its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We refer to the shrunken estimators of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (19) and (20) as cov and kcov respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It should be noted that a variety of other well-performing es- timators remain, including, but not limited to, Markov [1] or spectral scaling [20] supported by alternative inverse covari- ance shrinkage GLASSO method [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In the intent of limiting the scope of this work to the evaluation of a novel hierarchical regressor, however, the afore-presented prevailing covariance approximation methods are favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Figure 4 provides a visual illustration of the encompassed techniques along with their as- sociated topological covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Evaluation method The accuracy evaluation of hierarchical forecasting perfor- mances requires the consideration of an important principle that common forecasting methods are exempt from, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', the struc- tural scale differences inherent to hierarchical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In- deed, by its nature, hierarchical forecasting creates outputs of increasing orders of magnitudes, typically characterized by the aggregation levels of the tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', k-levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It consequently be- comes crucial to take these hierarchically-impended scale dif- ferences into account when undertaking the accuracy perfor- mance evaluation of hierarchical forecasts, else these would consistently produce poorer performances for the top levels of the aggregation, where predicted values possess larger magni- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This is commonly done by treating each aggregation level of the tree separately first, then evaluating the relative per-level performance of the reconciliation phase over the base fore- cast, allowing the removal of scale differences between aggre- gation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, a relative performance evaluation does not allow the comparison of approaches across case studies nor the distinctive performances of forecasting and reconcili- ation phases, which is why we propose to complement relative performance evaluations with measures based on structurally- scaled errors to provide an evaluation method more suited to the evaluation of hierarchical learning regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Relative measures The prevailing approach employed to evaluate hierarchical forecasting accuracy consists in scaling the accuracy perfor- mance of the reconciliation phase over a reference base fore- cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This can be done by exploiting either Relative Mean Squared Error (RelMSE) [2] or Relative Root Mean Squared Error (RRMSE) [1, 17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Both depict the improvement of a given reconciliation approach compared to base forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We favor RelMSE over RRMSE to align with the commonly em- ployed Mean Squared Error (MSE) loss function of machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The RelMSE can be expressed as RelMSEk = MSEk MSEbase k − 1, (26) where the RelMSEk is computed for each aggregation level k and k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A negative entry describes a percentage improvement of the reconciled forecast over the base forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The MSEk is computed as the average error of all prediction steps of a given aggregation level k from MSEk j = 1 h h � t=1 e2 k j,t, (27) MSEk = 1 Nk Nk � j=1 MSEk j, (28) where ek j,t = yk j,t − ˆyk j,t is the forecast error at a starting ref- erence time t ∈ Rh of an node k j with k being the aggregation level of the hierarchy possessing Nk elements and j the node ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The starting reference time t points to the very first time step considered in the hierarchy and is employed to anchor the nomenclature of temporal as well as spatio-temporal hierar- chies, which usually encase time frames of [t, t+m], in similar notations as spatial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Measures based on scaled errors An alternative way of removing the inherent structural- scale differences present in hierarchical structures is producing structurally-scaled errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This can be achieved by dividing the error vector et = yt − ˆyt by the structural vector κstr, where each element contains the number of nodes contributing to the forecasted error of that aggregation level, such that κstr = S 1m, (29) estr t = et ⊘ κstr, (30) where ⊘ is a Hadamard division and estr t is the structurally scaled error vector at a time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The hierarchy illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 1, for instance, gives κstr = (6, 3, 3, 1, 1, 1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Structurally-scaled errors can then be employed in any given evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We consequently define the Mean 10 Structurally-Scaled Square Error (MS3E) as MS3Ek j = 1 h h � t=1 estr k j,t 2, (31) which can be averaged either per aggregation level or over the entire hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical learning While traditional hierarchical forecasting approaches have treated forecasting and reconciliation phases separately, we pro- pose to unify these steps under a singular machine-learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To introduce our approach in a step-wise manner, let us first provide a comprehensive overview of the diverse ways machine learning may be employed within the frame of hier- archical forecasting, supported by the illustrative schemes pro- vided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We start by detailing the forecasting phase composing hier- archical forecasting with machine learning and continue with the description of the reconciliation step and its subsequent ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical forecasting with machine learning Machine learning regressors employed for hierarchical fore- casting can here be employed in one of three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Independent forecasting First, with independent models each forecasting a unique node of the hierarchy, see (1a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The models leverage data information uniquely related to the considered node and do not exchange information with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' They produce typical independent (base) forecasts of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Notable variations of this process involve either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (i) exploiting transfer learning across the models to allow exchange of information throughout the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The work of Sagheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' [27] pre- cisely employed such a scheme using a top-down approach to determine coefficients of the lower-level models as proportions of the learnt top-level one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This process secures the coherency of the forecasted tree all the while providing privacy protec- tion of data from one site to another, as transferred model co- efficients retrieve data sharing dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Or (ii) by employ- ing a unique single-output model for the forecasting of each node of the tree, namely a multivariate learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows one model to gather more information as it learns from a much larger database than independent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, the dis- advantage of this approach originates from the generalization intention of the learnt model applied to what could be, very dif- ferent forecasted processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', heterogeneous buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This is why this approach works best when considering processes exhibiting similar characteristics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', time series range, and typical patterns, which are generally obtained through a prior clustering phase [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In addition, because the approach relies on the formulation of a unique model, any miss-specification could drastically impact the performance of the forecast, con- sequently making its design a key consideration for scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The loss function of independent forecasting regressors are typically designed around a given error metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' mean squared error, describing the differences between forecasted and true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Typically Lb�Y, �Y|Θ� = 1 h h � t=1 �yt − ˆyt �2, (32) where Lb denotes the mean square loss function between the predicted independent base forecast set �Y subject to a set of parameters Θ and a set of observed values Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Multi-task forecasting Second, by taking the concept of multivariate regressors even further, a multi-task regressor can be contemplated, see (1b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The regressor now produces a hierarchical, dependent, forecast of the tree as a single vector output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The model notably accepts features from the bottom layer of the tree for spatial hierarchies, as aggregate levels would provide redundant infor- mation already present in the tree leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, temporal hierarchies typically benefit from the inclusion of aggregate- level features, allowing them to exploit important information about the time series at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Requirements for coherency are, however, not included with such a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The loss function of multi-task learners is similar to single- task ones other than considering vector rather than point errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', Lh�Y, �Y|Θ� = 1 h h � t=1 � yt − ˆyt �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (33) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical forecasting This takes us to our third and last approach, crystallizing the intention and concepts behind the contributions of our work, namely, hierarchical forecasting, see (1c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This technique builds on the aforementioned multi-task forecasting model while extending it with the inclusion of a coherency- informed learning process thanks to a custom loss function em- ploying established coherency taxonomy from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The coherency loss function is formulated as the difference be- tween the predicted values ˆy and its reconciled counterpart ˜y, following the reconciliation constraint of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The co- herency loss function Lc can consequently be expressed as Lc�Y, �Y|Θ� = 1 h h � t=1 � ˆyt − S �S TΣ−1S �−1S TΣ−1ˆyt �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (34) To combine both accuracy and coherency in the learning pro- cess of the regressor, the coherency loss is added to the hi- erarchical loss function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (33) forming the hierarchical-coherent loss function Lhc, Lhc�Y, �Y|Θ� = αLh t + (1 − α)Lc t , (35) where α ∈ [0, 1] weights the hierarchical loss against the co- herency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This avoids the over-adjustment of weights during ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='the training of the regressor due to the addition of the coherency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ỹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='independent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(base) forecast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='coherent ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Coherency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Hierarchical Learning Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Independent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Forecasting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(1a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Multi-task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Forecasting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(1b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='yh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Hierarchical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Forecasting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Coherency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(1c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='yh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Soft-constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Reconciliation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Coherency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(2a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Hard-constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Reconciliation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='(2b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='ỹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='Figure 5: Hierarchical forecasting methods with highlighted traditional approach steps (top left) and proposed hierarchical learning method (top right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Forecasting possibilities employing machine learning (here illustrated with the acronym BB, standing for Black Box) are illustrated (1a-c) along with two reconciliation approaches (2a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Forecasting methods encompass independent forecasting (1a), multi-task forecasting (1b), and our proposed hierarchical learning method (1c), working as a combined forecasting and reconciliation learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Reconciliation approaches presented cover our machine learning method employed as a soft- constrained coherency enforcement over the base forecast (2a) and the traditional hard-constrained coherency enforcing method (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' loss to the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We typically set α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='75 for hierar- chical forecasting to favor accuracy learning of produced pre- dictions over coherency, yet this parameter should commonly be tuned by hyper-parameter optimization in the validation pro- cess of the model development, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4 for implementa- tion details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The method regroups numerous key advantages of ma- chine learning-based forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' With large and rich multi- dimensional data to learn from the regressor effectively makes use of all the information provided by the most detailed layer of the hierarchy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', the tree leaves, all the while incorporating hierarchical structure information as a soft-constrained learning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Loss function augmentation via regularization and penalty methods has grown to become the most popular way of introducing constraints in deep learning [37, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Al- though the approach comes at the price of sacrificing hard con- straints, it has been shown that soft-constrained penalty meth- ods perform well in practice and often exceed hard constraint methods [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In addition, machine learning approaches are powerful at capturing non-linear relationships in the tar- geted predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In particular, deep-learning methods are known for effective and automatic feature extraction from the data, thus reducing the need for guesswork and heuristics, which could provide a much-needed solution to the problem of non-identifiability of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Its disadvantages are similar to those of hierarchical forecast- ing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' By relying on a unique model, architecture considerations become paramount for the accurate performance of the regressor and consequently require careful, tailored tun- ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' with hyper-parameter grid-search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Reconciliation with machine learning While our proposed hierarchical learning approach (1c) blurs the limit between the traditionally delineated forecasting and reconciliation steps, it can also be employed as a classic recon- ciliation step, see (2a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Proposed as a soft-constrained coherency regressor, the machine learning model now takes the entire base forecast ˆy as input and outputs a coherency- informed forecast ˆyh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The weighting coefficient α presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (34) can here be set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25 to favor coherent outputs for ex- ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The evaluation of such a scheme, lays, however, outside the scope of this work, as our contribution targets hierarchi- cal forecasting performance evaluation on varying dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This setup rather showcases the flexibility of our approach as applicable to both the forecasting and reconciliation phases of traditional hierarchical forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' For hard-constrained reconciliation, optimal reconciliation is considered, see (2b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It imposes coherency to its input forecast and can be employed a posteriori to the hierarchical learning step (1c) for eventual non-coherent outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In addi- tion, as an established reconciliation method, it provides a good benchmark to evaluate the performance of our proposed method to both forecasting (1c) and reconciliation (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Implementation This section details the implementation-related details of our study, namely, considered case studies, hierarchical structures, and predictive-learning setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='190242 Figure 6: Hierarchical spatial tree structure of the Fox site from Case Study 2 Table 1: Characteristics of assembled hierarchy per case study Characteristics Spatial Temporal Spatiotemporal Case study 1 n [#] 383 37 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='171 m [#] 192 24 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='608 horizon [hours] 1 24 24 Case study 2 n [#] 140 37 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='998 m [#] 133 24 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='200 horizon [hours] 1 24 24 Case Study 1 considers a total of 225 homes located in the Netherlands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' a European region under the K¨oppen climate classification index [42] Cfb which describes mild temperate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' fully humid and warm summer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Anonymized measure- ments are gathered from smart-meters collected by energy dis- tributor Eneco at resolutions of 10 seconds over a period of 3 years starting from January 1st 2019 to the 2nd of August 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Weather data is assembled from publicly available Royal Netherlands Meteorological Institute (KNMI) weather stations measurements [43], that are paired to each building thanks to a geo-localization process using 4 (over the 6) ZIP code dig- its;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' an aggregation level that allows no anonymized user to be geographically isolated nor identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Case Study 2 employs the open data set from the Building Data Genome project 2 (BDG2) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This open set was se- lected to allow reproducibility of our method while putting for- ward a benchmark for hierarchical forecasting in the building sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The BDG2 includes 3053 energy meters from 1636 non-residential buildings grouped by site located in Europe and, principally, North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The set covers two full years (2016–2017) at an hourly resolution with multi-meter building measurements paired with site weather data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchies 24H_0 6H_0 3H_0 1H_0 1H_1 1H_2 3H_1 1H_3 1H_4 1H_5 6H_1 3H_2 1H_6 1H_7 1H_8 3H_3 1H_9 1H_10 1H_11 6H_2 3H_4 1H_12 1H_13 1H_14 3H_5 1H_15 1H_16 1H_17 6H_4 3H_6 1H_18 1H_19 1H_20 3H_7 1H_21 1H_22 1H_23 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1932 Figure 7: Hierarchical temporal tree structure for day-ahead forecasts Spatial hierarchies are defined by hierarchically clustering the prediction target time series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', electricity demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This step is carried out employing the Ward variance minimization algorithm [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The obtained hierarchy is reduced in size by cutting the tree using a defined distance threshold over visual inspection of the derived dendrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In this way, hierarchi- cal structures located below the defined distance threshold will be clustered together, effectively reducing the number of con- nection nodes of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Figure 6 illustrates the attained re- duced tree of the Fox site of case study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Temporal hierar- chies are considered with a horizon of one day (tree root sam- pling frequency) while reaching down to granularities of hourly sampling intervals (tree leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Aggregation levels encompass sampling frequencies every 6 and 3 hours, resulting in a tree with sampling frequencies of 1 day, 6 hours, 3 hours and 1 hour per k-level, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spatio-temporal trees are then obtained as a result of the dimensional combination of spa- tial and temporal hierarchies, as detailed under Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To limit the exponential explosion in tree size from dimensional combination, spatial trees are limited to 50 leaves in case study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Table 1 details the different characteristics of the considered hierarchies per case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Model learning setup In both case studies, we proceed to resample the time-series to hourly intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Time-series with no cumulative missing values larger than 2 hours are considered and smaller gaps are interpolated via a moving average using a window size of 8 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Feature engineering Data sets are then treated per dimensional batches, namely, per site, sub-site sample, or building for spatial, spatio- temporal, and temporal dimensional hierarchies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Features are selected based on their Maximum Information Co- efficient (MIC) [45] computed in relation to the learning target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' MIC is a powerful indicator that captured a wide range of as- sociations both functional and not while providing a score that roughly equals the coefficient of determination (R2) of the data relative to the regression function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It ranges between values of 0 and 1, where 0 implies statistical independence and 1 a com- pletely noiseless relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The advantage of using MIC for feature engineering over the more commonly employed person 13 Σ1 Σ2 Σ3 Σn−2 n splits scale fit & transform train test 1 batch 2 3 n-2 n-1 coherency loss Σ Σid Σ1 Σ2 Σn−3 Σn−2 scale transform Σ estimation Figure 8: Data partitioning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' transformation and covariance matrix estimation setup correlation indicator [46] is that it captures non-linear relation- ships present in the data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' which deep-learning models are pop- ularly capable of detecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We retain features exhibiting MIC values higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25, as electric loads can typically become quite volatile and impede MIC values with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Additionally, to feed the learner with the most relevant his- torical information of the predicted target, we select the 3 top auto-correlation values per temporal aggregation level above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25 as model input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' If no target auto-correlation value is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25, we consider the most recent historical informa- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', tk −1 where tk is the first k-level time-step value of the predicted horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Both MIC and autocorrelation selection thresholds are set- tings that should typically be included in the hyper-parameter optimization of the model validation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' While evaluating the performance of hierarchical regressors over three varying dimensional considerations and two different case studies, this work considers the tuning of these thresholds to lay outside of its scope, as such computations rapidly become excessively bur- densome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Data partitioning and transformation Training and testing sets are then defined employing Time- SeriesSplit, a times-series cross-validator of the sklearn pack- age [47], with equal test-size in a rolling window setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We next proceed to standard normalize the data per batch using batch-specific available historical information such that each batch-scaler is first fitted to the current, and past, training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The fitted-scaler is then employed to transform batch-specific test sets, as depicted by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This process avoids data leak- age situations, which refers to the inadvertent use of data from test sets, or more generally data not available during inference while training a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This typically occurs when the data is normalized prior to partitioning for cross-validation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', by performing smoothing or normalization over the whole series before partitioning for training and testing [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' While it benefits the performance of (deep) neural networks to normalize input features and predicted target, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', from un- scaled yx to scaled yz, this shatters the hierarchical relationship of the regressors’ outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' thus, affecting the soundness of the coherency loss-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To integrate the coherency loss func- tion in such a setting, hierarchical relationships of predicted val- ues ˆyz are restored by reverse transformation prior to coherency loss calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The obtained reversed-scaled prediction ˆyx is reconciled to ˜yx following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (12) and is finally re-scaled to ˜yz to calculate the coherency loss function against its original predicted self ˆyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Coherency settings The estimation of the covariance matrix is performed over test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' For the first batch training, we employ the identity covariance estimate id as no forecasts are yet available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Each batch training i then comes with a new covariance matrix esti- mate Σi that is employed in the coherency loss function of the next training set i + 1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This setup echoes the adap- tive covariance matrix estimation proposed by [19] employed for temporal hierarchies, anchored here quite organically in the learning process of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Designing hierarchical regressors We select deep neural network regressors to best serve the benchmarking of hierarchical-coherent forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Such machine-learning regressors possess well-developed packages supporting custom implementations that serve our approach well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The regressor is structured as a series of sequential layers decreasing proportionally in size, from initial input layer size to the desired output dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The optimal number of lay- ers is selected from monitored test-set prediction performances while step-wise increasing the network’s depths starting from shallow 1-layer perceptrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This allows the selected architec- ture to serve an ”as simple as possible yet as complex as nec- essary” design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Model hyper-parameters are later tuned over a concise grid encompassing loss function parameter α, acti- vation functions, and dropout fraction, further improving the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' These tests resulted in the design of a deep neural network of 3 layers, leveraging sigmoid activa- tion functions and dropout ratios of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2 on all but the last layer favoring a linear activation and no dropouts, and a retained α coefficient value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='75 The presented models of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3 were implemented in Python using the Keras package [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Results We describe the outcome of the implementation here over spatial, temporal and spatio-temporal hierarchical structures per case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In particular, we evaluate the accuracy and co- herency of the forecasted building loads outlined in an anno- tated heatmap and bar plot respectively, where the presented coherency loss relates solely to the output of the forecasting method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', reconciliation referred to None, as reconciled fore- casts all possess null coherency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Evaluated forecasting methods cover the independent (base), multi-task, and hierar- chical forecasting methods presented under Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hier- archical forecasting and reconciliation methods each consider the covariance approximations presented under Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', or- dinary least square (id), structural (str), heterogeneous vari- ance (hvar), homogeneous variance (svar), shrunken covari- ance (cov) and shrunken k-level covariance (kcov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Necessary 14 0 72000 144000 [kWh] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='783e+05 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='196 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='75 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='68 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='41 None 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='122e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='141e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='045e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='146e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='042e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='077e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='105e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='051e+05 base multitask hierarchical - id hierarchical - str hierarchical - hvar hierarchical - svar hierarchical - cov hierarchical - kcov Forecasting method id str hvar svar cov kcov Reconciliation method 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='703e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='142e+05 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='046e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='147e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='042e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='078e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='105e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='051e+05 405000 410000 415000 420000 425000 430000 MS3E [kWh] Figure 9: Spatial hierarchy forecasting performance of case study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To allow the differentiation of performances across the heatmap, extreme values were cut off from the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' computational resources inherent to the forecasting methods are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Case study 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spatial Performances of spatial hierarchical forecasts are presented under Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 9, where illustrated hierarchical losses showcase svar as the best hierarchical forecast performer, with and with- out reconciliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The lowest hierarchical MS3E originates from base forecast reconciled with id covariance matrix ap- proximation, while hvar and kcov also notably perform quite poorly for this forecasting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Overall, the performance of the forecasts seems to rely more on the selected forecasting method rather than their reconciliation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Coherency losses seem in line with expected results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' base forecast is showcased as the most incoherent outcome, holding coherency errors ranging up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='783e5 kWh, while multi-task and hierarchical regressors score MS3Es of 36 kWh and 16 kWh (on average) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Temporal Temporal hierarchical forecasting performances, on the other hand, portray a much different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' As illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 10, it is here the base and multi-task regressors that possess the lowest hierarchical losses, with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='232e6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='116e6 kWh re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The poorer performer without reconciliation in this setup is svar, with an MS3E of up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='23e6 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Extreme poor performances are noticeable for the cov and kcov reconcil- iations of id and svar forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Overall, the perfor- mance of the forecasting methods here seems also more driven by the considered forecasting method than reconciliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In terms of coherency, the base forecast surprisingly exhibits the most coherent outputs with an MS3E of 7925 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It 0 108000 216000 [kWh] 7925 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='922e+05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='535e+05 4.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='527e+06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='974e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='652e+06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='23e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='233e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='532e+06 base multitask hierarchical - id hierarchical - str hierarchical - hvar hierarchical - svar hierarchical - cov hierarchical - kcov 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='283e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='295e+06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='00 MS3E [kWh] 1e6 Figure 10: Temporal hierarchy forecasting performance of 40 buildings from case study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To allow the differentiation of performances across the heatmap, extreme values were cut off from the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' is followed by str, and all other hierarchical forecasts which compare considerably worst featuring inconsistency errors of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='682e4 kWh and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='691e5 kWh (on average) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spatio-temporal Finally, spatio-temporal forecasting performances exposed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 11 reveal contrasting outcomes compared to previous hi- erarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' First, all cov and kcov reconciliations here perform extremely poorly, irrespective of the forecasting method em- ployed, with hierarchical losses ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='57e6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='62e6 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Similarly to the temporal hierarchy, base and multi-task forecasts perform overall better than hierarchical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The multi-task regressor without reconciliation is show- cased as the best performer in this setup with an MS3E of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='187e5 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It can notably be observed here that all hier- archical and multi-task forecast reconciliations do not improve the accuracy of their original forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Additionally, exposed performances here display a much stronger dependency on the considered reconciliation approach than forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Concerning coherency losses, spatio-temporal hierarchies produce two distinct performances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' where base and multi-task forecasts exhibit inconsistencies of 1 order of magnitude lower than all hierarchical ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='255e4 kWh against 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='878e5 kWh on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Case study 2 Concerning case study 2, the spatial hierarchical forecasting performance presented under Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 12, depicts noticeable varia- tions from case study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='187e+05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='905e+05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='083e+05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='954e+05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='907e+05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='925e+05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='275e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='166e+06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='278e+06 350000 400000 450000 500000 550000 600000 650000 MS3E [kWh] Figure 11: Spatio-temporal hierarchy forecasting performance of 41 buildings from case study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To allow the differentiation of performances across the heatmap, extreme values were cut off from the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0 80 160 240 [kWh] 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1558 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3183 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='98 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='02 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='18 None 615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1 1179 1231 1285 1285 1188 1717 1663 base multitask hierarchical - id hierarchical - str hierarchical - hvar hierarchical - svar hierarchical - cov hierarchical - kcov Forecasting method id str hvar svar cov kcov Reconciliation method 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='6 1179 1233 1277 1288 1190 1721 1664 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 1179 1232 1283 1285 1188 1721 1664 636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4 1180 1230 1279 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+page_content='005e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='808e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='543e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='004e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='566e+04 base multitask hierarchical - id hierarchical - str hierarchical - hvar hierarchical - svar hierarchical - cov hierarchical - kcov Forecasting method id str hvar svar cov kcov Reconciliation method 1618 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='327e+04 3.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='397e+04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='469e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='751e+04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='424e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='442e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='609e+04 1520 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='362e+04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='514e+04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='466e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='814e+04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='563e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='372e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='854e+04 10000 20000 30000 40000 50000 60000 MS3E [kWh] Figure 13: Temporal hierarchy forecasting performance of 66 buildings from the Fox site of case study 2 Multi-task forecasts followed by structural, str, hierarchical ones both produce the most coherent outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Surprisingly, while the multi-task forecast is trained without coherency in- formation, its forecast displays the best coherency performance in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Overall, the best forecast accuracy is obtained from base forecasting reconciled with the cov approximation, while the worst performer for this scenario is the cov hierarchical fore- casting with either kcov or hvar covariance approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It displays hierarchical MS3Es ranging from 611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='727e3 kWh and coherency MS3Es varying between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='156 and 245 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Temporal The averaged temporal hierarchical forecast performance of 66 buildings from the Fox site is exposed under Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Fore- casting performances are overall significantly worse than those of spatial-hierarchies, with hierarchical MS3Es now ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='164e3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='327e4 kWh, while coherency losses fluctuate from 180 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='213e4 kWh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' an order of magnitude about 3 times higher than temporal trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Here, the best- performing forecast belongs to the multi-task forecast with no reconciliation, which also displays the highest inconsistency score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The lowest performing forecast interestingly resides with the id reconciliation of that same forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The most coherent forecast produced for temporal-trees pe- culiarly originate from base forecasts, which neither share information across the hierarchy, nor possess coherency- knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Other hierarchical forecasts produce coherency losses ranging between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='120e3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='416e4 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Spatio-temporal Lastly, the forecast performance of spatio-temporal struc- tures considering 50 buildings from the Fox site is presented un- der Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Similarly to the temporal-tree, hierarchical losses 16 0 30000 60000 [kWh] 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='603e+04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='567e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='952e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='881e+04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='72e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='909e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='764e+04 None 1692 1485 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='335e+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='077e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='939e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='576e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='957e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='869e+04 base multitask hierarchical - id hierarchical - str hierarchical - hvar hierarchical - svar hierarchical - cov hierarchical - kcov 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='711e+05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='57e+09 2119 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='749e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='164e+04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='689e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='696e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='435e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='557e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='416e+04 10000 20000 30000 40000 50000 60000 70000 80000 MS3E [kWh] Figure 14: Spatio-temporal hierarchy forecasting performance of 50 buildings from the Fox site of case study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' To allow the differentiation of performances across the heatmap, extreme values were cut off from the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Table 2: Averaged computing times (in seconds) of evaluated forecasting meth- ods tree size forecasting method n base multi-task hierarchical Case study 1 14,171 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='6 392 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3 383 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='9 90 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='6 37 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2 12 13 Case study 2 1,998 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 70 77 140 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='5 70 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='6 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2 20 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3 display much poorer performances compared to their spatial an- tecedent, with MS3Es ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='485e3 and extreme 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='57e9 kWh values, while coherency losses vary from 391 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='603e4 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Mirroring the results from temporal-hierarchies, the forecast- ing technique withholding the lowest hierarchical loss is the multi-task learner without reconciliation which is also charac- terized by the highest coherency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' A series of extreme poor performers are identified as a result of the cov reconciliation over all hierarchical-learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Contrary to temporal-tree, recon- ciled forecasts performances here seem driven by the reconcil- iation method rather than the considered forecasting technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Coherency scores display overall poor performances across all hierarchical and multi-task learners with losses ranging 2 orders of magnitude higher than the best case base regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Computational prospects Computational performances of forecasting approaches are here considered, providing a complete overview of evaluated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Table 2 presents the computation time of each fore- casting method averaged over all training batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Two antici- pated findings can be noted from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' First, the computing time is positively correlated to the size of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' One exception seems to deviate from that rule in case study 2, between tree sizes of 1,998 and 140, which dis- play relatively close computing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Second, smaller regres- sors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e, base, train faster than larger ones, namely, multi-task and hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Both these observations can be explained by the increasing number of weights to update in the larger regres- sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The more weights to update, the longer the training will take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Although independent regressors seem attractive due to their noticeably faster computing times, it should be noted that the displayed performances depict only the average computing time of a unique independent regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Should such regressors not be trained and tested in a distributed computational setup, then these numbers would need to be multiplied by the hierarchy size to obtain an appropriate estimation of the required computing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Discussion Although presented case studies bear varying results, these also display a number of commonalities supporting interpreta- tion and analysis, which are here discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hierarchical-coherency value Unifying the forecast of hierarchical structures under one re- gressor possesses attractive data-efficient prospects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', cross- tree information exchange combined with embedded-structural learning provided from coherency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, produced outcomes from hierarchical-coherent learners were only found to bring added value in one setting, namely, the spatial hierar- chy of case study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This can be explained by the similarities in building loads of case study 1, which encompassed time series of similar patterns and dynamics, all originating from residen- tial constructions, while case study 2 included a broader collec- tion of construction types covering offices, college classrooms, lodging, warehouses, and parkings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Such profile diversities are challenging to learn from limited measurements, particularly for a large model involving considerable numbers of regression weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It can consequently be found that while the results of the spa- tial hierarchy of case study 1 are promising, these unveil, in fact, important challenges hierarchical forecasting must face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' An efficient but arduous learning process Although the outcome of hierarchical learning demonstrated promising performances, identified in the spatial hierarchy of case study 1, the resulting number of weights to update and pos- sibly conflicting forecasted outputs can become burdensome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', as unveiled by the performance of the spatial hierarchy of case study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, with hierarchical regressors growing in size, their number of neuron connections increases by an expo- nential factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This renders the learning process of these models laborious as more data should support the learning of larger number of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Additionally, multi-output regressors are faced with the challenging task of predicting numerous out- comes which might exhibit highly different, possibly antipodal, dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This also affects the learning process, which might 17 2017-02-03 2017-02-13 2017-02-23 2017-03-05 2017-03-15 2017-03-25 2017-04-04 2017-04-14 2017-04-24 2017-05-04 2017-05-14 2017-05-24 2017-06-03 2017-06-13 forecasted date 0 100 200 300 400 500 600 electric loads [kWh] Figure 15: Illustration of faulty coherent-learning from normalized trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The predicted (red) versus true (black) electric loads of the Fox assembly Lakeisha temporal hierarchical tree showcase the mirrored top-level forecast predicted in the negative domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' struggle to identify these discrepancies from limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Induced coherency over accuracy Overall, temporal hierarchies of the considered case studies were seen to perform significantly worse than spatial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This significant change can be attributed to the combination of two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' First, the longer forecasting horizon of temporal trees compared to spatial ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', 24 hours against 1, implies that forecasts must rely on fewer data and less recent information while dealing with higher uncertainties, thus negatively affect- ing their performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Secondly, building electrical loads are endowed with a periodicity that falls precisely on the forecasted horizon of 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This consequently leads to little variations in the forecasted element of its hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' And, while this char- acteristic is desirable for ordinary forecasting, the addition of the coherency-loss function, although weighted by the α coeffi- cient - see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (35), may push the regressor to produce constant predictions, tailored more to coherency than accuracy, thus re- sulting in unrealistic and inaccurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Faulty coherent-learning from normalized trees In some settings, hierarchical-coherent learning displayed particularly poor performances from extreme hierarchical and coherency errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', temporal and spatio-temporal hierar- chies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Following further inspection, it was noticed that these poor performers all withheld abnormal top-level forecasts which mirrored their expected true values in the negative do- main, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' These undesirable, yet pecu- liarly common, results can be traced back to the normaliza- tion of the target hierarchical time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, while neu- ral networks benefit from normalized targets, serving fair and balanced learning across the network’s weights, this also shat- ters the coherency structure of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The existing setup, de- tailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='2, proceeds to tackle this issue by reverse- transforming these target values prior to the coherency con- straint computation and re-scaling them for coherency loss cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This ensures both loss functions, namely hierarchical and coherency, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' (33) and (34) respectively, to oper- ate on akin normalized time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' However, coherency learn- ing can eventually produce adjustments larger than the origi- nal normalization ranges, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', lowering the top-level forecast ˆyz fully into the negative domain such that the reverse stan- dard transformation ˆyx = ˆyz · u + s, where u and s refer to the mean and standard deviation of the fitted time series respec- tively, also produces a fully negative reverse-scaled ˆyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This evidently improper outcome consequently negatively impacts both the learning and the forecasting performance of the regres- sor and should be dealt with in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Conclusion Ensuring coherent previsions of the future is crucial to sup- port better informed and aligned decision-making processes across hierarchical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' And while previous works have attempted to exploit spatio-temporal hierarchical reconciliation using disparate steps [2, 9, 31, 33], no common formulation of multi-dimensional hierarchical structures had, to this date, been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Furthermore, traditional hierarchical forecasts use disjointed forecasting and reconciliation processes that inher- ently deprive forecasting algorithms of (i) the benefits of infor- mation transfer across (hierarchical) models, as well as (ii) cap- italizing on the coherency requirements of the produced fore- cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This paper proposes a solution to these shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' First, by formally defining multi-dimensional hierarchi- cal structures, it extends conventional hierarchical forecasting methods, allowing the exploitation of spatio-temporal struc- tures unified under a common frame, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', a unique summation and covariance matrix resulting from spatio-temporal function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Second, rather than considering reconciliation a posteriori to forecasting, this work brings together independent forecast- ing models into a unique machine-learning regressor embedded with coherency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' This provides the regressor with (i) a global overview of information across its hierarchy, permit- ting a cross-dimensional and data-rich learning process, while (ii) learning coherency-requirements as a soft constraint thanks to a custom hierarchical-coherent loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The approach can notably be tuned thanks to an adjustable α coefficient to ei- ther consider multi-task, hierarchical or only reconciliation in its learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Coherency of the produced hierarchical forecasts can then be enforced as a hard constraint using es- tablished reconciliation technics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The outcome is a unified and coherent forecast across all examined dimensions, granting a common view of the future serving aligned and better decision- making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' The approach provides a data-driven solution to as- semble diverging parts of an organization and blend informa- tion from varying sources, hierarchy levels, or scales [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Third, we evaluated our approach on two different case stud- ies, across all hierarchical dimensions, considering established state-of-the-art reconciliation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Results revealed spa- tial hierarchies to perform best while temporal and spatiotem- 18 poral structures suffered from coinciding forecasted horizon with the periodicity of electric loads from buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Although the value potential of hierarchical-coherent learning was ob- served in case study 1, the performances of the approach were quite disparate in other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In this regard, a comprehen- sive analysis was reported revealing important challenges the approach faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In particular, dealing with predicted outputs of conflicting trends while fitting an exponentially large number of weights to the model is a recurring fragility of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Additionally, correcting faulty coherency training from normal- ized tree structures is another frailty future work may tackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Finally, to encourage knowledge dissemination we ren- der our work fully replicable by open-sourcing developed python implementations under the public GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='com/JulienLeprince/hierarchicallearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Outlooks and future work This paper proposes a novel hierarchical learning method yielding important implications for forecasting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Indeed, by directly forecasting hierarchies this work opens the door to leveraging multi-scale and multi-frequency measurement infor- mation driving improved forecast accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' It notably ex- pands and unites traditionally disjointed methods together pro- viding a path toward a novel generation of forecasting regres- sors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Meanwhile, numerous directions for future work can already be distinguished, guiding attempts to tackle uncovered obsta- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' As such, the curse of dimensionality endowed from larger, unified hierarchical models can notably be undertaken by investigating distributed and connected models working as a hybrid solution between independent, but tractable regres- sors and extensive hierarchical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In addition, varying na- tive multi-output machine learning algorithms may be exam- ined such as ensemble decision trees, Gaussian processes, K- Neighbors, long short-term memory neural networks, and sup- port vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' In particular, algorithms that deal with different ranges of target values can naturally tackle issues with coherency learning due to scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Finally, comparing hierar- chical learning performances against established models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=', grey- or white-box, that benefit from the inclusion of domain expertise to tackle targeted behaviors, such as seasonality, ad- vances another interesting endeavor for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' CRediT authorship contribution statement Julien Leprince: Conceptualization, Methodology, Soft- ware, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review and editing, Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Hen- rik Madsen: Methodology, Supervision, Validation, Writing - review and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Jan Kloppenborg Møller: Methodology, Supervision, Validation, Writing - review and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Wim Zeiler: Supervision, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Acknowledgments This work is funded by the Dutch Research Council (NWO), in the context of the call for Energy System Integration & Big Data (ESI-bida).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' We gratefully acknowledge the support and contribution of Eneco with particular thanks to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Kaus- tav Basu and Rik van der Vlist for this research, as well as SEM4Cities, funded by Innovation Fund Denmark (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' 0143-0004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Nystrup, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Lindstrøm, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Pinson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content=' Madsen, Temporal hierarchies with autocorrelation for load forecasting, European Journal of Opera- tional Research 280 (3) (2020) 876–888.' 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https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} +page_content='com/fchollet/keras 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFOT4oBgHgl3EQf8zQL/content/2301.12967v1.pdf'} diff --git a/XtE2T4oBgHgl3EQfuwhB/vector_store/index.faiss b/XtE2T4oBgHgl3EQfuwhB/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d68fe47f1c9b97e42a556dc2754fd98c4ff442d7 --- /dev/null +++ b/XtE2T4oBgHgl3EQfuwhB/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3fa9fcccb8774131f4a5a60c83816096b364d217fab098b29d2a0180fa544c5b +size 17956909 diff --git a/Z9E4T4oBgHgl3EQfnw0i/vector_store/index.pkl b/Z9E4T4oBgHgl3EQfnw0i/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4be9dadea3ec687e280d0dcbc045307fc7a6c394 --- /dev/null +++ b/Z9E4T4oBgHgl3EQfnw0i/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1e0f2ae00bd5dd7edad786116d19f87667ef8ec391f6edfd7023fdb8c7e27185 +size 81574 diff --git a/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/2301.12977v1.pdf.txt b/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/2301.12977v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a40563c4cfe2480767557ae449f05cb29589dbbe --- /dev/null +++ b/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/2301.12977v1.pdf.txt @@ -0,0 +1,1736 @@ +arXiv:2301.12977v1 [cs.LO] 30 Jan 2023 +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +ANTOINE MOTTET, TOM´AˇS NAGY, AND MICHAEL PINSKER +Abstract. We consider constraint satisfaction problems whose relations are defined in first-order +logic over any uniform hypergraph satisfying certain weak abstract structural conditions. +Our +main result is a P/NP-complete complexity dichotomy for such CSPs. Surprisingly, the large class +of structures under consideration falls into a mixed regime where neither the classical complexity +reduction to finite-domain CSPs can be used as a black box, nor does the class exhibit order prop- +erties, known to prevent the application of this reduction. We introduce an algorithmic technique +inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based +on symmetries that depend on a linear order that is external to the structures under consideration. +1. Introduction +The Constraint Satisfaction Problem (CSP) over a relational structure A in a finite signature, +denoted by CSP(A), is the problem of deciding satisfiability of a given conjunction of atomic +formulas in A. Any satisfying assignment of values to the variables is called a solution, and A is +called the template of CSP(A). +Some well-known computational problems like SAT or graph colorability can be formulated as +finite-domain CSPs, i.e., as CSPs with finite templates. It was conjectured by Feder and Vardi that +a finite-domain CSP is always polynomial-time solvable or NP-complete [30], and this conjecture +was confirmed independently by Bulatov and Zhuk recently [29, 45, 46]. Moreover, the border +between tractability and NP-completeness for finite-domain CSPs is determined by the identities +that are satisfied by the polymorphisms of the template. On the other hand, many computational +problems, even as simple as the digraph acyclicity problem, can be formulated as CSPs only with +an infinite template. +Infinite-domain CSPs also play an important role in several branches of +computer science, e.g., they are used for the study of finite-domain promise CSPs [2] and they find +applications in artificial intelligence [12, 13, 10, 17, 7, 18, 19]. +A full complexity classification of infinite-domain CSPs can never be achieved since every compu- +tational decision problem is polynomial-time Turing-equivalent to some infinite-domain CSP [11]. +Both proofs of the finite-domain CSP dichotomy theorem rely on the fact that the complexity of +CSP of a finite template depends only on its polymorphisms. This is known to be true also for +infinite templates which are ω-categorical, i.e., every instance of the CSP of such instance has only +finitely many solutions up to automorphisms [23]. The sole assumption of ω-categoricity is still +insufficient to assess the complexity of the CSP [31, 32] since the set of possible solutions of any +instance up to automorphisms does not need to be algorithmically enumerable. A natural way +to achieve this is to additionally require every solution to be described by the relations holding +on its image (homogeneity) and that every solution must only be verified locally on subsets of a +fixed size (finite boundedness). For templates whose relations are first-order definable in a finitely +bounded homogeneous structure (so-called first-order reducts of this structure), a generalization of +the original dichotomy conjecture for finite-domain CSPs was formulated by Bodirsky and Pinsker +This research was funded in whole or in part by the Austrian Science Fund (FWF) [P 32337, I 5948]. +For +the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted +Manuscript (AAM) version arising from this submission. +1 + +2 +A. MOTTET, T. NAGY, AND M. PINSKER +in 2011 [26]. The modern formulation of the conjecture based on recent progress [4, 3, 5] is the +following: +Conjecture 1. Let A be a CSP template which is a first-order reduct of a finitely bounded homo- +geneous structure. Then one of the following applies. +• The clone of polymorphisms of A has a uniformly continuous minion homomorphism to the +clone of projections P, and CSP(A) is NP-complete. +• The clone of polymorphisms of A has no uniformly continuous minion homomorphism to +the clone of projections P, and CSP(A) is in P. +The first item of Conjecture 1 corresponds to the clone of polymorphisms being essentially struc- +tureless, while in the second item, the clone is supposed to have a rich algebraic structure witnessing +the tractability of the CSP. Since, within the range of Conjecture 1, a solution to CSP(A) can be +described by specifying coherent local solutions to instances of bounded size, CSP of any template +within the range of the conjecture is in NP. Moreover, if the clone of polymorphisms of such a +template has a uniformly continuous minion homomorphism to the clone of projections, then the +CSP in NP-hard [5]. It is also known that in order to prove Conjecture 1, one can replace the as- +sumption of the template A being a first-order reduct of a finitely bounded homogeneous structure +by A being a model-complete core of such template. +Conjecture 1 has been confirmed for many subclasses: for example for CSPs of all structures +first-order definable in finitely bounded homogeneous graphs [24, 20], in (Q, <) [16], in any unary +structure [22], in the random poset [34], in the random tournament [38], or in the homogeneous +branching C-relation [14], as well as for all CSPs in the class MMSNP [19], for CSPs of representa- +tions of some relational algebras [17] and for CSPs of ω-categorical monadically stable structures +[28]. +There are two regimes in which the above-mentioned complexity classifications were proven. In +the first regime, the aim is to show that already the clone of canonical polymorphisms admits a +rich algebraic structure, which yields an efficient many-one reduction to a tractable finite-domain +CSP [21, 22]. +In order to prove that the structures under consideration have canonical polymorphisms enabling +an efficient reduction of their CSPs to tractable finite-domain CSPs, a demanding case distinction +had to be done until recently. Today, at least two general approaches that avoid this case distinction +are available: the theory of smooth approximations of Mottet and Pinsker [38] and the work of +Bodirsky and Bodor on the unique interpolation property [10]. +In the second regime, the standard action of the canonical polymorphisms of the structure on +orbits of tuples is trivial, and therefore ad hoc algorithms that do not use the standard reduction +to the finite-domain CSP mentioned above need to be given to confirm Conjecture 1. In [38], a +non-standard action of the canonical polymorphisms is used to explain the known algorithms in +the case of temporal CSPs, but the method does not seem to generalize. It was remarked in [10] +that the boundary between the two regimes is roughly drawn by whether or not the template has +the strict order property (SOP) (for details on the SOP, see [33]). Indeed, in all of the above- +mentioned complexity classifications, it was possible to confirm Conjecture 1 for first-order reducts +of structures without the SOP within the first regime. +However, in [41, Example 1], a worrying example of a first-order reduct A of the universal +homogeneous 3-uniform hypergraph H is given. Even though H does not have the SOP, the structure +A should have a polynomial-time solvable CSP according to Conjecture 1 but it does not possess +canonical polymorphisms that would allow us to use the standard reduction to a tractable finite- +domain CSP. The polymorphisms of A depend on a linear order on the domain of H even though +the hypergraph H is not ordered. This surprising discovery implies that new algorithmic techniques +are needed to solve CSPs of first-order reducts of finitely bounded homogeneous hypergraphs. + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +3 +1.1. Results. In this article, we will use the above-mentioned theory of smooth approximations +to confirm Conjecture 1 for first-order reducts of certain finitely bounded homogeneous ℓ-uniform +hypergraphs (where ℓ ≥ 3). +In particular, we show that the templates from [41, Example 1] +have polynomial-time solvable CSPs. For the precise definitions of all the mentioned concepts, +see Section 2. +Theorem 2. Let ℓ ≥ 3, let H be a finitely bounded homogeneous ℓ-uniform hypergraph whose +expansion with a freely added linear order < is a Ramsey structure and whose automorphism group +is n-“primitive” for every n ≥ 1 and let A be a first-order reduct of H. Then precisely one of the +following applies. +(1) The clone of polymorphisms of A has a uniformly continuous minion homomorphism to the +clone of projections P, and CSP(A) is NP-complete. +(2) The clone of polymorphisms of A has no uniformly continuous minion homomorphism to +the clone of projections P, and CSP(A) is in P. +The complexity classification from Theorem 2 is of particular interest for the following reasons: +• New algorithms are needed to prove Theorem 2. Since proving Conjecture 1 would in par- +ticular give an algorithm solving all tractable finite-domain CSPs, it seems likely that the +methods existing in the finite either have to be used as a black box or have to be adapted +in the infinite setting. The black box method from [22] does not work for the class of struc- +tures considered in this paper, so we resort to the second option and introduce algorithmic +techniques inspired by Zhuk’s algorithm for finite-domain CSPs. These techniques are then +coupled with the classical reduction to finite-domain CSPs, resulting in an intriguing inter- +play between infinitary and finitary methods. +• The result depends only on some general properties of the automorphism group of the +base structure (n-“primitivity”) and on the fact that the base structure has a particular +Ramsey expansion. Moreover, the base structures, i.e., ℓ-uniform hypergraphs satisfying +the properties from Theorem 2, are not classified and a complete classification seems to be +demanding if not hopeless (some 3-uniform hypergraphs satisfying our assumptions were +classified in [1]). This is the first classification where no structural results about the base +structures are known. +• In [38], the scalability of the theory of smooth approximations, i.e., the fact that this theory +does not require us to analyze all first-order reducts of the particular structure, was claimed +to be one of the main contributions of this theory compared to the archaic case-distinction +method from the early literature on the subject. +Theorem 2 provides us with the first +complexity classification using smooth approximations that truly stands by this promise. +By [44], even for the universal homogeneous ℓ-uniform hypergraph, the number of first-order +reducts of this hypergraph grows with ℓ, putting an exhaustive case analysis out of reach. +Finally, in Section 8, we obtain as an easy consequence of the proof of Theorem 2 a classification +of first-order expansions of the finitely bounded homogeneous hypergraphs under consideration +whose CSP is solvable by local consistency methods. +2. Preliminaries +For k ≥ 1, we write [k] for the set {1, . . . , k}. A tuple is called injective if its entries are pairwise +distinct. +2.1. CSPs and Relational Width. A CSP instance over a set A is a pair I = (V, C), where +V is a finite set of variables, and C is a set of constraints; each constraint C ∈ C is a subset of +AU for some non-empty U ⊆ V (U is called the scope of C). For a relational structure A, we say +that I is an instance of CSP(A) if for every C ∈ C with scope U, there exists an enumeration +u1, . . . , uk of the elements of U and a k-ary relation R of A such that for all f : U → A we have + +4 +A. MOTTET, T. NAGY, AND M. PINSKER +f ∈ C ⇔ (f(u1), . . . , f(uk)) ∈ R. A mapping s: V → A is a solution of the instance I if we have +s|U ∈ C for every C ∈ C with scope U. Given a constraint C ⊆ AU and a tuple v ∈ U k for some +k ≥ 1, the projection of C onto v is defined by projv(C) := {f(v): f ∈ C}. Let U ⊆ V. We define +the restriction of I to U to be an instance I |U = (U, C |U) where the set of constraints CU contains +for every C ∈ C the constraint C|U = {g|U | g ∈ C}. +We denote by CSPInj(A) the restriction of CSP(A) to those instances of CSP(A) where for every +constraint C and for every pair of distinct variables u, v in its scope, proj(u,v)(C) ⊆ {(a, b) ∈ A2 | +a ̸= b}. +Definition 3. Let 1 ≤ m ≤ n. We say that an instance I = (V, C) is (m, n)-minimal if both of +the following hold: +• every non-empty subset of at most m variables in V is contained in the scope of some +constraint in I; +• for every at most n-element tuple of variables v and any two constraints C1, C2 ∈ C whose +scopes contain all variables of v, the projections of C1 and C2 onto v coincide. +For m ≥ 1, we say that an instance is m-minimal if it is (m, m)-minimal. We say that an instance +I of the CSP is non-trivial if it does not contain any empty constraint. Otherwise, I is trivial. +For all 1 ≤ m ≤ n and for every instance I of a CSP(A) for some finite structure A, an (m, n)- +minimal instance with the same solution set as I can be computed from I in polynomial time. +The same holds for any ω-categorical structure A (see the next section for the definition of ω- +categoricity, and see e.g., Section 2.3 in [40] for a description of the (m, n)-minimality algorithm in +this setting). The resulting instance I′ is called the (m, n)-minimal instance equivalent to I and +the algorithm that computes this instance is called the (m, n)-minimality algorithm. Note that the +instance I′ is not necessarily an instance of CSP(A). However, I′ is an instance of CSP(A′) where +A′ is the expansion of A by all at most n-ary relations pp-definable in A. Moreover, CSP(A′) has +the same complexity as CSP(A). +If I is m-minimal and v is a tuple of variables of length at most m, then by definition there exists +a constraint of I whose scope contains v, and all the constraints who do have the same projection +on v. We write projv(I) for this projection, and call it the projection of I onto v. +Definition 4. Let 1 ≤ m ≤ n and let A be a relational structure. We say that CSP(A) has relational +width (m, n) if every non-trivial (m, n)-minimal instance of CSP(A) has a solution. CSP(A) has +bounded width if it has relational width (m, n) for some natural numbers m ≤ n. We say that +CSPInj has relational width (m, n) if every non-trivial (m, n)-minimal instance of CSPInj has a +solution. +2.2. Hypergraphs and model-theoretic notions. A relational structure B is homogeneous if +every partial isomorphism between finite induced substructures of B extends to an automorphism +of B. It is finitely bounded if there exists a finite set F of forbidden finite substructures such that +the age of B, i.e., the set of its finite substructures up to isomorphism, consists precisely of those +structures in the signature of B which do not embed any member of F. For a finitely bounded +structure B, we will denote by bB the cardinality of the biggest forbidden substructure of B. +Let ℓ ≥ 2. +A structure H = (H; E) is an ℓ-uniform hypergraph if the relation E (called a +hyperedge) is of arity ℓ, contains only injective tuples and is fully symmetric, i.e., for any tuple in +E, all tuples obtained by permuting its components are in E as well. +Let ℓ ≥ 2. The universal homogeneous ℓ-uniform hypergraph is the up to isomorphism unique +countably infinite homogeneous structure that is universal for the class of all finite ℓ-uniform +hypergraphs, i.e., it contains every such hypergraph as an induced substructure. +A homogeneous structure B is Ramsey if its age satisfies a certain structural analogue of Ramsey’s +theorem – the concrete definition will not be needed, we only need its consequence from [27] which +is stated later in this section. The universal homogeneous ℓ-uniform hypergraph is not Ramsey for + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +5 +any ℓ but it has a finitely bounded homogeneous Ramsey expansion – if we add a linear order freely +(i.e., so that the new age consists of the structures from the old age ordered in all possible ways +and the resulting structure is homogeneous), the resulting structure will be homogeneous, finitely +bounded and Ramsey – this follows, e.g., from the Neˇsetˇril-R¨odl theorem [43]. +Definition 5 (“Primitivity”). Let A be a set and n ≥ 1. A permutation group G acting on A is +n-“primitive” if for every orbit O ⊆ An of G , every G -invariant equivalence relation on O containing +some pair (a, b) with a, b disjoint is full. +Example 6. The automorphism group of the universal homogeneous ℓ-uniform hypergraph H is +n-“primitive” for any n ≥ 1. Indeed, let n ≥ 1, let O be an orbit of n-tuples under Aut(H) and let ∼ +be an equivalence relation on O containing (a, b) such that a, b are disjoint. Let c, d ∈ O be arbitrary. +We define X to be an ℓ-uniform hypergraph over 3n elements {xj +i | i ∈ [n], j ∈ [3]} such that the +following holds. The hypergraph induced by (xj +1, . . . , xj +n, xj+1 +1 +, . . . , xj+1 +n +) is isomorphic to the struc- +ture induced by (a, b) in H for every j ∈ [2] and the hypergraph induced by (x1 +1, . . . , x1 +n, x3 +1, . . . , x3 +n) +is isomorphic to the structure induced by (c, d) in H. By the universality of H, X embeds to H. By +the homogeneity of H, we can assume that the embedding maps (x1 +1, . . . , x1 +n, x3 +1, . . . , x3 +n) to (c, d). +By the transitivity of ∼, c ∼ d. +In the whole paper, we fix ℓ ≥ 3 and a finitely bounded homogeneous ℓ-uniform +hypergraph H whose expansion with a freely added linear order < is a Ramsey +structure and whose automorphism group is n-“primitive” for every n ≥ 1. +Note that not every finitely bounded homogeneous ℓ-uniform hypergraph satisfies our assump- +tions. However, the universal homogeneous ℓ-uniform hypergraph does satisfy the assumptions for +every ℓ ≥ 3. Additionally, for every fixed ℓ ≥ 3 and n > ℓ, there exists a homogeneous ℓ-uniform +hypergraph that is universal for the class of Kℓ +n-free hypergraphs, where Kℓ +n is the ℓ-hypergraph +on n vertices whose every ℓ-element subset forms a hyperedge; these hypergraphs also satisfy our +assumptions. +For n ≥ 1, we denote by In the set of injective n-tuples in H and we write I := Iℓ. We write N +for the complement of the hyperedge relation E in I and we call it the non-hyperedge relation. +A permutation group is oligomorphic if it has only finitely many orbits in its componentwise +action on n-tuples of elements for all n ≥ 1. A countable relational structure is ω-categorical if +its automorphism group is oligomorphic. A first-order reduct of a structure B is a structure A +on the same domain whose relations are first-order definable without parameters from B. Every +first-order reduct of a finitely bounded homogeneous structure is ω-categorical. A first-order reduct +of a relational structure B is a first-order expansion of B if it has among its relations all relations +from the signature of B. +A primitive-positive (pp-)formula is a first-order formula built only from atomic formulae, exis- +tential quantifiers, and conjunction. A relation is pp-definable in a relational structure A if it is +first-order definable by a pp-formula in A. +We say that an ω-categorical relational structure has no algebraicity if none of its elements is +first-order definable using other elements as parameters. Note that H has no algebraicity. To see +this, suppose that there exists a0 ∈ H that is first-order definable using elements a1, . . . , an ∈ H +as parameters. Let us define ordered ℓ-uniform hypergraphs X = ({x0, . . . , xn}, EX, yi for every i ∈ [n]. It follows that X +and Y embeds to (H, <) by embeddings eX and eY and by the homogeneity, we may suppose that +eX(ai) = eY (bi) = xi for every i ∈ [n]. Hence, eX(x0), eY (y0) and a0 satisfy the same first-order +formulas over (H, {{ai} | i ∈ [n]}) but eX(x0) ̸= eY (y0), a contradiction. + +6 +A. MOTTET, T. NAGY, AND M. PINSKER +2.3. Universal algebra. A polymorphism of a relational structure A is a homomorphism from +some finite power of A to A. The set of all polymorphisms of a structure A, denoted by Pol(A), +is a function clone, i.e., a set of finitary operations on a fixed set which contains all projections +and which is closed under arbitrary compositions. A relation is pp-definable in an ω-categorical +relational structure A if, and only if, it is invariant under Pol(A). +For a function clone C , we denote the domain of its functions by C; we say that C acts on C. +The clone C also naturally acts (componentwise) on Cl for any l ≥ 1, on any invariant subset S of +C (by restriction), and on the classes of any invariant equivalence relation ∼ on an invariant subset +S of C (by its action on representatives of the classes). We write C ↷ Cl, C ↷ S and C ↷ S/∼ +for these actions. Any action C ↷ S/∼ is called a subfactor of C , and we also call the pair (S, ∼) +a subfactor. A subfactor (S, ∼) is minimal if the equivalence relation ∼ has at least two classes +and no proper subset of S that intersects at least two ∼-classes is invariant under C . +For n ≥ 1, a k-ary operation f defined on the domain C of a permutation group G is n-canonical +with respect to G if for all a1, . . . , ak ∈ Cn and all α1, . . . , αk ∈ G there exists β ∈ G such that +f(a1, . . . , ak) = β ◦ f(α1(a1), . . . , αk(ak)). In particular, f induces an operation on the set Cn/G +of orbits of n-tuples under G . If all functions of a function clone C are n-canonical with respect +to G , then C acts on Cn/G and we write C n/G for this action; if G is oligomorphic then C n/G is +a function clone on a finite set. A function that is n-canonical with respect to G for all n ≥ 1 is +called canonical with respect to G . We say that a function is diagonally canonical if it satisfies the +definition of n-canonicity in the case α1 = · · · = αk for every n ≥ 1. +We write GC to denote the largest permutation group contained in a function clone C , and say +that C is oligomorphic if GC is oligomorphic. +For a set of functions F over the same fixed set C we write F for the set of those functions g +such that for all finite subsets F ⊆ C, there exists a function in F which agrees with g on F. For +k ≥ 1, for k-ary functions f, g and for a permutation group G such that f, g and G act on the same +domain, we say that f locally interpolates g modulo G if g ∈ {β ◦ f(α1, . . . , αk) | β, α1, . . . , αk ∈ G }. +Similarly, we say that f diagonally interpolates g modulo G if f locally interpolates g with α1 = +· · · = αk. If G is the automorphism group of a Ramsey structure in the sense of [9], then every +function on its domain locally (diagonally) interpolates a canonical (diagonally canonical) function +modulo G [27, 25]. We say that a clone D locally interpolates a clone C modulo a permutation +group G if for every g ∈ D there exists f ∈ C such that g locally interpolates f modulo G . A clone +C is a model-complete core if its unary functions are equal to GC . Hence, for every n ≥ 1, every +orbit of n-tuples under GC is preserved by C . +A structure A is called a model-complete core if its polymorphism clone is a model-complete core. +By [8], for any ω-categorical structure A, there exists an up to isomorphism unique ω-categorical +model-complete core A′ and the CSPs of A and A′ are the same computational problem. In any +ω-categorical model-complete core A, all orbits of n-tuples with respect to the automorphism group +Aut(A) are pp-definable, for all n ≥ 1. +A weak near-unanimity (WNU) operation of arity n ≥ 2 is an operation w satisfying the equation +w(x, . . . , x, y) = · · · = w(y, x, . . . , x) for all x, y from its domain. A ternary operation m is a minority +operation if it satisfies m(x, x, y) = m(x, y, x) = m(y, x, x) = y for all x, y from its domain. A binary +operation f on a two element domain is a semilattice operation if f(x, y) = f(y, x) = f(x, x) = x +and f(y, y) = y for some enumeration {x, y} of its domain. +A function is idempotent if it satisfies f(x, . . . , x) = x for every x from its domain. A function +clone is idempotent if all of its functions are. For a function f of arity n ≥ 1 and for i ∈ [n], we say +that the i-th variable of f is essential if there exist a1, . . . , an, a′ +i from the domain of f such that +f(a1, . . . , an) ̸= f(a1, . . . , ai−1, a′ +i, ai+1, . . . , an). A function is called essentially unary if at most +one of its variables is essential, otherwise is the function essential. +An arity-preserving map ξ : C → D between function clones is called a minion homomorphism +if it preserves compositions with projections, i.e., it satisfies ξ(f ◦(π1, . . . , πn)) = ξ(f)◦(π1, . . . , πn) + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +7 +for all n, m ≥ 1 and all n-ary f ∈ C and m-ary projections π1, . . . , πn. +An arity-preserving +map ξ is called a clone homomorphism if it preserves projections, i.e., maps every projection in +C to the corresponding projection in D, and compositions, i.e., it satisfies ξ(f ◦ (g1, . . . , gn)) = +ξ(f) ◦ (ξ(g1), . . . , ξ(gn)) for all n, m ≥ 1 and all n-ary f ∈ C and m-ary g1, . . . , gn ∈ C . +We +say that a function clone C is equationally trivial if it has a clone homomorphism to the clone +P, and equationally non-trivial otherwise. We say that C is equationally affine if it has a clone +homomorphism to an affine clone, i.e., a clone of affine maps over a finite module. It is known that +a finite idempotent clone is either equationally affine or it contains WNU operations of all arities +n ≥ 3 ([36], this stronger version is attributed to E. Kiss in [35, Theorem 2.8], a different proof can +be found in [47]). +If C , D are function clones and D has a finite domain, then a clone (or minion) homomorphism +ξ : C → D is uniformly continuous if for all n ≥ 1 there exists a finite subset F of Cn such that +ξ(f) = ξ(g) for all n-ary f, g ∈ C which agree on F. +For a first-order reduct A of H, we will consider the following two subclones of Pol(A): +• C H,inj +A +is the clone of those polymorphisms of A which preserve the equivalence of orbits of +injective tuples under Aut(H). +• C A,inj +A +is the clone of those polymorphisms of A which preserve the equivalence of orbits of +injective tuples under Aut(A). +3. Overview of the proof of Theorem 2 +Let A be a model-complete core of a first-order reduct A′ of H. By [5], it is enough to prove +that Theorem 2 holds for A since there exists a uniformly continuous minion homomorphism from +Pol(A′) to Pol(A) and from Pol(A) to Pol(A′). By Proposition 10, we can assume that A is itself a +first-order reduct of H. +By applying some folklore results, as well as a compactness argument, we finally obtain in Proposition 13 +that Pol(A) contains an injective operation with certain properties unless Pol(A) admits a uniformly +continuous clone homomorphism to P. This polymorphism witnesses that I is a binary absorbing +subuniverse of Hℓ. In the rest of the paper, we will prove that assuming that I is absorbing, then +CSP(A) is tractable if, and only if, C H,inj +A +↷ {E, N} is equationally non-trivial. +Let us therefore first suppose that C H,inj +A +↷ {E, N} is equationally non-trivial. In this case, +CSPInj(A) can be solved by the reduction to a tractable finite domain CSP from [21, 22]. We will +show that CSP(A) can be reduced to CSPInj(A). +By the classification of clones on a two-element domain from [42], C H,inj +A +↷ {E, N} is either equa- +tionally non-affine, or it consists of affine maps over Z2. In the first case, CSPInj(A) has relational +width (2ℓ, max(3ℓ, bH)) by an easy modification of the proof of Theorem 2 in [40]. Now, Theorem 2 +follows from Corollary 15 (details in Section 5.1). In the second case, CSPInj(A) amounts to solving +linear equations over Z2. In this situation, we can apply the following algorithm – for more details, +see Section 5.2. +Let I be an instance of CSP(A). Our algorithm transforms I into an equi-satisfiable instance I′ +that is sufficiently minimal, such that the solution set of a certain relaxation of I′ is subdirect on +all projections to an ℓ-tuple of pairwise distinct variables, and that additionally satisfies a condition +which we call inj-irreducibility, inspired by Zhuk’s notion of irreducibility [45, 46]. We then prove +that any satisfiable instance satisfying those properties has an injective solution. This step is to be +compared with the case of absorbing reductions in Zhuk’s algorithm, and in particular with Theorem +5.5 in [46], in which it is proved that any sufficiently minimal and irreducible instance that has +a solution also has a solution where an arbitrary variable is constrained to belong an absorbing +subuniverse. Since in our setting I is an absorbing subuniverse of Hℓ, this fully establishes a parallel +between the present work and [46]. + +8 +A. MOTTET, T. NAGY, AND M. PINSKER +If C H,inj +A +↷ {E, N} is equationally trivial, our goal is to prove that CSP(A) is NP-hard. The first +step is to establish that C A,inj +A +is equationally trivial as well and that C A,inj +A +⊆ C H,inj +A +(Lemma 24). +Moreover, Proposition 36 in [38] implies that there exists k ≥ ℓ such that the action of C A,inj +A +on +orbits of injective k-tuples under Aut(A) is equationally trivial. Therefore, there exists a naked +set (S, ∼) for this action. +A naked set of C A,inj +A +↷ Ik/Aut(A) consists of an invariant subset +S ⊆ Ik/Aut(A) and an invariant equivalence relation ∼ on S such that ∼ has at least two equivalence +classes and such that C A,inj +A +↷ Ik/Aut(A) acts on S/∼ by projections. By classical results in finite +clone theory, the existence of such a naked set is equivalent to the existence of a clone homomorphism +from C A,inj +A +↷ Ik/Aut(A) to P, which extends to a uniformly continuous clone homomorphism +C A,inj +A +→ P. +Now, we can employ the theory of smooth approximations to extend this homomorphism further +and obtain a uniformly continuous clone homomorphism Pol(A) → P. We recall below the relevant +definitions from the theory of smooth approximations. +Definition 7 (Smooth approximations). Let A be a set and let ∼ be an equivalence relation on +S ⊆ A. We say that an equivalence relation η on a set S′ with S ⊆ S′ ⊆ A approximates ∼ if the +restriction of η to S is a refinement of ∼. η is called an approximation of ∼. +For a permutation group G acting on A and leaving the ∼-classes invariant as well as η, we say +that the approximation η is smooth if every equivalence class C of ∼ intersects some equivalence +class C′ of η such that C ∩C′ contains a G -orbit. η is very smooth if orbit-equivalence with respect +to G is a refinement of η on S. +Recall that H has no algebraicity and hence, the hypotheses of the loop lemma of smooth +approximations [38, Theorem 10] are met and we may use the following reformulation of the lemma +to our situation. +Theorem 8. Let k ≥ 1 and suppose that C A,inj +A +↷ Ik/Aut(A) is equationally trivial. Then there +exists a naked set (S, ∼) of C A,inj +A +↷ Ik/Aut(A) with Aut(A)-invariant ∼-classes such that one of +the following holds: +• ∼ is approximated by a Pol(A)-invariant equivalence relation that is very smooth with respect +to Aut(A); +• every Pol(A)-invariant binary symmetric relation R ⊆ (Ik)2 that contains a pair (a, b) ∈ S2 +such that a ̸= b and such that a ̸∼ b contains a pseudo-loop modulo Aut(A), i.e., a pair +(c, c′) where c, c′ belong to the same orbit under Aut(A). +In [38, Theorem 10], the first item of the statement gives an approximation ∼ that is not very +smooth, but presmooth. By [38, Lemma 8], under the assumption that Pol(A) preserves I2 and +that Aut(A) is n-“primitive”, every presmooth approximation is very smooth; this justifies our +reformulation above. +Suppose that the first case of Theorem 8 applies. By a minor modification of the smooth ap- +proximation toolbox (Lemma 25), this implies that Pol(A) has a uniformly continuous clone homo- +morphism to C A,inj +A +↷ Ik/Aut(A) and hence to the clone of projection. +Suppose now that the second case of Theorem 8 applies.By [38, Lemma 13], Pol(A) contains a +weakly commutative function, i.e., a binary operation f with the property that f(a, b) ∼ f(b, a) +holds for all a, b ∈ Ik such that f(a, b) and f(b, a) are in S and disjoint. It follows from a fairly +involved compactness argument in Lemma 27 that C H,inj +A +↷ {E, N} contains a semilattice operation +and in particular, is equationally non-trivial, which is a contradiction. +4. Model-Complete Cores and Injective Polymorphisms +In this section, we first prove some basic facts about model-complete cores of first-order reducts +of H. In the second part, we prove that such first-order reducts that are model-complete cores and + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +9 +that are equationally non-trivial have binary injective polymorphisms acting as a projection or as +a semilattice operation on {E, N}. These binary injections will play an important in the algorithm +for tractable CSPs in Section 5. +4.1. Model-complete cores. Let G be a permutation group and let g: G → G be a function. We +say that g is range-rigid with respect to G if all orbits of tuples under G that intersect the range +of g are invariant under g. We will use the following theorem to understand the model-complete +cores of first-order reducts of A. +Theorem 9 ([37]). Let A be a first-order reduct of a homogeneous Ramsey structure B and let A′ +be its model-complete core. Then A′ is a first-order reduct of a homogeneous Ramsey substructure +B′ of B. +Moreover, there exists g ∈ End(A) that is range-rigid with respect to Aut(B) and such that the +age of B′ is equal to the age of the structure induced by the range of g in B. +Proposition 10. Let A be a first-order reduct of H. Then the model-complete core of A is a one- +element structure or a first-order reduct of H. Moreover, if A is a model-complete core that is a +first-order reduct of H and not of (H, =), then the range of every f ∈ End(A) intersects every orbit +under Aut(H, <). +Proof. Using Theorem 9, we obtain that the model-complete core A′ of A is a first-order reduct +of a homogeneous Ramsey substructure B′ of (H, <). Moreover, there exists g ∈ End(A) which is +range-rigid with respect to Aut(H, <) and such that the age of the structure induced by the range +of g in (H, <) is equal to the range of B′. +If the range of g is contains just one element, B′ is a one-element structure. Otherwise, the range +contains at least two elements a < b and by the range-rigidity, it then contains infinitely many +elements. In particular, it contains a hyperedge or a non-hyperedge. +If the range contains only hyperedges, then g sends every ℓ-tuple in N to an ℓ-tuple ordered in +the same way as the original tuple that lies in the hyperedge relation E. It follows that for all +injective tuples a, b of the same length, there exists an automorphism α of H and an embedding +from the range of g into B′ such that e ◦ g ◦ α(a) = b. It follows that Aut(A′) is the full symmetric +group on the domain of A′ and hence, A′ is a first-order reduct of (A′, =) which is isomorphic to +(H, =). If the range of g contains only ℓ-tuples in N, A′ is a first-order reduct of (A′, =) by the +same argument where the roles of E and N are switched. Finally, if the range of g contains both a +hyperedge as well as a non-hyperedge, it follows from the range-rigidity of g that B′ is isomorphic +to (H, <) and A′ is isomorphic to A. In particular, A′ is a first-order reduct of H. +Suppose now that A is a model-complete core that is a first-order reduct of H but not of (H, =) +and let f ∈ End(A). Suppose that the range of f does not intersect every orbit under Aut(H, <). +By Lemma 15 and Lemma 11 in [37] applied to End(A), the range-rigid function g does not intersect +every orbit under Aut(H, <) either and we obtain a contradiction with the previous paragraph. +□ +4.2. Injective binary polymorphisms. +Lemma 11. Let A be a first-order reduct of a finitely bounded homogeneous hypergraph H that is a +model-complete core. If Pol(A) does not have a uniformly continuous clone homomorphism to P, +then it contains a binary essential operation. +Proof. It follows from Corollary 6.9 in [6] that Pol(A) contains a ternary essential operation. More- +over, the binary relation O := {(a, b) | a ̸= b ∈ H} is an orbit under Aut(H) that is free, i.e., for +every (c, d) ∈ H2 there exists a ∈ H such that (a, c), (a, d) ∈ O. Now, the lemma follows directly +from Proposition 22 in [38]. +□ +Note that for every binary injective operation f on H, there exists an embedding e of the +substructure induced by the range of f into H such that f ′ := e ◦ f acts lexicographically on the + +10 +A. MOTTET, T. NAGY, AND M. PINSKER +order, i.e., f ′(x, y) < f ′(x′, y′) if x < x′ or x = x′, y < y′. To see this, let us define an ordered +hypergraph (Y, EY, p2(x, y), +• the ranges of p1 and p2 are disjoint and independent as substructures of H. +To show the existence of p1, p2, assume without loss of generality that g1 acts as the first pro- +jection on {E, N}. Then the function g2(x, y) := g1(y, x) acts as the second projection on {E, N}. +By the remark above Lemma 12, we may assume that both g1 and g2 act lexicographically on the +order <. Finally, by Theorem 26, we may compose g1 and g2 with automorphisms of (H, <) and +obtain binary injections g′ +1, g′ +2 that still satisfy all the assumptions above and whose ranges are +disjoint and induce in (H, <) substructures that are independent. +Let us define a linear order <∗ on U := im(g′ +1) ∪ im(g′ +2) as follows. We set u <∗ v if one of the +following holds. +• u < v and u, v ∈ im(g′ +i) for some i ∈ [2], or +• u = g′ +i(x1, y1), v = g′ +j(x2, y2) for some i ̸= j ∈ [2], x1, x2, y1, y2 ∈ H and one of the following +holds +– i = 1, j = 2, x2 ≤ y2 and u ≤ g′ +1(x2, y2), or +– i = 1, j = 2, x1 ≤ y1 and g′ +2(x1, y1) ≤ v, or +– i = 2, j = 1, x2 > y2 and u ≤ g′ +2(x2, y2), or +– i = 2, j = 1, x1 > y1 and g′ +1(x1, y1) ≤ v. +It is easy to verify that <∗ is a linear order on U. Let us define a hyperedge relation E∗ on +U as the restriction of the relation E to U. It follows that (U, E∗) is isomorphic to the structure + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +23 +induced by the union of the ranges of g′ +1 and g′ +2 in H and hence, (U, E∗, <∗) embeds to (H, <) by +an embedding e since (H, <) is universal for the class of all ℓ-uniform linearly ordered hypergraphs +(X, 1, then moreover yi−1 < xi. +Suppose that within some increasing diagonal order type B, we have that h acts as a constant +function on {E, N}. Pick such B such that nB is minimal. Assume without loss of generality +that the constant value of h on B is E. We claim that A has an endomorphism onto a clique, +which contradicts the conjunction of the assumptions that A is a model-complete core and not a +reduct of (H, =). We prove this claim by showing that any finite injective tuple can be mapped +to a clique by an endomorphism of A. Suppose that m ≥ 1 and that (a1, . . . , am) is an injective +m-tuple of elements of H which does not induce a clique, i.e., there is a subtuple of length ℓ +which is not an element of E. Then m ≥ ℓ, and we may assume without loss of generality that +(a1, . . . , aℓ) ̸∈ E. By applying a self-embedding e1 of H, we obtain an increasing tuple (x1, . . . , xm) +such that (x1, . . . , xℓ) ∈ N. +Applying an appropriate self-embedding e2 of H to (a1, . . . , am), +we moreover obtain an increasing tuple (y1, . . . , ym) such that ((x1, . . . , xℓ), (y1, . . . , yℓ)) ∈ B and +such that yi−1 < xi < yi for all i > ℓ. Then the increasing diagonal order type C for any pair +((xi1, . . . , xiℓ), (yi1, . . . , yiℓ)) of subtuples, both increasing, has the property that nC ≤ nB (note +that in order to compute nC, the entries of both tuples receive the indices 1 to ℓ). If nC < nB, then +h acts idempotently on {E, N} within C, by the minimality of nB; if on the other hand nC = nB, +then B = C and h acts as a constant with value E. It follows that applying the endomorphism +h(e1(x), e2(x)) to (a1, . . . , am), one obtains a tuple which has strictly more subtuples in E than +(a1, . . . , am). The claim follows. +We may thus henceforth assume that within each increasing diagonal order type B, we have that +h acts in an idempotent fashion on {E, N}. Thus, within each such type, h acts as a semilattice +operation or as a projection on {E, N}. +Let 0 ≤ j ≤ ℓ, and consider the diagonal order type T given by a pair (c, d) of increasing +injective j-tuples of elements of H; we call j the length of T. In the following, we shall say that T +is categorical if for all pairs (a, b) of increasing injective ℓ-tuples which extend (c, d) (we mean any +extension, not just end-extension) the corresponding diagonal order type is one where h behaves +like the first projection on, or if a similar statement holds for the second projection, or for the +semilattice behaviour (both semilattice behaviours are considered the same here). Note that for +length j = 0 every T is non-categorical, by the behaviours on U × V and V × U. Note also that +for length j = ℓ every T is trivially categorical. We claim that there exists T of length j = ℓ − 1 +which is not categorical. Suppose otherwise, and take any T which is categorical and implies the +behaviour as the second projection; this exists by the behaviour on V × U. Let (c, d) be a pair +of injective j-tuples which are ordered such that they represent the diagonal order type T. Let +(a1, b1) be obtained from (c, d) by extending both increasing tuples by a single element c′ and d′ +at the end, respectively, in such a way that c′ < d′. Let (c1, d1) be tuple obtained from (a1, b1) by +taking away the first components. Then the order type represented by (c1, d1) is categorical but +not for the first projection. We continue in this fashion until we arrive at a pair (aℓ, bℓ) in U × V , +a contradiction. +In the following, we assume that there exists T of length j = ℓ − 1 which extends to diagonal +order types where h behaves like different projections; the other case (projection + semilattice) +is handled similarly. We show by induction on m ≥ 1 that for all tuples a, b ∈ Im such that for +every i ∈ {1, . . . , m} at most one of the tuples ai, bi is in N there exists u ∈ Pol(A) such that +u(a, b) ∈ Em. A standard compactness argument then implies that C H,inj +A +contains a function such +that C H,inj +A +↷ {E, N} is a semilattice operation. +The base case m = 1 is clearly achieved by applying an appropriate projection. For the induction +step, let a, b ∈ Im for some m ≥ 2. Since Pol(A) contains p1, we may assume that the kernels of a +and b are identical. We may then also assume that ai, aj induce distinct sets whenever 1 ≤ i, j ≤ m +and i ̸= j, for otherwise we are done by the induction hypothesis. By the induction hypothesis, we + +AN ORDER OUT OF NOWHERE: +A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS +25 +may assume that all components of a are in E except for the second, and all components of b are in +E except for the first. It is sufficient to show that there exists an increasing diagonal order type on +the pair (a, b) (i.e., increasing diagonal order types for each of the pairs (a1, b1), . . . , (am, bm) which +are consistent with the kernels of a and b) such that h behaves like the first projection within the +order type of (a1, b1) and like the second within the order type of (a2, b2). This, however, is obvious +by our assumption. +□ +8. Bounded width +In this section, we prove a characterization of first-order expansions of H whose CSPs have +bounded width. +Theorem 28. Let A be a first-order expansion of H. Then precisely one of the following applies. +(1) The clone Pol(A) has a uniformly continuous minion homomorphism to the clone of affine +maps over a finite module. +(2) The clone Pol(A) has no uniformly continuous minion homomorphism to the clone of affine +maps over a finite module, and CSP(A) has relational width (2ℓ, max(3ℓ, bH)). +We prove Theorem 28 in a similar way as Theorem 2. We prove that if A is a first-order expansion +of H which is a model-complete core then CSP(A) has bounded width if, and only if, C H,inj +A +↷ +{E, N} is equationally non-affine. If C H,inj +A +↷ {E, N} is equationally non-affine, the result follows +from Corollary 15. +Let us therefore suppose that C H,inj +A +↷ {E, N} is equationally affine. Moreover, we may assume +that C H,inj +A +↷ {E, N} is equationally non-trivial as otherwise, Pol(A) has a uniformly continuous +homomorphism to the clone of projections by the proof of Theorem 2. We apply the second loop +lemma of smooth approximations [38, Theorem 11]. +Theorem 29. Let k ≥ 1 and suppose that C H,inj +A +↷ {E, N} is equationally non-trivial. Let (S, ∼) +be a minimal subfactor of C H,inj +A +↷ {E, N} with Aut(H)-invariant ∼-classes. Then one of the +following holds: +• ∼ is approximated by a Pol(A)-invariant equivalence relation that is very smooth with respect +to Aut(H); +• every C H,inj +A +↷ {E, N}-invariant binary symmetric relation R ⊆ I2 that contains a pair +(a, b) ∈ S2 such that a ̸= b and such that a ̸∼ b contains a pseudo-loop modulo Aut(H), i.e., +a pair (c, c′) where c, c′ belong to the same orbit under Aut(H). +In the formulation of the first item of Theorem 29, we are using [38, Lemma 8]. If the first +case of the theorem applies, i.e., the equivalence relation (S, ∼) on whose classes C H,inj +A +acts by a +function from a clone M of affine maps over a finite module is approximated by a Pol(A)-invariant +equivalence relation that is very smooth with respect to Aut(H), Lemma 24 (with k = ℓ, C A,inj +A += +C H,inj +A +) implies that Pol(A) has a uniformly continuous clone homomorphism to C H,inj +A +↷ (S, ∼) +and hence to M . +If the second case of Theorem 29 applies, we get a weakly commutative function by [38, Lemma +13], and the same argument mentioned at the end of Section 3 gives that C H,inj +A +↷ {E, N} contains +a semilattice operation. +In particular, C H,inj +A +↷ {E, N} is equationally non-affine, which is a +contradiction. +References +[1] Reza Akhtar and Alistair H. Lachlan. On countable homogeneous 3-hypergraphs. Archive for Mathematical Logic, +34:331–344, 1995. + +26 +A. MOTTET, T. NAGY, AND M. PINSKER +[2] Libor Barto. 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In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th In- +ternational Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbr¨ucken, +Germany (Virtual Conference), volume 168 of LIPIcs, pages 131:1–131:17. Schloss Dagstuhl - Leibniz-Zentrum +f¨ur Informatik, 2020. +[32] Pierre Gillibert, Julius Jonuˇsas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. When symmetries +are not enough: a hierarchy of hard constraint satisfaction problems. SIAM Journal on Computing, 51(2):175– +213, 2022. +[33] Karim Khanaki. Dividing lines in unstable theories and subclasses of baire 1 functions. Archive for Mathematical +Logic, 61(7-8):977–993, 2022. +[34] Michael Kompatscher and Trung Van Pham. A complexity dichotomy for poset constraint satisfaction. IfCoLog +Journal of Logics and their Applications (FLAP), 5(8):1663–1696, 2018. 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In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Au- +tomata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), +volume 198 of LIPIcs, pages 138:1–138:20. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2021. +[40] Antoine Mottet, Tom´aˇs Nagy, Michael Pinsker, and Micha�l Wrona. When symmetries are enough: collapsing +the bounded width hierarchy for infinite-domain CSPs. arxiv:2102.07531, 2022. +[41] Michael Pinsker. Current challenges in infinite-domain constraint satisfaction: Dilemmas of the infinite sheep. In +2022 IEEE 52nd International Symposium on Multiple-Valued Logic (ISMVL), pages 80–87, Los Alamitos, CA, +USA, 2022. IEEE Computer Society. +[42] Emil L. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematics Studies, 5, 1941. +[43] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. Ramsey classes of set systems. Journal of Combinatorial Theory, Series A, +34(2):183–201, 1983. +[44] Simon Thomas. Reducts of random hypergraphs. Annals of Pure and Applied Logic, 80(2):165–193, 1996. +[45] Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on +Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331–342. IEEE +Computer Society, 2017. +[46] Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. Journal of the ACM, 67(5):30:1–30:78, 2020. +[47] Dmitriy Zhuk. Strong subalgebras and the constraint satisfaction problem. Journal of Multiple-Valued Logic and +Soft Computing, 36(4-5):455–504, 2021. + +28 +A. MOTTET, T. NAGY, AND M. PINSKER +Hamburg University of Technology, Research Group on Theoretical Computer Science, Germany +Email address: antoine.mottet@tuhh.de +URL: http://amottet.github.io/~mottet/ +Institut f¨ur Diskrete Mathematik und Geometrie, FG Algebra, TU Wien +Email address: tomas.nagy@email.com +URL: http://dmg.tuwien.ac.at/nagy/ +Institut f¨ur Diskrete Mathematik und Geometrie, FG Algebra, TU Wien +Email address: marula@gmx.at +URL: http://dmg.tuwien.ac.at/pinsker/ + diff --git a/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/load_file.txt b/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ddc47fae19e2ae59a3ec021027438fe05f003e7 --- /dev/null +++ b/Z9FOT4oBgHgl3EQf_TTE/content/tmp_files/load_file.txt @@ -0,0 +1,1530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf,len=1529 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='12977v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='LO] 30 Jan 2023 AN ORDER OUT OF NOWHERE: A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS ANTOINE MOTTET, TOM´AˇS NAGY, AND MICHAEL PINSKER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We consider constraint satisfaction problems whose relations are defined in first-order logic over any uniform hypergraph satisfying certain weak abstract structural conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Our main result is a P/NP-complete complexity dichotomy for such CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Surprisingly, the large class of structures under consideration falls into a mixed regime where neither the classical complexity reduction to finite-domain CSPs can be used as a black box, nor does the class exhibit order prop- erties, known to prevent the application of this reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Introduction The Constraint Satisfaction Problem (CSP) over a relational structure A in a finite signature, denoted by CSP(A), is the problem of deciding satisfiability of a given conjunction of atomic formulas in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Any satisfying assignment of values to the variables is called a solution, and A is called the template of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Some well-known computational problems like SAT or graph colorability can be formulated as finite-domain CSPs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', as CSPs with finite templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' It was conjectured by Feder and Vardi that a finite-domain CSP is always polynomial-time solvable or NP-complete [30], and this conjecture was confirmed independently by Bulatov and Zhuk recently [29, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Moreover, the border between tractability and NP-completeness for finite-domain CSPs is determined by the identities that are satisfied by the polymorphisms of the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' On the other hand, many computational problems, even as simple as the digraph acyclicity problem, can be formulated as CSPs only with an infinite template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Infinite-domain CSPs also play an important role in several branches of computer science, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', they are used for the study of finite-domain promise CSPs [2] and they find applications in artificial intelligence [12, 13, 10, 17, 7, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A full complexity classification of infinite-domain CSPs can never be achieved since every compu- tational decision problem is polynomial-time Turing-equivalent to some infinite-domain CSP [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Both proofs of the finite-domain CSP dichotomy theorem rely on the fact that the complexity of CSP of a finite template depends only on its polymorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' This is known to be true also for infinite templates which are ω-categorical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', every instance of the CSP of such instance has only finitely many solutions up to automorphisms [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The sole assumption of ω-categoricity is still insufficient to assess the complexity of the CSP [31, 32] since the set of possible solutions of any instance up to automorphisms does not need to be algorithmically enumerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A natural way to achieve this is to additionally require every solution to be described by the relations holding on its image (homogeneity) and that every solution must only be verified locally on subsets of a fixed size (finite boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For templates whose relations are first-order definable in a finitely bounded homogeneous structure (so-called first-order reducts of this structure), a generalization of the original dichotomy conjecture for finite-domain CSPs was formulated by Bodirsky and Pinsker This research was funded in whole or in part by the Austrian Science Fund (FWF) [P 32337, I 5948].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' MOTTET, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' NAGY, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' PINSKER in 2011 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The modern formulation of the conjecture based on recent progress [4, 3, 5] is the following: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let A be a CSP template which is a first-order reduct of a finitely bounded homo- geneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Then one of the following applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The clone of polymorphisms of A has a uniformly continuous minion homomorphism to the clone of projections P, and CSP(A) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The clone of polymorphisms of A has no uniformly continuous minion homomorphism to the clone of projections P, and CSP(A) is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The first item of Conjecture 1 corresponds to the clone of polymorphisms being essentially struc- tureless, while in the second item, the clone is supposed to have a rich algebraic structure witnessing the tractability of the CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Since, within the range of Conjecture 1, a solution to CSP(A) can be described by specifying coherent local solutions to instances of bounded size, CSP of any template within the range of the conjecture is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Moreover, if the clone of polymorphisms of such a template has a uniformly continuous minion homomorphism to the clone of projections, then the CSP in NP-hard [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' It is also known that in order to prove Conjecture 1, one can replace the as- sumption of the template A being a first-order reduct of a finitely bounded homogeneous structure by A being a model-complete core of such template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Conjecture 1 has been confirmed for many subclasses: for example for CSPs of all structures first-order definable in finitely bounded homogeneous graphs [24, 20], in (Q, <) [16], in any unary structure [22], in the random poset [34], in the random tournament [38], or in the homogeneous branching C-relation [14], as well as for all CSPs in the class MMSNP [19], for CSPs of representa- tions of some relational algebras [17] and for CSPs of ω-categorical monadically stable structures [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' There are two regimes in which the above-mentioned complexity classifications were proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In the first regime, the aim is to show that already the clone of canonical polymorphisms admits a rich algebraic structure, which yields an efficient many-one reduction to a tractable finite-domain CSP [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In order to prove that the structures under consideration have canonical polymorphisms enabling an efficient reduction of their CSPs to tractable finite-domain CSPs, a demanding case distinction had to be done until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Today, at least two general approaches that avoid this case distinction are available: the theory of smooth approximations of Mottet and Pinsker [38] and the work of Bodirsky and Bodor on the unique interpolation property [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In the second regime, the standard action of the canonical polymorphisms of the structure on orbits of tuples is trivial, and therefore ad hoc algorithms that do not use the standard reduction to the finite-domain CSP mentioned above need to be given to confirm Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In [38], a non-standard action of the canonical polymorphisms is used to explain the known algorithms in the case of temporal CSPs, but the method does not seem to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' It was remarked in [10] that the boundary between the two regimes is roughly drawn by whether or not the template has the strict order property (SOP) (for details on the SOP, see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Indeed, in all of the above- mentioned complexity classifications, it was possible to confirm Conjecture 1 for first-order reducts of structures without the SOP within the first regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' However, in [41, Example 1], a worrying example of a first-order reduct A of the universal homogeneous 3-uniform hypergraph H is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Even though H does not have the SOP, the structure A should have a polynomial-time solvable CSP according to Conjecture 1 but it does not possess canonical polymorphisms that would allow us to use the standard reduction to a tractable finite- domain CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The polymorphisms of A depend on a linear order on the domain of H even though the hypergraph H is not ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' This surprising discovery implies that new algorithmic techniques are needed to solve CSPs of first-order reducts of finitely bounded homogeneous hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' AN ORDER OUT OF NOWHERE: A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In this article, we will use the above-mentioned theory of smooth approximations to confirm Conjecture 1 for first-order reducts of certain finitely bounded homogeneous ℓ-uniform hypergraphs (where ℓ ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In particular, we show that the templates from [41, Example 1] have polynomial-time solvable CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For the precise definitions of all the mentioned concepts, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let ℓ ≥ 3, let H be a finitely bounded homogeneous ℓ-uniform hypergraph whose expansion with a freely added linear order < is a Ramsey structure and whose automorphism group is n-“primitive” for every n ≥ 1 and let A be a first-order reduct of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Then precisely one of the following applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' (1) The clone of polymorphisms of A has a uniformly continuous minion homomorphism to the clone of projections P, and CSP(A) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' (2) The clone of polymorphisms of A has no uniformly continuous minion homomorphism to the clone of projections P, and CSP(A) is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The complexity classification from Theorem 2 is of particular interest for the following reasons: New algorithms are needed to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Since proving Conjecture 1 would in par- ticular give an algorithm solving all tractable finite-domain CSPs, it seems likely that the methods existing in the finite either have to be used as a black box or have to be adapted in the infinite setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The black box method from [22] does not work for the class of struc- tures considered in this paper, so we resort to the second option and introduce algorithmic techniques inspired by Zhuk’s algorithm for finite-domain CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' These techniques are then coupled with the classical reduction to finite-domain CSPs, resulting in an intriguing inter- play between infinitary and finitary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The result depends only on some general properties of the automorphism group of the base structure (n-“primitivity”) and on the fact that the base structure has a particular Ramsey expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Moreover, the base structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', ℓ-uniform hypergraphs satisfying the properties from Theorem 2, are not classified and a complete classification seems to be demanding if not hopeless (some 3-uniform hypergraphs satisfying our assumptions were classified in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' This is the first classification where no structural results about the base structures are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In [38], the scalability of the theory of smooth approximations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', the fact that this theory does not require us to analyze all first-order reducts of the particular structure, was claimed to be one of the main contributions of this theory compared to the archaic case-distinction method from the early literature on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Theorem 2 provides us with the first complexity classification using smooth approximations that truly stands by this promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' By [44], even for the universal homogeneous ℓ-uniform hypergraph, the number of first-order reducts of this hypergraph grows with ℓ, putting an exhaustive case analysis out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Finally, in Section 8, we obtain as an easy consequence of the proof of Theorem 2 a classification of first-order expansions of the finitely bounded homogeneous hypergraphs under consideration whose CSP is solvable by local consistency methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Preliminaries For k ≥ 1, we write [k] for the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A tuple is called injective if its entries are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' CSPs and Relational Width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A CSP instance over a set A is a pair I = (V, C), where V is a finite set of variables, and C is a set of constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' each constraint C ∈ C is a subset of AU for some non-empty U ⊆ V (U is called the scope of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For a relational structure A, we say that I is an instance of CSP(A) if for every C ∈ C with scope U, there exists an enumeration u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , uk of the elements of U and a k-ary relation R of A such that for all f : U → A we have 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' MOTTET, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' NAGY, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' PINSKER f ∈ C ⇔ (f(u1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , f(uk)) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A mapping s: V → A is a solution of the instance I if we have s|U ∈ C for every C ∈ C with scope U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Given a constraint C ⊆ AU and a tuple v ∈ U k for some k ≥ 1, the projection of C onto v is defined by projv(C) := {f(v): f ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let U ⊆ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We define the restriction of I to U to be an instance I |U = (U, C |U) where the set of constraints CU contains for every C ∈ C the constraint C|U = {g|U | g ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We denote by CSPInj(A) the restriction of CSP(A) to those instances of CSP(A) where for every constraint C and for every pair of distinct variables u, v in its scope, proj(u,v)(C) ⊆ {(a, b) ∈ A2 | a ̸= b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let 1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We say that an instance I = (V, C) is (m, n)-minimal if both of the following hold: every non-empty subset of at most m variables in V is contained in the scope of some constraint in I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' for every at most n-element tuple of variables v and any two constraints C1, C2 ∈ C whose scopes contain all variables of v, the projections of C1 and C2 onto v coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For m ≥ 1, we say that an instance is m-minimal if it is (m, m)-minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We say that an instance I of the CSP is non-trivial if it does not contain any empty constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Otherwise, I is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For all 1 ≤ m ≤ n and for every instance I of a CSP(A) for some finite structure A, an (m, n)- minimal instance with the same solution set as I can be computed from I in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The same holds for any ω-categorical structure A (see the next section for the definition of ω- categoricity, and see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='3 in [40] for a description of the (m, n)-minimality algorithm in this setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The resulting instance I′ is called the (m, n)-minimal instance equivalent to I and the algorithm that computes this instance is called the (m, n)-minimality algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Note that the instance I′ is not necessarily an instance of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' However, I′ is an instance of CSP(A′) where A′ is the expansion of A by all at most n-ary relations pp-definable in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Moreover, CSP(A′) has the same complexity as CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' If I is m-minimal and v is a tuple of variables of length at most m, then by definition there exists a constraint of I whose scope contains v, and all the constraints who do have the same projection on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We write projv(I) for this projection, and call it the projection of I onto v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let 1 ≤ m ≤ n and let A be a relational structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We say that CSP(A) has relational width (m, n) if every non-trivial (m, n)-minimal instance of CSP(A) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' CSP(A) has bounded width if it has relational width (m, n) for some natural numbers m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We say that CSPInj has relational width (m, n) if every non-trivial (m, n)-minimal instance of CSPInj has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Hypergraphs and model-theoretic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A relational structure B is homogeneous if every partial isomorphism between finite induced substructures of B extends to an automorphism of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' It is finitely bounded if there exists a finite set F of forbidden finite substructures such that the age of B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', the set of its finite substructures up to isomorphism, consists precisely of those structures in the signature of B which do not embed any member of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For a finitely bounded structure B, we will denote by bB the cardinality of the biggest forbidden substructure of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A structure H = (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' E) is an ℓ-uniform hypergraph if the relation E (called a hyperedge) is of arity ℓ, contains only injective tuples and is fully symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', for any tuple in E, all tuples obtained by permuting its components are in E as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The universal homogeneous ℓ-uniform hypergraph is the up to isomorphism unique countably infinite homogeneous structure that is universal for the class of all finite ℓ-uniform hypergraphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', it contains every such hypergraph as an induced substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A homogeneous structure B is Ramsey if its age satisfies a certain structural analogue of Ramsey’s theorem – the concrete definition will not be needed, we only need its consequence from [27] which is stated later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The universal homogeneous ℓ-uniform hypergraph is not Ramsey for AN ORDER OUT OF NOWHERE: A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS 5 any ℓ but it has a finitely bounded homogeneous Ramsey expansion – if we add a linear order freely (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', so that the new age consists of the structures from the old age ordered in all possible ways and the resulting structure is homogeneous), the resulting structure will be homogeneous, finitely bounded and Ramsey – this follows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=', from the Neˇsetˇril-R¨odl theorem [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Definition 5 (“Primitivity”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let A be a set and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A permutation group G acting on A is n-“primitive” if for every orbit O ⊆ An of G , every G -invariant equivalence relation on O containing some pair (a, b) with a, b disjoint is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The automorphism group of the universal homogeneous ℓ-uniform hypergraph H is n-“primitive” for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Indeed, let n ≥ 1, let O be an orbit of n-tuples under Aut(H) and let ∼ be an equivalence relation on O containing (a, b) such that a, b are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let c, d ∈ O be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We define X to be an ℓ-uniform hypergraph over 3n elements {xj i | i ∈ [n], j ∈ [3]} such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' The hypergraph induced by (xj 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , xj n, xj+1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , xj+1 n ) is isomorphic to the struc- ture induced by (a, b) in H for every j ∈ [2] and the hypergraph induced by (x1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , x1 n, x3 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , x3 n) is isomorphic to the structure induced by (c, d) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' By the universality of H, X embeds to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' By the homogeneity of H, we can assume that the embedding maps (x1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , x1 n, x3 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , x3 n) to (c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' By the transitivity of ∼, c ∼ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' In the whole paper, we fix ℓ ≥ 3 and a finitely bounded homogeneous ℓ-uniform hypergraph H whose expansion with a freely added linear order < is a Ramsey structure and whose automorphism group is n-“primitive” for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Note that not every finitely bounded homogeneous ℓ-uniform hypergraph satisfies our assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' However, the universal homogeneous ℓ-uniform hypergraph does satisfy the assumptions for every ℓ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Additionally, for every fixed ℓ ≥ 3 and n > ℓ, there exists a homogeneous ℓ-uniform hypergraph that is universal for the class of Kℓ n-free hypergraphs, where Kℓ n is the ℓ-hypergraph on n vertices whose every ℓ-element subset forms a hyperedge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' these hypergraphs also satisfy our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' For n ≥ 1, we denote by In the set of injective n-tuples in H and we write I := Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We write N for the complement of the hyperedge relation E in I and we call it the non-hyperedge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A permutation group is oligomorphic if it has only finitely many orbits in its componentwise action on n-tuples of elements for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A countable relational structure is ω-categorical if its automorphism group is oligomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A first-order reduct of a structure B is a structure A on the same domain whose relations are first-order definable without parameters from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Every first-order reduct of a finitely bounded homogeneous structure is ω-categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A first-order reduct of a relational structure B is a first-order expansion of B if it has among its relations all relations from the signature of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A primitive-positive (pp-)formula is a first-order formula built only from atomic formulae, exis- tential quantifiers, and conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' A relation is pp-definable in a relational structure A if it is first-order definable by a pp-formula in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' We say that an ω-categorical relational structure has no algebraicity if none of its elements is first-order definable using other elements as parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Note that H has no algebraicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' To see this, suppose that there exists a0 ∈ H that is first-order definable using elements a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , an ∈ H as parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' Let us define ordered ℓ-uniform hypergraphs X = ({x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FOT4oBgHgl3EQf_TTE/content/2301.12977v1.pdf'} +page_content=' , xn}, EX, 0.6 +max response box + response map +xoq indino +Search area +KF predict box +kalman filte +Calculate location + max-value box +scores +NMS +iV + candidate boxes +Select from the candidate boxes.. +/ +Ground Trurh +Candidate Boxes +Max Response Box +Kalman Filter Predict-box +Output BoxarXiv Template +A PREPRINT +Table 1: PyTorch pseudo code for our method +PyTorch Pseudo Code +# self.kf_num = 40; self.nms_thre = 0.8; self.conf=0.6 +If self.use_kf: +kf_pre_box = self.kalmanfilter.predict() +if box_iou(kf_pre_box, max_response_box) > self.conf: +output_box = max_response_box +else: +pred_boxes, conf = decode_muti_boxes(output, self.kf_num) +pred_boxes, conf = NMS(pred_boxes, response, self.nms_thre) +location_score = box_iou(kf_pre_box, pred_boxes) +location_score = torch.mul(location_score, conf) +index = torch.argmax(location_score) +output_box = pred_boxes[index] +# update kalmanfilter +If self.use_kf: +self.kalmanfilter.update(output_box) +Table 2: The use of information by different post-processing paradigms +Single Frame Information +Image Sequence Information +paradigm +Max response value +Candidate information +Candidate information +Other information +Detection-based +✓ +Motion-based (ours) +✓ +✓ +✓ +Please note that we do not directly use the box predicted by the Kalman filter as the output box. The reason is that the +box predicted by the Kalman filter has certain inaccuracies. We only use the prediction box as a constraint to select +one of the candidate boxes generated by the tracker that best matches the position where the current target should be. +Kalman filter can be used to establish the correlation between frames of the target to realize the interaction of position +information between frames and make the tracker know where its target is likely to be. +3.4 +Comparison with detection-based paradigm +The detection-based post-processing paradigm selects the bounding box of the maximum response value in the search +area of each frame to locate the target through the powerful discriminant ability of the neural network. The detection of +each frame is independent of each other, and it does not pay attention to the information of non- maximum response +value points in the response value. +Our motion-based post-processing paradigm not only uses the maximum response value to locate the target, but also +uses the motion information of the target in the sequence to constrain the position of the bounding box. When the +bounding box drifts, the candidate frame information is reused to retrieve the tracked target. The degree of information +utilization is shown in Table 2. The illustration of utilization of target past status information is in Figure 1. The +illustration of utilization of candidate box information is in Figure 2. +4 +Experiment +4.1 +Implementation Details +We replaced Stark’s CornerHead with CenterHead and transferred the remaining parameters, and then trained 50 epochs +according to the paradigm of OSTrack. Then we set kf_num to 40 on the OSTrack, Starks, Starkst on the UOT100. +Set kf_num to 30 on the OSTrack on the UTB180. Set kf_num to 40 on the Starks and Starkst on the UTB180. Set +nms_thre to 0.8 and conf to 0.6 on each tracker and both datasets. Starkst update interval is 100. The selected response +map is the original response map without hanning window. On the TransT, we set kf_num to 20, nms_thre to 0.8, conf +5 + +arXiv Template +A PREPRINT +Figure 2: The original tracker only uses the maximum response value of each frame to locate the target, so its track is +discrete in the tracking. Although significant errors occur in the trajectory, the detection-based paradigm does not have +the function of trajectory self-correction. Our method uses the target state information in past frames to effectively +constrain the tracker’s tracking position in each frame. Even if there is a slight tracking drift, it can still be effectively +corrected. +to 0.6 and the selected response map has been multiplied by the Hanning window on both datasets. Other trackers +evaluated on UOT100 and UTB180 were set according to their original parameters. +The device we used is CPU: i9-12900KF. GPU: GTX-3090Ti. The version of Pytorch we used is 1.7.1. +4.2 +Dataset and Evaluation indicators +UOT100 dataset benchmark is the first typical Underwater Object Tracking dataset, which is mainly used to reflect +the challenges faced by underwater object tracking, such as image degradation, similar targets, large deformation and +other issues. In order to verify the effectiveness of our method in dealing with similar target challenges, we extract 28 +subsets of UOT 100 with similar target interference challenges as similar subsets of UOT. [See appendix for details of +division]. At the same time, we use the remaining 78 sequences as the complement of similar subsets to verify whether +our strategy will reduce the performance of the tracker on non-similar problem sequences. +UTB180 dataset benchmark is a high-quality Underwater Object Tracking dataset. It reflects challenges in underwater +object tracking such as Unclear Water, Occlusion, Similar Objects. We also use its 116 similar object subsets to prove +that our method is effective. The other 64 non-similar object subsets are used to explore the performance of our method +on non-similar problem sequences. We follow the One Pass Evaluation (OPE) protocol most commonly used in single +object tracking to evaluate the tracker, and the AUC, precision, and norm-precision are used to evaluate the tracker. +We follow the One Pass Evaluation (OPE) protocol most commonly used in single object tracking to evaluate the tracker, +and the AUC, precision, and norm-precision are used to evaluate the tracker. +4.3 +Effectiveness of our method +We select SOTA performance trackers such as OSTrack[23], Stark[19] and TransT[20] to verify the effectiveness of +our strategy. Since the Corner-Head of the original Stark only outputs a box end-to-end, we replace CornerHead with +CenterHead to obtain multiple candidate boxes. +6 + +#frame:222 +Center point position at intervals of five frames +Center point position at intervals of five frames after 222 frames +Trajectory after frame 222 + Trajectory before frame 222 +OSTrack+ours +Ground Truth +Original OSTrackarXiv Template +A PREPRINT +Table 3: Boost of our method on different trackers in UOT100 +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +68.40 +63.93 +86.48 +65.19 +58.51 +83.62 +67.00 +59.05 +82.49 +64.39 +57.00 +80.70 +Original +66.88 +62.11 +84.55 +62.65 +54.10 +78.69 +66.39 +57.46 +80.65 +63.75 +56.27 +79.85 +Boost +1.52 +1.82 +1.93 +2.54 +4.41 +4.93 +0.61 +1.59 +1.84 +0.64 +0.73 +0.85 +Table 4: Boost of our method on different trackers in UTB180 +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +64.13 +56.99 +73.82 +58.92 +49.99 +69.92 +60.61 +53.57 +71.64 +58.89 +51.82 +67.67 +Original +63.03 +56.52 +72.61 +52.72 +44.07 +62.09 +54.90 +47.79 +64.47 +57.52 +50.30 +66.13 +Boost +1.10 +0.47 +1.21 +6.20 +5.92 +7.83 +5.71 +5.78 +7.17 +1.37 +1.52 +1.54 +The results are shown in the Table 3 and Table 4. On UOT100 and UTB180, our method can effectively improve the +performance of the tracker. Besides, Our method only needs 0.008s to filter candidate boxes once. Considering that our +method is only used when the IOU of the Kalman filter prediction box and the maximum response box is less than the +threshold, the average consumption time will be less. +To further prove whether our method can effectively improve the discriminability of the tracker for similar objects +around the target, we verified the effectiveness of the method in the similar object subsets of UOT100 and UTB180. +As shown in Table 5 and Table 6, our strategy can effectively improve the performance of the tracker when dealing +with similar target interference. We noticed that Starks has shown significant improvement. The AUC has improved by +more than 6% on average and the accuracy has improved by more than 9% on average. This is because Stark extracts +the semantic features of the target through backbones and sends them to the Transformer for fusion. However, similar +semantic features alone cannot distinguish between targets and similar objects. For example, a ball can represent a +basketball or a tennis ball, so Starks is more likely to be disturbed by similar semantics and tracking drift, which is why +our method has the highest performance improvement for Starks. Starkst introduces a second variable template to bring +online updating adaptability to the tracker, which can enhance the appearance discrimination of the tracker for similar +targets. But our method still has gains. In OStrack, our method brings 3% performance improvement on average. In +TransT, our method also improves the performance on MLP-based prediction head. +In order to verify whether our method will affect the general underwater tracking, we also tested the performance of +four trackers on the complements of similar sequences in UOT100 and UTB180. As shown in Table 7 and Table 8, +Starks, Starkst and TransT have shown good adaptability in tracking performance of non-similar problem sequences. +In contrast, in the UTB100 non-similar problem sequences, the performance of OSTrack decreases by more than 2%. +Since the performance is still good after the degradation, we did not further explore the cause of the performance +degradation. However, the performance degradation on non-similar sequences will limit the promotion of our method +on the Open-air tracking dataset (which is also what we hope to solve in the next stage). +4.4 +Comparison of our method with SOTA trackers +We evaluated 14 SOTA trackers (NeighborTrack (represented by OSTrack-N)[36], OSTrack[23], MixFormer[21], +AiATrack[22], ToMP[27], KeepTrack[26], Stark[19], TransT[20], TrDimp[37], SiamBAN-ACM[38], Dimp[25], +Table 5: Boost of our method on different trackers in UOT100’s similarity subset +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +63.27 +59.57 +77.65 +60.70 +54.31 +74.63 +64.18 +56.32 +76.19 +58.05 +50.33 +68.55 +Original +58.81 +54.33 +72.08 +54.72 +45.14 +65.08 +59.95 +51.48 +70.15 +57.12 +50.02 +67.74 +Boost +4.46 +5.24 +5.54 +5.98 +9.17 +9.55 +4.23 +4.84 +6.04 +0.93 +0.31 +0.81 +7 + +arXiv Template +A PREPRINT +Table 6: Boost of our method on different trackers in UTB180 similarity subset +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +57.70 +52.31 +65.80 +51.99 +45.47 +61.14 +55.34 +51.07 +65.24 +51.89 +47.68 +59.28 +Original +54.78 +50.29 +62.50 +42.68 +36.87 +49.63 +47.16 +42.64 +54.91 +49.88 +45.39 +56.94 +Boost +2.92 +2.02 +3.30 +9.31 +8.60 +11.51 +8.18 +8.43 +10.33 +2.01 +2.29 +2.34 +Table 7: Boost of our method on different trackers in complements of similar sequences in UOT100 +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +70.38 +66.11 +89.92 +67.07 +60.73 +87.26 +68.15 +60.60 +84.86 +67.01 +59.93 +85.26 +Original +69.88 +65.49 +89.23 +65.75 +58.01 +83.92 +68.91 +60.19 +84.67 +66.23 +59.03 +84.28 +Boost +0.50 +0.62 +0.69 +1.32 +2.72 +3.34 +-0.76 +0.41 +0.19 +0.78 +0.90 +0.98 +SiamBAN[16], ATOM[24], SiamCAR[39]) on UOT100 and UTB180 dataset. We compared the results with our +method results. +The results of comparison of our method (applied to OSTrack256) and SOTA performance trackers on the UOT100 and +UTB180 are shown in Table 9 and Table 10. Our method (applied to OSTrack256) achieved SOTA performance on +the UOT100. In addition, we found that the Open-air SOTA performance tracker still has a very good performance in +terms of precision and regularization precision on the sequence of non-similar problems of UOT100 and UTB180. This +means that in underwater target tracking, the improvement of strong feature representation ability is not the most urgent +task to improve the performance of the underwater tracker. We need to further consider the performance improvement +in edge cases (such as similar targets, occulsion, unclear water, out of view and so on) during tracking. +4.5 +Comparison of our method with other improvement strategies +We compared our method with two other classical tracker enhancement methods (NeighborTrack (Represented by +OSTrack-N)[36], Alpha-Refine[40]). Specifically, we applied these enhancement methods to OSTrack and evaluate the +performance of the tracker on the UOT100 and UTB180. +The NeighborTrack is designed to deal with occlusion problem. The UOT100 dataset does not focus on requiring the +tracker to deal with occlusion challenge, so the NeighborTrack does not perform very well. In the UTB180, there are 92 +sequences reflecting occlusion problem, which improves the performance of NeighborTrack. In addition, we found +that our method is not in conflict with the NeighborTrack method, and they can jointly enhance the performance of the +tracker. Besides, our method can also be combined with the Alpha-Refine module to jointly improve the performance +of the tracker. In conclusion, our method has good compatibility, and can be inserted into the tracker to improve the +tracker performance together with other methods. +Table 8: Boost of our method on different trackers in complements of similar sequences in UTB180 +Tracker +OSTrack256 +Starks+CenterHead+ep50 +Starkst+CenterHead+ep50 +TransT +method +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +Ours +75.79 +65.46 +88.35 +71.47 +58.19 +85.84 +70.17 +58.10 +83.22 +71.57 +59.32 +82.89 +Original +77.97 +67.83 +90.92 +70.94 +57.11 +84.68 +68.91 +57.12 +81.81 +71.37 +59.20 +82.79 +Boost +-2.18 +-2.37 +-2.57 +0.53 +1.08 +1.16 +1.26 +0.98 +1.41 +0.20 +0.12 +0.10 +8 + +arXiv Template +A PREPRINT +Table 9: Comparisons of our method (applied to OSTrack256) with SOTA performance trackers on the UOT100. The +best two results are shown in red and blue fonts. +UOT100(106) +UOT100’s Similary Subset (28) +Complements of Similar Subsets (78) +Tracker +Source +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +ATOM +CVPR2019 +54.79 +44.24 +68.76 +48.63 +41.42 +59.84 +57.48 +46.03 +73.53 +Dimp50 +ICCV2019 +59.82 +48.90 +75.39 +56.59 +45.62 +68.80 +61.42 +50.91 +78.38 +SiamCAR +CVPR2020 +53.55 +45.96 +69.40 +44.14 +40.88 +54.78 +57.32 +48.44 +75.06 +SiamBAN +CVPR2020 +56.72 +50.71 +73.41 +53.76 +49.56 +67.64 +57.92 +51.89 +76.11 +SiamBAN-ACM +CVPR2021 +61.43 +51.89 +75.01 +55.84 +48.07 +65.70 +63.73 +54.05 +78.82 +TrDimp +CVPR2021 +61.19 +51.04 +77.47 +54.90 +44.42 +66.32 +63.66 +54.17 +81.92 +TransT +CVPR2021 +63.75 +56.27 +79.85 +57.12 +50.02 +67.74 +66.23 +59.03 +84.28 +Stark-ST101 +ICCV2021 +66.33 +58.12 +82.66 +57.50 +49.82 +68.03 +69.63 +61.68 +88.04 +Stark-S50 +ICCV2021 +63.40 +55.29 +77.38 +52.86 +41.72 +58.75 +67.35 +60.66 +84.27 +KeepTrack +ICCV2021 +60.04 +51.20 +78.05 +54.48 +46.29 +66.84 +63.02 +53.66 +82.37 +ToMP50 +CVPR2022 +66.84 +58.55 +82.82 +61.94 +53.69 +73.80 +68.87 +60.92 +86.47 +AiaTrack +ECCV2022 +65.31 +57.57 +83.06 +59.67 +52.24 +72.43 +67.48 +60.04 +87.02 +MixFormer +CVPR2022 +66.20 +59.81 +83.50 +58.97 +50.89 +69.78 +68.94 +63.55 +88.63 +OSTrack384 +ECCV2022 +66.96 +62.322 +84.69 +60.19 +54.82 +73.18 +69.45 +65.60 +88.90 +OSTrack256 +ECCV2022 +66.88 +62.11 +84.55 +58.81 +54.33 +72.08 +69.88 +65.49 +89.23 +OSTrack-N +ArXiv2022 +67.32 +62.56 +85.23 +59.82 +55.25 +73.63 +70.15 +65.79 +89.66 +OSTrack256+ours +68.40 +63.93 +86.48 +63.27 +59.57 +77.65 +70.38 +66.11 +89.92 +OSTrack256+ours+N +68.52 +64.00 +86.75 +63.65 +59.86 +78.85 +70.40 +66.09 +89.88 +Table 10: Comparisons of our method (applied to OSTrack256) with SOTA performance trackers on the UTB180. The +best two results are shown in red and blue fonts. +UTB180 (180) +UTB Similary Subset (116) +Complements of Similar Subsets (64) +Tracker +Source +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +ATOM +CVPR2019 +47.49 +35.23 +55.16 +42.02 +33.66 +48.25 +57.40 +38.09 +67.32 +Dimp50 +ICCV2019 +50.52 +37.51 +58.49 +43.41 +34.48 +49.33 +63.39 +43.01 +75.08 +SiamCAR +CVPR2020 +49.80 +40.84 +60.07 +46.62 +41.71 +56.27 +55.57 +39.25 +66.94 +SiamBAN +CVPR2020 +56.95 +46.95 +68.22 +54.29 +47.11 +64.53 +61.77 +46.64 +74.90 +SiamBAN-ACM +CVPR2021 +56.97 +46.97 +66.74 +52.61 +45.50 +60.69 +54.88 +49.63 +77.61 +TrDimp +CVPR2021 +59.00 +47.52 +68.65 +52.03 +43.49 +59.66 +71.62 +54.82 +84.93 +TransT +CVPR2021 +57.52 +50.30 +66.13 +49.88 +45.39 +56.94 +71.37 +59.20 +82.79 +Stark-ST50 +ICCV2021 +55.86 +4845 +64.63 +45.75 +40.85 +52.13 +74.18 +62.21 +87.27 +Stark-S50 +ICCV2021 +53.89 +45.62 +61.33 +43.08 +37.68 +47.84 +73.47 +60.02 +85.78 +KeepTrack +ICCV2021 +54.90 +43.49 +64.06 +45.71 +37.78 +52.52 +71.55 +53.82 +84.99 +ToMP50 +CVPR2022 +61.14 +53.40 +70.82 +54.39 +48.72 +62.45 +73.36 +61.87 +85.99 +AiaTrack +ECCV2022 +62.01 +52.94 +72.11 +55.14 +48.13 +63.39 +74.47 +61.67 +87.91 +MixFormer +CVPR2022 +57.44 +50.60 +65.69 +46.37 +41.01 +51.96 +77.49 +68.00 +90.58 +OSTrack384 +ECCV2022 +62.04 +57.18 +70.99 +52.14 +48.79 +59.10 +79.98 +72.40 +92.55 +OSTrack256 +ECCV2022 +63.03 +56.52 +72.61 +54.78 +50.29 +62.50 +77.97 +67.83 +90.92 +OSTrack-N +ArXiv2022 +64.53 +58.10 +74.16 +58.21 +53.85 +66.29 +75.99 +65.79 +88.42 +OSTrack256+ours +64.13 +56.99 +73.82 +57.70 +52.31 +65.80 +75.79 +65.46 +88.35 +OSTrack256+ours+N +65.33 +58.29 +74.89 +59.61 +54.33 +67.66 +75.70 +65.45 +88.00 +5 +Discussion +5.1 +What does our method actually do? +We visualized some of the tracking results for OSTrack and Stark+CenterHead, as shown in the figure. Our method +mainly helps the tracker to suppress tracking drift under similar interference of the same kind, thus improving the +performance of the tracker. +To further demonstrate our method can suppress tracking frame drift, we show some pictures with the maximum +response frame different from the final output frame and the candidate frame information of the current frame. As +shown in Figure 4, in the tracking process, although the tracker will generate a large number of meaningless candidate +boxes, each target like a template in the search area can still be accurately located. Our method focuses on how to find +the accurate target location among the candidate frames when the maximum response frame drifts. In typical cases, the +Kalman filter prediction frame is usually close to the target truth box, so the most appropriate candidate frame can be +selected by calculating the IOU and response score of the prediction frame and candidate frame. At the same time, the +screened candidate box can also maximize the use of the powerful discriminant ability of neural network to obtain an +accurate target location and scale estimation. +9 + +arXiv Template +A PREPRINT +Table 11: Comparisons of our method with other improvement strategies on the UOT100 +UOT100 (106) +UOT100’s Similary Subset (28) +Complements of Similar Subsets (78) +Tracker +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +OSTrack +66.88 +62.11 +84.55 +58.81 +54.33 +72.08 +69.88 +65.49 +89.23 +OSTrack+ours +68.40 +63.93 +86.48 +63.27 +59.57 +77.65 +70.38 +66.11 +89.92 +OSTrack+N +67.32 +62.56 +85.23 +59.82 +55.25 +73.63 +70.15 +65.79 +89.66 +OSTrack+ours+N +68.52 +64.00 +86.75 +63.65 +59.86 +78.85 +70.40 +66.09 +89.88 +OSTrack+AR +67.32 +60.87 +83.90 +58.53 +53.49 +70.29 +70.62 +64.09 +88.97 +OSTrack+ours+AR +68.36 +61.93 +85.08 +62.05 +57.18 +74.59 +70.82 +64.23 +89.11 +Table 12: Comparisons of our method with other improvement strategies on the UTB180 +UTB180 (180) +UTB Similary Subset (116) +Complements of Similar Subsets (64) +Tracker +AUC +P +P-Norm +AUC +P +P-Norm +AUC +P +P-Norm +OSTrack+ours +64.13 +56.99 +73.82 +57.70 +52.31 +65.80 +75.79 +65.46 +88.35 +OSTrack+N +64.53 +58.10 +74.16 +58.21 +53.85 +66.29 +75.99 +65.79 +88.42 +OSTrack+ours+N +65.33 +58.29 +74.89 +59.61 +54.33 +67.66 +75.70 +65.45 +88.00 +OSTrack+AR +64.88 +59.60 +74.73 +58.74 +55.05 +66.83 +75.99 +67.83 +89.05 +OSTrack+ours+AR +65.55 +59.42 +75.32 +59.96 +55.07 +68.07 +75.67 +67.31 +88.46 +5.2 +Lack of instance-level discrimination for close similar targets +In the experiment, we found that most target trackers lack the instance-level discrimination ability for similar targets of +the same kind that are close to them. +As shown in Figure 5, when two similar targets approach, the tracker will regard them as a larger target and predict a +large box that covers both targets. This is because the tracker mainly obtains the final bounding box by matching the +features in the template with the features in the search area. Each feature matched successfully is considered to be part +of the unique output box. They are combined to form a large bounding box containing multiple similar objects. On the +one hand, this bounding box will reduce the performance of the tracker and may lead to tracking drift. On the other +hand, it will contaminate the Kalman filter in our method and affect the effectiveness of our method. +We find that in Object Detection, the training data of the detector contains a large number of similar targets and their +labels, which can naturally distinguish target instances after training. At present, the paradigm of tracker post-processing +is close to that of detector post-processing. Therefore, can we construct a training set by using a Mosaic-like method +or using an unsupervised training paradigm to make the tracker also have the instance-level discrimination ability for +similar targets? +Figure 3: Visualization comparison examples of our method results on different trackers. +10 + +#frame: 1 ++frame:516 +JerkbaitBites +#frame: 468 +e: 470 +#frame: 960 +fram +#fram +e: 518 +le: 623 +MonsterCreature2 +#frame:404 +#frame: +#frame:135 +#f +WhiteShark +Ground Truth +OSTrack +OsTrack+ours +Starks+CenterHead +Starks+CenterHead+oursarXiv Template +A PREPRINT +Figure 4: Example of our method self-correcting process. +Figure 5: Examples of trackers lacking instance-level discrimination ability. +5.3 +Disadvantages of our method +Because our method only considers the normal form of using Kalman filter to eliminate potential interference targets in +the tracking process, and lacks the processing of edge conditions, our method is actually vulnerable to interference. +For example, when the Kalman filter uses an inaccurate tracking bounding box to update the state, it will continuously +accumulate errors. Finally, the contaminated Kalman filter will screen out the wrong candidate box, reducing the +performance of the tracker. +In fact, we believe that when using the underwater tracker on an underwater vehicle, various edge conditions need to +be considered to obtain the performance improvement of the tracker, such as considering the low response value of +the target due to occlusion or out-of-view, the high confidence update scheme for template update (if available ) and +target state update (such as our method) and their correction mechanism. Moreover, tracking is usually continuous. In +a general tracking process, the IOU between the two frames is usually greater than a certain threshold. In addition, +the tracker does not only perform tasks on the underwater robot. Generally, the vehicle may carry Object Detection +algorithms as detectors, matching algorithms in MOT, etc. The effective use of external information to improve the +performance of the tracker is also a meaningful issue. +In general, we believe that the design of an underwater tracker with multiple strategies is helpful to improve the +operational efficiency of underwater vehicles. And we hope our method can provide a reference for the design of +underwater trackers to address similar object challenges of the same kind. +6 +Conclusion +In this paper, considering the problem of marine organism swarming in underwater target tracking, a simple motion- +based post-processing strategy using Kalman filter to eliminate similar interference near the target is proposed. Our +11 + +JerkbaitBites +me: 841 +e: 832 +MonsterCreature2 +#frame:138 +#frame +#frame: 304 +119 +122 +147 +#frame: ++frame: +#frame: +MississippiFish +#frame: 53 +rame: 111 + PinkFish +Ground Truth +Candidate boxes +Max response box +Kalman Filter predict-box +Output box#frame:137 +#frame:147 +#frame:162 +#frame:180 +#frame:290 +ColourChangingSquid +CressiGuillaumeNeri2 +ram +Rocketman +Stark-ST +Ground Truth +OSTrack +MixFormer +AiaTrack +TOMParXiv Template +A PREPRINT +Table 13: Sequence names of UOT100 similar subsets +UOT100 similarity subsets +ArmyDiver1, ArmyDiver2, ArmyDiver3, ClickerAndTarget, ColourChangingSquid, CrabTrap, +CressiGuillaumeNeri1, CressiGuillaumeNeri2, Diving360Degree2, FightingEels2, GarryFish, +GiantCuttlefish2, GreenMoreyEel1, GreenMoreyEel3, JerkbaitBites, MantaRescue1, MantaRescue2, +MantaRescue3, MantaRescue4, MississippiFish, MonsterCreature2, MythBusters, PinkFish, +Rocketman, ScubaDiving1, ScubaDiving2, SharkSuckers2, WhiteShark +strategy reuses the information of the candidate target boxes and their response values in the tracker response graph, +without additional training for the tracker. +We have proved the effectiveness of our method in dealing with similar target challenges on multiple trackers, and +explained why our method is effective. In addition, we also evaluate the performance of the open-air tracker in +underwater target tracking in the past three years. Our strategy combined with the OSTrack tracker has SOTA +performance in underwater object tracking. Our strategy is also compared with other tracker strategies, and it is proved +that our method and other methods such as NeighborTrack can jointly improve the tracker performance. +Finally, we analyze the shortcomings of existing trackers in dealing with similar problems of underwater targets and the +inadequacy of our method. +7 +Appendix +More details can be found in Table 13. +References +[1] Heng Fan, Liting Lin, Fan Yang, Peng Chu, Ge Deng, Sijia Yu, Hexin Bai, Yong Xu, Chunyuan Liao, and Haibin +Ling. Lasot: A high-quality benchmark for large-scale single object tracking. In Proceedings of the IEEE/CVF +conference on computer vision and pattern recognition, pages 5374–5383, 2019. +[2] Lianghua Huang, Xin Zhao, and Kaiqi Huang. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 5289–5298, 2021. +14 + diff --git a/_NAzT4oBgHgl3EQfhPyg/content/tmp_files/load_file.txt b/_NAzT4oBgHgl3EQfhPyg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..24f0036f03837b6f1b9e94592528d7917af45b84 --- /dev/null +++ b/_NAzT4oBgHgl3EQfhPyg/content/tmp_files/load_file.txt @@ -0,0 +1,1124 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf,len=1123 +page_content='MOTION-BASED POST-PROCESSING: USING KALMAN FILTER TO EXCLUDE SIMILAR TARGETS IN UNDERWATER OBJECT TRACKING A PREPRINT YunFeng Li Harbin Engineering University liyunfeng@hrbeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='cn Bo Wang Harbin Engineering University cv_heu@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='com Ye Li Harbin Engineering University liye@hrbeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='cn Wei Huo Harbin Engineering University weihuo@hrbeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='cn ZhuoYan Liu Harbin Engineering University liuzhuoyan@hrbeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='cn January 5, 2023 ABSTRACT Visual tracker includes network and post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Despite the color distortion and low contrast of underwater images, advanced trackers can still be very competitive in underwater object tracking because deep learning empowers the networks to discriminate the appearance features of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' However, underwater object tracking also faces another problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Underwater targets such as fish and dolphins, usually appear in groups, and creatures of the same species usually have similar expressions of appearance features, so it is challenging to distinguish the weak differences characteristics only by the network itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The existing detection-based post-processing only reflects the results of single frame detection, but cannot locate real targets among similar targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In this paper, we propose a new post-processing strategy based on motion, which uses Kalman filter (KF) to maintain the motion information of the target and exclude similar targets around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Specifically, we use the KF predicted box and the candidate boxes in the response map and their confidence to calculate the candidate location score to find the real target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our method does not change the network structure, nor does it perform additional training for the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' It can be quickly applied to other tracking fields with similar target problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We improved SOTA trackers based on our method, and proved the effectiveness of our method on UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The AUC of our method for OSTrack on similar subsequences is improved by more than 3% on average, and the precision and normalization precision are improved by more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='5% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' It has been proved that our method has good compatibility in dealing with similar target problems and can enhance performance of the tracker together with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' More details can be found in: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='com/LiYunfengLYF/KF_in_underwater_trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Keywords Underwater Object Tracking · Similar Object Tracking · Post-processing strategy · Kalman Filter 1 Introduction The goal of Single Object Tracking (SOT) is to find the position and scale of the target in each frame by using a Template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='The single object tracker is designed to track all kinds, but only one target box is output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In recent years, the performance of trackers has been greatly improved with the powerful feature expression ability of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Though single object trackers support all kinds of target tracking, the existing large object tracking dataset benchmarks such as LaSOT[1], GOT10k[2], TrackingNet[3] mainly focus on target tracking in open-air scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As a branch of SOT, Underwater Object Tracking (UOT) has to deal with the unique challenges of underwater environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the one hand, underwater images usually have problems such as color cast, low contrast, and low visibility, which affect the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='01482v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='CV] 4 Jan 2023 arXiv Template A PREPRINT performance of the open-air tracker in underwater tracking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the other hand, in the tasks of marine biological observation and behavior analysis, organisms such as fish and dolphins often appear in groups, and it is difficult to distinguish similar marine organisms from each other in terms of appearance characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Therefore, it is challenging for open-air trackers to identify similar underwater targets of the same kind based on their appearance characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Before the presentation of the UOT100 dataset[4], due to the lack of unified underwater object tracking dataset benchmark, people mainly verified the effectiveness of their trackers by selecting some video or image sequences that reflect the challenges of underwater tracking tasks on Fish4knowledge (F4K)[5] or Underwater Change Detection (UWCD)[6] or other self -built datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' After UOT100 and UTB180[7] were proposed, the researchers pay more attention to the development of high performance underwater trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The advanced open-air tracker with powerful feature representation capability can still maintain competitive performance when dealing with underwater image distortion, but are prone to tracking drift when dealing with multiple similar marine organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' At present, the main paradigm of the tracker is to extract features and fuse features (or jointly extract and fuse features) from the template image and the search area through neural network, and then use the prediction head to decode the fused features to obtain the bounding box of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The prediction head is mainly divided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' One type is the end-to-end direct output of bounding boxes (such as CornerHead), the other type locates the target through the response map and provides candidate boxes (such as CenterHead) or directly outputs candidate boxes and their scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Then the maximum response value is used to locate the target, which can also be called the detection-based post-processing paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The response map contains the target prediction boxes in the search area and their confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' However, the difference between the response values of two objects with similar appearance is very small, and the response value of the real target may be lower than the response value of similar objects with illumination changes, motion blur or occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Therefore, it is difficult to accurately locate the real target only using detection-based post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The tracker is composed of neural network and post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' When the neural network and its detection-based post processing cannot effectively distinguish the nuances of similar objects, we rethink how the visual tracking task itself should distinguish different similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We find that similar objects can be directly distinguished by their positions in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the image sequence, similar objects can be located by their own motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Therefore, a natural idea is to record the motion information of the target itself so as to exclude similar targets around through the movement and positioning of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We have noticed that the Tracking by Detection (TBD) paradigm in Multi Object Tracking (MOT) has provided a mature method, that is, maintaining the state and motion information of the target through Kalman filter, and constantly providing position constraints based on motion information in tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Therefore, we propose a simple and efficient motion-based post-processing paradigm to deal with similar problems in underwater object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We maintain a Kalman filter during the tracking process of the single target tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' When similar targets of the same kind appear in the search area, the prediction box of the Kalman filter is used as the target state constraint to match all candidate boxes of the tracker and remove the bounding boxes of potential similar targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our main work includes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We rethink how the underwater visual tracking itself should distinguish similar targets, and design a simple, effective, easy to expand, and easy to migrate motion based post-processing strategy with reference to the method of motion state maintenance in MOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' This strategy uses Kalman filter to maintain, update, and predict target motion information, and reuse candidate tracking boxes to eliminate interference from similar targets of the same kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We verified the effectiveness of our method on different trackers and comprehensively compared the performance of top-level Open-air trackers on UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our strategy was compared with other tracker enhancement strategies, which proves that our method is suitable for dealing with underwater similar object tracking and our method has good compatibility to work together with other enhancement methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We analyzed which aspect of our method has improved the performance of the tracker, and explained the shortcomings of our strategy and the existing trackers when dealing with similar target problems of the same kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='1 Single Object Tracking Single object trackers contain traditional methods, correlation filtering methods, Siamese-based methods, Transformer- based methods and Online-discrimination methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Traditional tracking methods include mean shift, particle filter, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Classical correlation filtering methods mainly include MOSSE[8], KCF[9], DSST[10], ECO[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Although their tracking accuracy and success rate are far less than the current trackers, these methods are still active in some specific tracking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 2 arXiv Template A PREPRINT Siamese-based trackers use the powerful representation ability of depth features to improve the tracking performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As the first Siamese-based tracker, SiamFC[12] extracts features from Template and Search area using AlexNet and computes their similarity to locate target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Inspired by the Anchor-based thought, SiamRPN[13] replaces the prediction head with an RPN network to fuse features to make the tracker perform faster and more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Then, deeper networks such as Resnet50[14], Mask branch[15] and so on are added to improve SiamRPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Later, inspired by the idea of Anchor-Free, trackers such as SiamBAN[16], Ocean[17] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, adding online branches[17] and attention mechanisms[18] are also used to improve the Siamese based trackers, which has produced great results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Transformer-based trackers have powerful global receiving field and global information integration capability, which is the weakness of Siamese-based trackers, further improving the performance of tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Stark[19]uses Transformer to fuse features and CornerHead to predict the bounding box end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' TransT[20] uses self-attention module and cross-attention module to construct Transformer-like network to fuse features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' MixFormer[21] uses the hybrid module of convolution and self-attention to extract and fuse features efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' AiAtrack[22] improves feature expression ability by introducing self-attention into the self-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' OSTrack[23] uses VIT to jointly extract and fuse template features and search area features, greatly improving the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Online discriminant-based trackers not only improve the expression ability of object appearance features through neural networks, but also improve the adaptability to appearance changes through self-updating of the object model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' ATOM[24] uses the classification and estimation module to classify the foreground and background and estimate the bounding box of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Dimp[25] introduces the discrimination of template background information and improves the identification ability of the tracker through online update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' KeepTrack[26] keeps track of all potential targets by building the Target Candidate Extraction Network and Association Network to distinguish between targets and distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' TOMP[27] improves the feature expression ability in DCF paradigm and the performance of the online tracker through the Transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='2 Underwater Object Tracking The single object tracker does not distinguish between categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' It is a natural idea to introduce the open-air tracker into underwater tracking tasks and improve it based on the characteristics of underwater tracking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Considering that the underwater tracker usually needs to be carried in the Autonomous Unmanned Vehicle (AUV) or Remote Operated Vehicle (ROV) to perform the actual ocean observation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' However, AUV usually has limited energy and computing power, so, fast, efficient and low-power correlation filtering methods play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The KCF algorithm through the adaptive appearance model and its tracking strategy[28], feature fusion and scale correction mechanism[29] to deal with the challenges of underwater tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' BACF is used to obtain more adaptive features, and combine scale estimation with the confidence -based update strategy to improve the performance of the tracker[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, TLD-based[31]trackers and Particle-Based[32][33] trackers are still used to track underwater targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' ROV can usually provide more energy and more powerful computing power, so it is feasible to use deep learning trackers on ROV to obtain more robust tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The reverse residual bottleneck block is added to SiamRPN++ to enhance the feature expression ability of the tracker to meet the challenge of underwater image degradation[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' UStark[35] uses an image adaptive enhancement head to predict a set of enhancement parameters, and uses an enhancement module to process input images to improve the performance of the Stark tracker under different underwater image distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3 Methods Our approach is based on three assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We think that the advanced features extracted by the neural network of the tracker are the combination of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Similar targets usually have similar response values, which means similar targets have similar values in the score graph generated by the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We believe that after training, the neural network has a strong ability to distinguish target features, which means that the neural network has enough assurance to confirm that the location and scale estimation of each generated high-confidence target are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We believe that tracking is a continuous process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The location and scale of the target in the past frame will affect the state of the target in the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3 arXiv Template A PREPRINT Figure 1: Visualization of our method in tracking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The Network input the Template and Search area and output the Response map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the Post processing, we first follow the detection-based post processing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' When encountering similar target interference and tracking drift, we design a new motion-based post processing method to correct tracking drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We did not make any changes to the tracker’s network itself, nor did we perform additional training on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='This means that our work can be quickly applied to other tracking fields with similar target problems We only reuse the state information of the target in the past frame and the candidate target box information of the tracker, which means that we can find the real target among the candidate boxes using the motion constraint brought by Kalman filter when tracking drift occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='1 Motion information estimation by using Kalman Filter Following the estimation model of the SORT(Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=', 2016) tracker, we set the target status as: x = [u, v, s, r, u′, v′, s′] (1) where u and v represent the horizontal and vertical positions of the target center respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' s represents the area of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the tracking process, we use Kalman filter to maintain the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The output bounding box in each frame is used to update the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the next frame, the Kalman filter is used to predict the state of the target and generate a prediction box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='2 Location score calculation We calculate the motion-based location score of candidate boxes and select the bounding box with the maximum value as the most appropriate candidate box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The location score is shown as: location_score = conf × IOU(pred_box, candidate_boxes) (2) where conf represents the response value of each candidate box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' pred_box represents the prediction box of Kalman filter in the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' candidate_boxes represents the response boxes whose response scores rank above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='3 Motion-based location paradigm Inspired by the target matching strategy in MOT, We designed a motion-based candidate boxes location strategy for the single target tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Specifically, the tracker initializes an additional Kalman filter in the initialization process to record, update and predict the motion information of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the tracking process, the tracker predicts a tracking box based on the maximum response value of the response map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Then the Kalman filter is used to predict the target position in this frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' If the IOU of the Kalman filter prediction box and the maximum-response box is greater than a threshold value, we consider this to be a correct tracking and process the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the contrary, we think that the target has drifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We extract other high response value points in the response map and obtain candidate boxes with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' After NMS, we calculate the similarity scores of candidate boxes, select the bounding box corresponding to the maximum value, and output it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our pseudo code is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4 Tracker 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Post processing Correct successfully Response map Template 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Network 个 Red, detection-based Blue, motion-based (ours) lou>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='6 max response box response map xoq indino Search area KF predict box kalman filte Calculate location max-value box scores NMS iV candidate boxes Select from the candidate boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='. / Ground Trurh Candidate Boxes Max Response Box Kalman Filter Predict-box Output BoxarXiv Template A PREPRINT Table 1: PyTorch pseudo code for our method PyTorch Pseudo Code # self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='kf_num = 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='nms_thre = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='conf=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='6 If self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='use_kf: kf_pre_box = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='kalmanfilter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='predict() if box_iou(kf_pre_box, max_response_box) > self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='conf: output_box = max_response_box else: pred_boxes, conf = decode_muti_boxes(output, self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='kf_num) pred_boxes, conf = NMS(pred_boxes, response, self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='nms_thre) location_score = box_iou(kf_pre_box, pred_boxes) location_score = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='mul(location_score, conf) index = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='argmax(location_score) output_box = pred_boxes[index] # update kalmanfilter If self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='use_kf: self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='kalmanfilter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='update(output_box) Table 2: The use of information by different post-processing paradigms Single Frame Information Image Sequence Information paradigm Max response value Candidate information Candidate information Other information Detection-based ✓ Motion-based (ours) ✓ ✓ ✓ Please note that we do not directly use the box predicted by the Kalman filter as the output box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The reason is that the box predicted by the Kalman filter has certain inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We only use the prediction box as a constraint to select one of the candidate boxes generated by the tracker that best matches the position where the current target should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Kalman filter can be used to establish the correlation between frames of the target to realize the interaction of position information between frames and make the tracker know where its target is likely to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='4 Comparison with detection-based paradigm The detection-based post-processing paradigm selects the bounding box of the maximum response value in the search area of each frame to locate the target through the powerful discriminant ability of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The detection of each frame is independent of each other, and it does not pay attention to the information of non- maximum response value points in the response value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our motion-based post-processing paradigm not only uses the maximum response value to locate the target, but also uses the motion information of the target in the sequence to constrain the position of the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' When the bounding box drifts, the candidate frame information is reused to retrieve the tracked target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The degree of information utilization is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The illustration of utilization of target past status information is in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The illustration of utilization of candidate box information is in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4 Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='1 Implementation Details We replaced Stark’s CornerHead with CenterHead and transferred the remaining parameters, and then trained 50 epochs according to the paradigm of OSTrack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Then we set kf_num to 40 on the OSTrack, Starks, Starkst on the UOT100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Set kf_num to 30 on the OSTrack on the UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Set kf_num to 40 on the Starks and Starkst on the UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Set nms_thre to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='8 and conf to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='6 on each tracker and both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Starkst update interval is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The selected response map is the original response map without hanning window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the TransT, we set kf_num to 20, nms_thre to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='8, conf 5 arXiv Template A PREPRINT Figure 2: The original tracker only uses the maximum response value of each frame to locate the target, so its track is discrete in the tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Although significant errors occur in the trajectory, the detection-based paradigm does not have the function of trajectory self-correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our method uses the target state information in past frames to effectively constrain the tracker’s tracking position in each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Even if there is a slight tracking drift, it can still be effectively corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='6 and the selected response map has been multiplied by the Hanning window on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Other trackers evaluated on UOT100 and UTB180 were set according to their original parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The device we used is CPU: i9-12900KF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' GPU: GTX-3090Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The version of Pytorch we used is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='2 Dataset and Evaluation indicators UOT100 dataset benchmark is the first typical Underwater Object Tracking dataset, which is mainly used to reflect the challenges faced by underwater object tracking, such as image degradation, similar targets, large deformation and other issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In order to verify the effectiveness of our method in dealing with similar target challenges, we extract 28 subsets of UOT 100 with similar target interference challenges as similar subsets of UOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' [See appendix for details of division].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' At the same time, we use the remaining 78 sequences as the complement of similar subsets to verify whether our strategy will reduce the performance of the tracker on non-similar problem sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' UTB180 dataset benchmark is a high-quality Underwater Object Tracking dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' It reflects challenges in underwater object tracking such as Unclear Water, Occlusion, Similar Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We also use its 116 similar object subsets to prove that our method is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The other 64 non-similar object subsets are used to explore the performance of our method on non-similar problem sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We follow the One Pass Evaluation (OPE) protocol most commonly used in single object tracking to evaluate the tracker, and the AUC, precision, and norm-precision are used to evaluate the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We follow the One Pass Evaluation (OPE) protocol most commonly used in single object tracking to evaluate the tracker, and the AUC, precision, and norm-precision are used to evaluate the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='3 Effectiveness of our method We select SOTA performance trackers such as OSTrack[23], Stark[19] and TransT[20] to verify the effectiveness of our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Since the Corner-Head of the original Stark only outputs a box end-to-end, we replace CornerHead with CenterHead to obtain multiple candidate boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 6 #frame:222 Center point position at intervals of five frames Center point position at intervals of five frames after 222 frames Trajectory after frame 222 Trajectory before frame 222 OSTrack+ours Ground Truth Original OSTrackarXiv Template A PREPRINT Table 3: Boost of our method on different trackers in UOT100 Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='40 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='48 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='19 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='51 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='62 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='00 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='05 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='49 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='39 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='70 Original 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='55 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='65 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='10 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='69 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='39 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='46 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='65 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='75 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='27 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='85 Boost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='85 Table 4: Boost of our method on different trackers in UTB180 Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='13 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='99 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='99 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='61 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='57 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='64 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='67 Original 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='03 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='52 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='61 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='72 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='07 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='09 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='90 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='79 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='47 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='52 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='30 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='13 Boost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='78 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='54 The results are shown in the Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On UOT100 and UTB180, our method can effectively improve the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Besides, Our method only needs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='008s to filter candidate boxes once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Considering that our method is only used when the IOU of the Kalman filter prediction box and the maximum response box is less than the threshold, the average consumption time will be less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' To further prove whether our method can effectively improve the discriminability of the tracker for similar objects around the target, we verified the effectiveness of the method in the similar object subsets of UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As shown in Table 5 and Table 6, our strategy can effectively improve the performance of the tracker when dealing with similar target interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We noticed that Starks has shown significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The AUC has improved by more than 6% on average and the accuracy has improved by more than 9% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' This is because Stark extracts the semantic features of the target through backbones and sends them to the Transformer for fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' However, similar semantic features alone cannot distinguish between targets and similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' For example, a ball can represent a basketball or a tennis ball, so Starks is more likely to be disturbed by similar semantics and tracking drift, which is why our method has the highest performance improvement for Starks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Starkst introduces a second variable template to bring online updating adaptability to the tracker, which can enhance the appearance discrimination of the tracker for similar targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' But our method still has gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In OStrack, our method brings 3% performance improvement on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In TransT, our method also improves the performance on MLP-based prediction head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In order to verify whether our method will affect the general underwater tracking, we also tested the performance of four trackers on the complements of similar sequences in UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As shown in Table 7 and Table 8, Starks, Starkst and TransT have shown good adaptability in tracking performance of non-similar problem sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In contrast, in the UTB100 non-similar problem sequences, the performance of OSTrack decreases by more than 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Since the performance is still good after the degradation, we did not further explore the cause of the performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' However, the performance degradation on non-similar sequences will limit the promotion of our method on the Open-air tracking dataset (which is also what we hope to solve in the next stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='4 Comparison of our method with SOTA trackers We evaluated 14 SOTA trackers (NeighborTrack (represented by OSTrack-N)[36], OSTrack[23], MixFormer[21], AiATrack[22], ToMP[27], KeepTrack[26], Stark[19], TransT[20], TrDimp[37], SiamBAN-ACM[38], Dimp[25], Table 5: Boost of our method on different trackers in UOT100’s similarity subset Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='27 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='57 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='65 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='70 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} 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70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='15 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='12 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='02 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='74 Boost 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='98 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='81 7 arXiv Template A PREPRINT Table 6: Boost of our method on different trackers in UTB180 similarity subset Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='70 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='31 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='80 51.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='43 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='34 Table 7: Boost of our method on different trackers in complements of similar sequences in UOT100 Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 70.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='98 SiamBAN[16], ATOM[24], SiamCAR[39]) on UOT100 and UTB180 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We compared the results with our method results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The results of comparison of our method (applied to OSTrack256) and SOTA performance trackers on the UOT100 and UTB180 are shown in Table 9 and Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our method (applied to OSTrack256) achieved SOTA performance on the UOT100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, we found that the Open-air SOTA performance tracker still has a very good performance in terms of precision and regularization precision on the sequence of non-similar problems of UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' This means that in underwater target tracking, the improvement of strong feature representation ability is not the most urgent task to improve the performance of the underwater tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We need to further consider the performance improvement in edge cases (such as similar targets, occulsion, unclear water, out of view and so on) during tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='5 Comparison of our method with other improvement strategies We compared our method with two other classical tracker enhancement methods (NeighborTrack (Represented by OSTrack-N)[36], Alpha-Refine[40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Specifically, we applied these enhancement methods to OSTrack and evaluate the performance of the tracker on the UOT100 and UTB180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The NeighborTrack is designed to deal with occlusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The UOT100 dataset does not focus on requiring the tracker to deal with occlusion challenge, so the NeighborTrack does not perform very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In the UTB180, there are 92 sequences reflecting occlusion problem, which improves the performance of NeighborTrack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, we found that our method is not in conflict with the NeighborTrack method, and they can jointly enhance the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Besides, our method can also be combined with the Alpha-Refine module to jointly improve the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In conclusion, our method has good compatibility, and can be inserted into the tracker to improve the tracker performance together with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Table 8: Boost of our method on different trackers in complements of similar sequences in UTB180 Tracker OSTrack256 Starks+CenterHead+ep50 Starkst+CenterHead+ep50 TransT method AUC P P-Norm AUC P P-Norm AUC P P-Norm AUC P P-Norm Ours 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='79 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='46 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='35 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='47 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='19 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='84 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='17 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='57 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='32 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 Original 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='97 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='83 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='94 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='68 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='91 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='12 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='81 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='37 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='20 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='79 Boost 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='10 8 arXiv Template A PREPRINT Table 9: Comparisons of our method (applied to OSTrack256) with SOTA performance trackers on the UOT100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The best two results are shown in red and blue fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' UOT100(106) UOT100’s Similary Subset (28) Complements of Similar Subsets (78) Tracker Source AUC P P-Norm AUC P P-Norm AUC P P-Norm ATOM CVPR2019 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='79 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='24 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='76 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='63 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='42 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='84 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='48 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='03 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='53 Dimp50 ICCV2019 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='90 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='39 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='59 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='62 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='80 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='42 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='91 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='38 SiamCAR CVPR2020 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='55 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='96 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='40 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='14 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='78 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='32 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='44 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='06 SiamBAN CVPR2020 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='72 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='71 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='41 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='76 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='56 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='64 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 SiamBAN-ACM CVPR2021 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='43 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='01 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='84 48.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='61 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='33 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='66 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='70 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='45 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='00 5 Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='1 What does our method actually do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We visualized some of the tracking results for OSTrack and Stark+CenterHead, as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our method mainly helps the tracker to suppress tracking drift under similar interference of the same kind, thus improving the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' To further demonstrate our method can suppress tracking frame drift, we show some pictures with the maximum response frame different from the final output frame and the candidate frame information of the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As shown in Figure 4, in the tracking process, although the tracker will generate a large number of meaningless candidate boxes, each target like a template in the search area can still be accurately located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our method focuses on how to find the accurate target location among the candidate frames when the maximum response frame drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In typical cases, the Kalman filter prediction frame is usually close to the target truth box, so the most appropriate candidate frame can be selected by calculating the IOU and response score of the prediction frame and candidate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' At the same time, the screened candidate box can also maximize the use of the powerful discriminant ability of neural network to obtain an accurate target location and scale estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 9 arXiv Template A PREPRINT Table 11: Comparisons of our method with other improvement strategies on the UOT100 UOT100 (106) UOT100’s Similary Subset (28) Complements of Similar Subsets (78) Tracker AUC P P-Norm AUC P P-Norm AUC P P-Norm OSTrack 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='55 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='81 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='33 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='08 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='49 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='23 OSTrack+ours 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='40 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='48 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='27 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='57 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='65 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='38 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='92 OSTrack+N 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='32 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='56 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='23 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='25 73.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='65 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='86 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='85 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='40 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='09 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 OSTrack+AR 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='32 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='87 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='90 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='53 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='49 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='29 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='62 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='09 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='97 OSTrack+ours+AR 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='36 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='93 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='08 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='05 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='18 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='59 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='82 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 Table 12: Comparisons of our method with other improvement strategies on the UTB180 UTB180 (180) UTB Similary Subset (116) Complements of Similar Subsets (64) Tracker AUC P P-Norm AUC P P-Norm AUC P P-Norm OSTrack+ours 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='13 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='99 73.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='79 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='42 OSTrack+ours+N 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='33 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='29 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='89 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='61 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='33 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='66 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='70 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='45 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='00 OSTrack+AR 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='88 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='60 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='73 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='74 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='05 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='83 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='99 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='83 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='05 OSTrack+ours+AR 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='55 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='42 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='32 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='96 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='07 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='07 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='67 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='31 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='2 Lack of instance-level discrimination for close similar targets In the experiment, we found that most target trackers lack the instance-level discrimination ability for similar targets of the same kind that are close to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' As shown in Figure 5, when two similar targets approach, the tracker will regard them as a larger target and predict a large box that covers both targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' This is because the tracker mainly obtains the final bounding box by matching the features in the template with the features in the search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Each feature matched successfully is considered to be part of the unique output box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' They are combined to form a large bounding box containing multiple similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the one hand, this bounding box will reduce the performance of the tracker and may lead to tracking drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' On the other hand, it will contaminate the Kalman filter in our method and affect the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We find that in Object Detection, the training data of the detector contains a large number of similar targets and their labels, which can naturally distinguish target instances after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' At present, the paradigm of tracker post-processing is close to that of detector post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Therefore, can we construct a training set by using a Mosaic-like method or using an unsupervised training paradigm to make the tracker also have the instance-level discrimination ability for similar targets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Figure 3: Visualization comparison examples of our method results on different trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 10 #frame: 1 +frame:516 JerkbaitBites #frame: 468 e: 470 #frame: 960 fram #fram e: 518 le: 623 MonsterCreature2 #frame:404 #frame: #frame:135 #f WhiteShark Ground Truth OSTrack OsTrack+ours Starks+CenterHead Starks+CenterHead+oursarXiv Template A PREPRINT Figure 4: Example of our method self-correcting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Figure 5: Examples of trackers lacking instance-level discrimination ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='3 Disadvantages of our method Because our method only considers the normal form of using Kalman filter to eliminate potential interference targets in the tracking process, and lacks the processing of edge conditions, our method is actually vulnerable to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' For example, when the Kalman filter uses an inaccurate tracking bounding box to update the state, it will continuously accumulate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Finally, the contaminated Kalman filter will screen out the wrong candidate box, reducing the performance of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In fact, we believe that when using the underwater tracker on an underwater vehicle, various edge conditions need to be considered to obtain the performance improvement of the tracker, such as considering the low response value of the target due to occlusion or out-of-view, the high confidence update scheme for template update (if available ) and target state update (such as our method) and their correction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Moreover, tracking is usually continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In a general tracking process, the IOU between the two frames is usually greater than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, the tracker does not only perform tasks on the underwater robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Generally, the vehicle may carry Object Detection algorithms as detectors, matching algorithms in MOT, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' The effective use of external information to improve the performance of the tracker is also a meaningful issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In general, we believe that the design of an underwater tracker with multiple strategies is helpful to improve the operational efficiency of underwater vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' And we hope our method can provide a reference for the design of underwater trackers to address similar object challenges of the same kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 6 Conclusion In this paper, considering the problem of marine organism swarming in underwater target tracking, a simple motion- based post-processing strategy using Kalman filter to eliminate similar interference near the target is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='JerkbaitBites ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='me: 841 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='e: 832 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='MonsterCreature2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame:138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame: 304 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='122 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='147 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='+frame: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='MississippiFish ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame: 53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='rame: 111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='PinkFish ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Ground Truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Candidate boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Max response box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Kalman Filter predict-box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Output box#frame:137 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame:147 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame:162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame:180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='#frame:290 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='ColourChangingSquid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='CressiGuillaumeNeri2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='ram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Rocketman ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Stark-ST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Ground Truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='OSTrack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='MixFormer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='AiaTrack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='TOMParXiv Template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='A PREPRINT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='Table 13: Sequence names of UOT100 similar subsets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='UOT100 similarity subsets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content='ArmyDiver1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' WhiteShark strategy reuses the information of the candidate target boxes and their response values in the tracker response graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' without additional training for the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' We have proved the effectiveness of our method in dealing with similar target challenges on multiple trackers, and explained why our method is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' In addition, we also evaluate the performance of the open-air tracker in underwater target tracking in the past three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our strategy combined with the OSTrack tracker has SOTA performance in underwater object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Our strategy is also compared with other tracker strategies, and it is proved that our method and other methods such as NeighborTrack can jointly improve the tracker performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' Finally, we analyze the shortcomings of existing trackers in dealing with similar problems of underwater targets and the inadequacy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfhPyg/content/2301.01482v1.pdf'} +page_content=' 7 Appendix More details can be found in Table 13.' 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(2023) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +V994 Her: A Unique Triply Eclipsing Sextuple Star System +P. Zasche1 ⋆, T. Borkovits2,3,4,5,6, R. Jayaraman7 , S. A. Rappaport7, M. Brož1, +D. Vokrouhlický1, I. B. Bíró2,3, T. Hegedüs2, Z. T. Kiss2, R. Uhlař8, +H. M. Schwengeler9, A. Pál4, M. Mašek10, S. B. Howell11, S. Dallaporta12, +U. Munari13, R. Gagliano14, T. Jacobs15, M. H. Kristiansen16,17, D. LaCourse18, +M. Omohundro19, I. Terentev20, A. Vanderburg21, Z. Henzl22,23, B. P. Powell24, +V. B. Kostov24,25,26 +1 Charles University, Faculty of Mathematics and Physics, Astronomical Institute, V Holešovičkách 2, Praha 8, +180 00, Czech Republic +2 Baja Astronomical Observatory of Szeged University, H-6500 Baja, Szegedi út, Kt. 766, Hungary +3 ELKH-SZTE Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út, Kt. 766, Hungary +4 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, +H-1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary +5 ELTE Gothard Astrophysical Observatory, H-9700 Szombathely, Szent Imre h. u. 112, Hungary +6 MTA-ELTE Exoplanet Research Group, H-9700 Szombathely, Szent Imre h. u. 112, Hungary +7 MIT Department of Physics and and MIT Kavli Institute for Astrophysics and Space Research, Cambridge, MA 02139, USA +8 Private Observatory, Pohoří 71, 254 01 Jílové u Prahy, Czech Republic +9 Citizen Scientist, Planet Hunter, Bottmingen, Switzerland +10 FZU - Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, CZ-182 21, Praha, Czech Republic +11 NASA Ames Research Center, Moffett Field, CA 94035, USA +12 ANS Collaboration, c/o Astronomical Observatory, 36012 Asiago (VI), Italy +13 INAF Astronomical Observatory of Padova, 36012 Asiago (VI), Italy +14 Amateur Astronomer, Glendale, AZ 85308 +15 Amateur Astronomer, 12812 SE 69th Place Bellevue, WA 98006, USA +16 3DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Lyngby, Denmark +17 Brorfelde Observatory, Observator Gyldenkernes Vej 7, DK-4340 Tżllżse, Denmark +18 Amateur Astronomer, 7507 52nd Place NE Marysville, WA 98270, USA +19 Citizen Scientist, c/o Zooniverse, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, +Oxford, OX1 3RH, UK +20 Citizen Scientist, Planet Hunter, Petrozavodsk, Russia +21 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA +22 Hvězdárna Jaroslava Trnky ve Slaném, Nosačická 1713, Slaný 1, 274 01, Czech Republic +23 Variable Star and Exoplanet Section, Czech Astronomical Society, Fričova 298, 251 65 Ondřejov, Czech Republic +24 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA +25 SETI Institute, 189 Bernardo Ave, Suite 200, Mountain View, CA 94043, USA +26 GSFC Sellers Exoplanet Environments Collaboration +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We report the discovery with TESS of a third set of eclipses from V994 Herculis (TIC 424508303), previously only +known as a doubly-eclipsing system. The key implication of this discovery and our analyses is that V994 Her is the +second fully-characterized (2+2) + 2 sextuple system, in which all three binaries eclipse. In this work, we use a +combination of ground-based observations and TESS data to analyze the eclipses of binaries A and B in order to +update the parameters of the inner quadruple’s orbit (with a derived period of 1062 ± 2 d). The eclipses of binary C +that were detected in the TESS data were also found in older ground-based observations, as well as in more recently +obtained observations. The eclipse timing variations of all three pairs were studied in order to detect the mutual +perturbations of their constituent stars, as well as those of the inner pairs in the (2+2) core. At the longest periods +they arise from apsidal motion, which may help constraining parameters of the component stars’ internal structure. +We also discuss the relative proximity of the periods of binaries A and B to a 3:2 mean motion resonance. This +work represents a step forward in the development of techniques to better understand and characterize multiple star +systems, especially those with multiple eclipsing components. +Key words: binaries: eclipsing – binaries: close – stars: individual: (TIC 424508303, V994 Her), sextuple system +⋆ E-mail: zasche@sirrah.troja.mff.cuni.cz +© 2023 The Authors +arXiv:2301.13521v1 [astro-ph.SR] 31 Jan 2023 + +2 +Zasche et al. +1 INTRODUCTION +Multiple star systems consisting of three or more stars are +estimated to make up at least 30% of binaries, based on a +statistical analysis of Kepler data in Borkovits et al. (2016). +However, only two thousand have been observed in detail, +and the number of systems known with multiplicities higher +than 5 is ≲ 50 (Tokovinin 2018). Understanding these high- +multiplicity systems is important, as they can shed new light +on many of the open questions that remain in currently- +accepted models of stellar formation and provide insight +into the dynamical interactions of multiple stars (Aarseth & +Mardling 2001). +V994 Herculis (V994 Her, TIC 424508303) is a bright, well- +studied quadruple system, with two known eclipsing binary +components (Lee et al. 2008). At the time, this was the first +known doubly-eclipsing quadruple system. The component +stars are bright and massive, and the periods of the two eclips- +ing binary components have been well-constrained: binary A +consists of a B8V and an A0V star, with a period of 2.083 +days; binary B consists of an A2V and an A4V star, with +a period of 1.420 days. These stars are young and occupy +a position near the zero-age main sequence (ZAMS) on the +Hertzsprung-Russell (H-R) diagram. +Martín-Ruiz et al. (2013) initially postulated, based on +analyses of photometric data, that this system could har- +bor another eclipsing binary. However, because the quality +of their data was rather poor, with relatively low photomet- +ric precision, their results were not conclusive enough. With +the advent of high-precision space-based survey missions such +as the Transiting Exoplanet Survey Satellite (TESS; Ricker +et al. 2015), we are able to conclusively confirm the presence +of a third set of eclipses using the TESS light curves. Table +1 contains basic information about V994 Her and a nearby +visual companion (separated by ≈ 1′′). +Zasche & Uhlař (2016) were the first to accurately con- +strain the period of the inner binaries’ (A and B) revolution +about their common center of mass (≃ 1060 days in their +study). Additionally, they argued that TIC 1685970000, a +faint (mV ∼ 8.8) neighbor some 1.1′′ away from V994 Her, is +also gravitationally bound to the main quadruple, making it +one of the few known quintuple star systems. This putative +close companion has been observed many times since its dis- +covery as a visual double in 1831, and these measurements +have been catalogued in the Washington Double Star Catalog +(WDS; Mason et al. 2001). The WDS calls the known quadru- +ple a “primary star,” and the fainter companion a “secondary +star.” However, any physical connection between these two +visually-close objects has not yet been conclusively proven; +further follow-up and analyses (such as those in Section 5.1) +can resolve this question. +In this paper, we introduce V994 Her as a bona fide triply +eclipsing sextuple star system, which we identified using +TESS data. In addition to the known set of two binaries, +the system also consists of a third binary of period 1.96 days, +and we demonstrate that the visual companion listed in the +WDS catalog is likely gravitationally bound to the primary +star. In Section 2, we describe all available observational data +and how they were prepared and used for the analysis. Then, +Section 3 provides detailed modelling of the available data, +while 4 discusses the results of our modeling. Finally, in Sec- +tion 5 we discuss the possible architecture of the whole system +Table 1. Archival properties of the V994 Her visual double star +Name +V994 Her +TYC 2110-1170-2 +TIC 424508303 +TIC 1685970000 +RA (J2000, deg) +276.941222 +276.941246 +Dec (J2000, deg) +24.697407 +24.697757 +TESS a +7.037 ± 0.017 +8.3949 ± 0.6 +Ba +7.136 ± 0.024 +V a +6.9599 ± 0.023 +Ja +6.948 ± 0.019 +Ha +6.999 ± 0.0036 +Ka +6.989 ± 0.023 +W1b +6.844 ± 0.07 +W2b +6.838 ± 0.02 +W3b +6.903 ± 0.018 +W4b +6.732 ± 0.067 +Gc +7.0966 +8.8761 +Gc +Bp +6.9898 +6.9842 +Gc +Rp +6.9358 +6.9869 +Parallaxc (mas) +3.43639 ± 0.08394 +3.48065 ± 0.09849 +PMc (RA, mas/yr) +5.5770 ± 0.0701 +4.8802 ± 0.1275 +PMc (Dec, mas/yr) +11.2675 ± 0.0765 +7.1568 ± 0.0834 +Notes: Magnitudes are from (a) TIC-8 catalog (Stassun et al. +2019). (b) WISE point source catalog (Cutri et al. 2021), (c) +Gaia DR3; PM stands for proper motion (Gaia Collaboration +et al. 2022). Some parameters for the visual companion TIC +1685970000 are difficult to come by, as the brighter primary star +is 1.1” away from it, making measurements difficult. +that we infer from our findings and comment on the proxim- +ity of the inner 2+2 component (binaries A and B) to their +mutual 3:2 mean motion resonance. +2 OBSERVATIONS OF V994 HER +2.1 TESS Observations +V994 Her was observed by TESS during Year 2 in Sector +26 (i.e., June 2020), and during Year 4 in Sectors 40 and +53 (i.e. July 2021 & June 2022). In Sector 26, this star was +observed at 2-minute cadence; this light curve was prepro- +cessed and detrended by the Science Processing Operations +Center (SPOC) pipeline (Jenkins et al. 2016), which is par- +tially based on that used for Kepler data. The detrended +SPOC light curve from Sector 26 is shown in Figure 1. For +the Year 4 observations, however, only the full-frame image +(FFI) data (at 10-minute cadence) are available. These data +were processed using the convolution-based differential im- +age analysis methods of the fitsh package (Pál 2012). V994 +Her’s triply eclipsing nature was identified both algorithmi- +cally and through a visual survey1 of all stars brighter than +13.5 mag in the TESS FFIs (for more information on the +latter initiative, see Kristiansen et al. 2022). +2.1.1 Three methods for disentangling +Using the TESS data, we applied three different methods to +disentangle the combined light curve into the three compo- +nent eclipsing signals: the time-domain iterative disentangle- +1 This search makes use of the LcTools desktop application +(Schmitt et al. 2019) to view and study light curves. +MNRAS 000, 1–?? (2023) + +V994 Herculis: A Triply Eclipsing Sextuple +3 +2010 +2014 +2018 +2022 +2026 +2030 +2034 +Time (BTJD) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux +2012 +2014 +2016 +2018 +2020 +0.93 +0.96 +0.99 +1.02 +1.05 +Secondary Eclipse (C) +Primary Eclipse (C) +Figure 1. The TESS Sector 26 light curve of TIC 424508303, aka V994 Her. The x-axis is plotted in Barycentric TESS Julian Day +(BTJD), which corresponds to BJD–2457000.0. The main plot shows the full 25-day light curve, which includes multiple eclipses from +the previously known eclipsing binaries A and B (Lee et al. 2008; Zasche & Uhlař 2016). It also contains relatively shallow eclipses from +the new binary C, discussed in this work. The inset panel shows a zoom-in on a roughly 9-d segment of the data. Three clearly visible +primary eclipses of the C binary are overplotted in blue, while the eclipse lost in a deeper eclipse from the A binary is indicated with a +blue line above its expected location. The (shallow) secondary eclipses of the C binary are overplotted in gold, with the gold line at BTJD +∼ 2017.8 indicating a secondary eclipse that is lost in one of the deeper eclipses from the “main” quadruple. +ment method, the Fourier-decomposition method, and the it- +erative phenomenological model method. Results for all three +methods are plotted side-by-side in Figure 2. +First, we used the method of time-domain iterative dis- +entanglement. This technique is a powerful tool for separat- +ing the light curves of strongly blended targets, and was de- +scribed in detail in Section 3 of Powell et al. (2021), where it +was applied for the first time to disentangle the blended light +curves of three eclipsing binaries. +To verify our results and compare the three methods, we +used two other methods to disentangle the light curves. The +second one is the Fourier-based iterative method, which was +also described in Section 3.1 of Powell et al. (2021). Such a +technique is suitable for these data because all the signals of +interest are strictly periodic over at least one sector of TESS +data, wherein movement on the longer outer orbit can be ne- +glected. The third and final method is based on iteratively +fitting the individual pairs with their respective phenomeno- +logical models and then subtracting these from the overall +light curve. After a few (usually two to five) iterative steps, +a shape for the eclipsing light curve of the C pair was clearly +obtained. The method itself and the code used here are de- +scribed in the Appendix of Pejcha et al. (2022). +Apart from the three eclipsing signals, the light curve also +exhibits an additional pulsation-like oscillation. Such a vari- +ation shows a periodicity of (PA − PB). This extra feature +is apparently not present in the Fourier-disentangled light +curve, as well as in the phenomenologically-disentangled one +(see the bottom panels of Figure 2). This may be due to the +subtraction of this signal as part of the disentanglement pro- +cess. Unfortunately, we have not yet been able to come up +with a coherent astrophysical explanation of this signal. +We found the first method of time-domain iterative disen- +tanglement as the most suitable for a subsequent analysis of +the individual light curves, which is discussed in Section 3. +This is mainly due to problematic fitting of outside-eclipse +parts of the light curves by the methods 2 and 3. +To derive the precise times of eclipses of each binary, we +used the result of the time-domain iterative disentanglement +method. These were calculated for each binary after subtrac- +tion of the light curves of the other two pairs. Eclipse times +of each binary, as observed in TESS, are presented in Tables +A1, A2, and A3. +2.2 Ground-based photometric Observations +2.2.1 Baja Astronomical Observatory, Hungary (2007) +V994 Her was observed with the 50-cm f/6 modified +Cassegrain Baja Astronomical Robotic Telescope (BART-1), +located at the Baja Astronomical Observatory in Hungary, +on 40 nights between 18 June 2007 and 9 October 2007. The +observations were carried out with a 4096×4096 Apogee Alta +U16 CCD camera, using a standard Johnson V filter. +MNRAS 000, 1–?? (2023) + +4 +Zasche et al. +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 1, pair A (2.083 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 2, pair A (2.083 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 3, pair A (2.083 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 1, pair B (1.420 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 2, pair B (1.420 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.85 +0.9 +0.95 +1 +Phase +Flux +method 3, pair B (1.420 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.96 +0.97 +0.98 +0.99 +1 +1.01 +Phase +Flux +method 1, pair C (1.960 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.96 +0.97 +0.98 +0.99 +1 +1.01 +Phase +Flux +method 2, pair C (1.960 d) +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.97 +0.98 +0.99 +1 +1.01 +Phase +Flux +method 3, pair C (1.960 d) +Figure 2. The disentangled and folded light curves of all three eclipsing binaries A, B, and C using different approaches: the time-domain +iterative disentanglement (i.e. method 1), the Fourier-decomposition (method 2), and the iterative phenomenological model methods +(method 3), respectively. For details see the text. +The original goal of this photometric monitoring of V994 +Her was to prove and publish for the first time the previously- +unknown doubly eclipsing nature of this system; however, Lee +et al. (2008) independently discovered and characterized this +system’s true nature. Thus, our team at the time chose to +not further analyze the data and simply published the de- +rived times of minima in Borkovits et al. (2011). However, we +make use of this archival photometric data set in the present +work, as it is especially useful for an additional constraining +of the apsidal advance rates of binaries A and B through the +complete lightcurve fittings and, also for checking the con- +stancy (or variability) of the eclipse depths within a one and +half decade-long interval. +2.2.2 Additional observations +V994 Her was monitored over several dozens of nights by R.U. +at his private observatory in the Czech Republic, as well as +remotely from northern Italy using three different telescopes: +a 34-mm refractor, a 150-mm reflector, and a 200-mm re- +flector. Some of these observations were obtained using fil- +tered photometry (usually with R or I filters), while others +were carried out without any filter. Due to the different in- +strumental setups of these instruments, the comparison stars +were different for each telescope; however, they were always +chosen to be adequately close to the target and of a similar +spectral type in order to minimize the effect of differential ex- +tinction during the nights. Additionally, four more nights of +data were obtained by the 250-mm F/(Ph)otometric Robotic +Atmospheric Monitor (FRAM) telescope CTA-N, located on +the island of La Palma, Spain (Prouza et al. 2019). We also +have data from one night of observations by M.M. at his +private observatory in the Czech Republic, using a 200-mm +reflector. From the combination of these datasets, more than +40 new times of eclipses for pair A were derived, and more +than 30 for pair B. Several new estimates for pair C were also +calculated; however, these are of lower quality due to the sig- +nificantly lower photometric amplitude of its variation. +Between 2002 June 10 and 2004 July 14, a total of 1170 +measurements in V -band and 653 in B-band were collected +for V994 Her by S.D. and U.M., using a 28-cm telescope lo- +cated in Cembra (Trento, Italy). This telescope was equipped +with an Optec SSP-5 photoelectric photometer and Johnson +B and V filters. The comparison and check stars were, respec- +tively, HIP 89975 (V = 6.978 mag, B − V = −0.095 mag) +and HIP 90637 (V = 5.862 mag, B − V = −0.099 mag). +These stars are nearly identical in B − V color to V994 Her +and are located nearby on the sky (≤2◦ angular separation). +From these data, we were also able to derive several times of +eclipses for both the A & B pairs. Moreover, we were able to +derive a rough value for the times of eclipse for pair C, which +allowed us to significantly improve our estimate of its orbital +period due to the increased time coverage. +All the previously-unpublished eclipse times are given in +Table A4. The minima presented in this work for the first +time, as well as the previously-published ones, were used for +a final fit of the data over the whole interval (covering more +than 30 years now). This is shown in Figure 3 for all three +pairs. +2.3 Other Catalogs +We queried the WDS Catalog for archival data on V994 +Her and its nearby visual companion (TIC 1685970000), with +MNRAS 000, 1–?? (2023) + +V994 Herculis: A Triply Eclipsing Sextuple +5 +-0.04 +-0.02 +0.00 +0.02 +0.04 +1990 +1995 +2000 +2005 +2010 +2015 +2020 +2025 +V994 Her A +T0=2459011.2085 P=2.0832649d +ETV [in days] +Calendar Year +-0.02 +0.00 +0.02 +50000 +55000 +60000 +Residual +BJD - 2400000 +-0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +1990 +1995 +2000 +2005 +2010 +2015 +2020 +2025 +V994 Her B +T0=2459010.6980 P=1.4200401d +ETV [in days] +Calendar Year +-0.02 +0.00 +0.02 +50000 +55000 +60000 +Residual +BJD - 2400000 +-0.20 +-0.18 +-0.16 +-0.14 +-0.12 +-0.10 +-0.08 +-0.06 +-0.04 +-0.02 +0.00 +0.02 +1990 +1995 +2000 +2005 +2010 +2015 +2020 +2025 +V994 Her C +T0=2459011.1245 P=1.960110d +ETV [in days] +Calendar Year +-0.02 +0.00 +0.02 +50000 +55000 +60000 +Residual +BJD - 2400000 +Figure 3. Eclipse timing variations (ETVs) of V994 Her collected +over the past three decades, with the TESS eclipses included. The +top, middle and bottom panels show the ETVs for binary A, B +and C, in that order; the red and blue points denote the primary +and secondary eclipses, respectively. The red and blue curves are +the photodynamical fitting models. The “divergence” of the ETVs +for the primary and secondary eclipses are due to apsidal motion. +measurements spanning from 1831 to 2015. This data con- +sisted of position angles and separations for the system be- +tween these years. Additionally, we calculated the position +angle and separation for the visual double using data from +Gaia DR3 (Gaia Collaboration et al. 2022). These data were +used in Section 5.1 to investigate whether or not the visual +double star is gravitationally bound. +Table 2. System parameters derived from speckle imaginga. +Parameter +Value +obs. date [JD] +2459710.018 ± 0.001 +position angle [deg] +357.5 ± 0.5 +separation [arc sec] +1.06 ± 0.01 +δV [mag]b +1.89 ± 0.2 +δI [mag]b +1.67 ± 0.2 +Notes: (a) Observations made at 562 nm and 832 nm with the +‘Alopeke speckle interferometric imager mounted on the Gemini +North 8-m telescope (Scott et al. 2021). (b) Difference in +magnitude between Image 1 (containing binaries A and B) and +Image 2 (likely hosting binary C). +2.4 Speckle observation +V994 Her was observed on 10 May 2022 using the, ‘Alopeke +speckle interferometric imager mounted on the Gemini North +8-m telescope (Scott et al. 2021). ‘Alopeke provides simulta- +neous speckle imaging in two bands (562 nm and 832 nm), +with output data products including a reconstructed image +and derived parameters for any detected close companion +stars. Three sets of 1000 × 0.06 sec exposures were collected; +these underwent Fourier analysis in the standard reduction +pipeline (Howell et al. 2011). +Figure 4 shows the image around V994 Her, with a bright +component to the South (hereafter ‘Image 1’) which hosts +binaries A and B. The fainter image to the North (hereafter +‘Image 2’) is 1.06′′ away, and we believe that this image hosts +binary C, as discussed in Section 5.1. At the 290 pc distance +to V994 Her (derived from the Gaia DR3 parallax), this cor- +responds to a spatial separation of ∼ 307 au. The middle +panel shows a zoom-in around Image 1, revealing that there +are no resolved components within it, down to a limiting res- +olution of ≲ 0.1′′. Both the image of the quadruple system to +the South (Image 1), and binary C (likely residing in Image 2) +remain unresolved into their component parts—respectively, +either the A and B binaries, and the primary and secondary +star in binary C—as these components are separated on the +sky by less than our 20 mas nominal angular resolution. This +value is the Gemini optical diffraction limit when Nyquist +sampled with 2×0.01′′ pixels. Our derived 5-σ contrast curves +for this observation, for both the 562 nm and 832 nm images, +are shown in the bottom panel. These curves will be further +discussed in Sect. 5.1 as we attempt to rule out the possibility +that binary C might actually be located in Image 1. +The system properties gleaned from the speckle observa- +tions are summarized in Table 2. +3 PHOTODYNAMICAL MODELING +We carried out a joint photodynamical modeling in which +we combined the three sectors of TESS data alongside the +2007 V band Baja light curves. As part of this, we also mod- +eled the eclipse timing variation (ETV) curves of all three +binaries, the radial velocity (RV) points of binaries A and +B obtained by Lee et al. (2008), and the net stellar spectral +energy distributions (SEDs). To prepare for this analysis, we +improved the Lightcurvefactory software package to al- +low it to handle hierarchical configurations of (2+2)+2 stars +in their entirety. Specifically, the updated code calculates the +MNRAS 000, 1–?? (2023) + +6 +Zasche et al. +Figure 4. Speckle imaging of V994 Her. North is up and East is to +the left. Top panel: 832 nm image of a 1.5′′ ×1.5′′ region near V994 +Her. We define the brighter feature to the South as ‘Image 1,’ which +contains binaries A and B. We label the ∼1.7 magnitude fainter +object to the North as ‘Image 2’. Middle panel: Same as top panel +but zoomed in around Image 1. Each pixel is 0.01′′ in size. Bottom +panel: 5-σ confidence level contrast curves (obtained at 562 nm +and 832 nm). The image spans angular scales from the diffraction +limit, near 20 mas, out to ∼ 1′′, the approximate end of speckle +coherence. The dotted black lines mark the detectable separation +distance of ∼ 0.06′′ of a source that is 2.2 magnitudes fainter than +Image 1 itself (i.e., the approximate brightness of binary C). +revolutions of the six bodies on their three inner orbits, the +middle orbit (i.e., of the quadruple), and the outer orbit (i.e., +of the sextuple). All five orbits may be considered either to +be purely Keplerian or, for tight systems, Lightcurvefac- +tory is able to take into account the mutual perturbations +of the constituent stars with numerical integration of the or- +bital motions. Moreover, any combinations of two-body or +multiple-body eclipses are also considered. The updated code +does not require the disentangling of the three eclipsing bi- +nary light curves; rather, they can be modeled in their ob- +served, blended form (e.g., as shown in Figure 1). Apart from +this improvement, the software package is functionally iden- +tical to that described in previous work (see, e.g., Borkovits +et al. 2019, 2021). +For the specific case of V994 Her, we find that bina- +ries A and B form a relatively wide 2+2 quadruple system +(PA−B/PB > PA−B/PA > 500). As a result, the gravitational +perturbations of the binary components in the 2+2 quadruple +are small and can be described by simple Keplerian orbits. +Therefore, we use a simple analytic Keplerian formalism in +order to calculate the stellar positions at any given time, with +the slight empirical modification of considering, for all three +binaries, a constant apsidal advance rate ( ˙ωA,B,C) and ref- +erence values for the argument of periastron (ωA,B,C) at a +specific epoch. A physical interpretation of the apsidal mo- +tion is discussed in Section 4.1. +In our joint photodynamical analysis, we optimized the +following parameters using a Markov Chain Monte Carlo +(MCMC) method: +(i) Orbit-related parameters: +– For all four orbits (three eclipsing pairs and the +quadruple A-B): The components of the eccentricity vec- +tors at epoch t0: (e sin ω)A,B,C,A−B, (e cos ω)A,B,C,A−B, and +the inclinations relative to the plane of the sky: iA, iB, iC, +iA−B. +– For the A–B orbit: the period PA−B and the periastron +passage time τA−B. +– For the three eclipsing pairs: the (constant) apsidal +advance rates: ˙ωA,B,C. +(ii) Stellar parameters: +– Six mass-related parameters: the masses of the pri- +maries (mAa,Ba,Ca), and the mass ratios of the three EBs +(qA,B,C), +– The metallicity of the system ([M/H]), +– The (logarithmic) age of the six coeval stars (log τ), +– The interstellar reddening E(B − V ), and +– The “extra light” contamination (ℓ) parameters. +A couple of other parameters were constrained instead of +being adjusted or held constant during our analyses: +(i) Orbits: +– The sidereal orbital periods of the inner binaries +(PA,B,C) and their respective orbital phases (derived using +the time of an arbitrary primary eclipse) were constrained +internally through the ETV curves. +– The systemic radial velocity of the whole sextuple sys- +tem (γ) is calculated a posteriori at the end of each trial +step by minimizing the value of χ2 +RV. +Note that the (2+2)+2 mode of Lightcurvefactory +MNRAS 000, 1–?? (2023) + +Image 2 +V994 Her +832 nm +Image 1V994 Her +Image 1 +832 nm +0.2"0 +V994 Her Contrast Curves +7 +562 nm +2 +832 nm +3 +4 +5 +6 +7 +8 +0.02 +0.05 +0.1 +0.2 +0.5 +1 +Angular Separation (arc sec)V994 Herculis: A Triply Eclipsing Sextuple +7 +requires the orbital elements of the outermost (AB-C) orbit. +In the present case, this orbit is completely unknown. We +do know, however, that it must be so wide that we do not +expect any observable variations in the positions of the six +stars arising from their motion along this orbit. Thus, we +chose the elements of this outmost orbit arbitrarily; we use +a circular orbit seen face-on with a period of ∼ 30 kyr with +its parameters kept fixed. +(ii) Stars: +– The radii and temperatures of the six stars were cal- +culated with the use of three linear interpolations from the +precomputed 3D PARSEC grids (the dimensions were metal- +licity, logarithmic age, and stellar mass). +– The distance of the system (needed for the SED fit- +ting) was calculated a posteriori at the end of each trial +step, by minimizing the value of χ2 +SED. For a detailed ex- +planation of this process, see Borkovits et al. (2020). +The atmospheric parameters of the stars were handled in a +similar manner as in our previous photodynamical studies. +We utilized a logarithmic limb-darkening law (Klinglesmith +& Sobieski 1970), for which the passband-dependent linear +and non-linear coefficients were interpolated in each trial step +via the tables from the original version of the Phoebe software +(Prša & Zwitter 2005). We set the (constant) gravity darken- +ing exponents for five radiative stars to β = 1; for the coolest, +solar-like component Cb, however, we used β = 0.32, which +is in line with the classical model of Lucy (1967) and is valid +for convective stars. +Prior to conducting our analysis, we performed some fur- +ther preparatory steps on the light curves. First, after disen- +tangling the three eclipsing binaries as in Section 2.1.1, we +found that the residual TESS light curves contained oscil- +lations with an amplitude of approximately 2% and a char- +acteristic period of (PA − PB). We removed this oscillation +from the light curves before performing the full photodynami- +cal analysis by subtracting the final residual light curve of the +iterative disentanglement process (which contained this peri- +odic variability) from the original TESS light curves for each +sector. Second, for the sake of equal sampling across sectors, +we binned the 2-min sector 26 TESS light curve to 10-min +bins, identical to the cadence time of the sector 40 and 53 +FFI light curves. We also binned the 2007 Baja photometry +to 10-min bins. Third, we noticed that the eclipse depths of +all three binaries in sector 26 were deeper by a few percent +than the corresponding eclipses in the sectors 40 and 53 data; +the depths in the latter two sectors were similar. As a result, +we assume that in sector 26, the ratio of contaminating light +is somewhat lower than in the other two sectors. Thus, for +the sector 26 light curve, we adjusted the amount of contam- +inating light independent of sectors 40 and 53. The effect of +slightly different flux contamination for the same star in data +from different TESS sectors has been studied previously in +the literature (see, e.g., Plachy et al. 2021.) +The median values and their 1-σ uncertainties (derived +from the MCMC calculation) for the orbital and physical +parameters of the sextuple system, as well as some derived +quantities, are tabulated in Table 3. Furthermore, a compar- +ison of the observed and model light curves are plotted in +Figure 5, while a similar comparison for the ETV curves is + 0.70 + 0.75 + 0.80 + 0.85 + 0.90 + 0.95 + 1.00 + 1.05 +Normalized Flux + -0.02 + 0.00 + 0.02 +59408 +59410 +59412 +Residual Flux +BJD - 2400000 +Figure 5. A section of the TESS sector 40 light curve (blue points) +after the removal of the oscillations with period (PA −PB), plotted +with the photodynamically fitted model light curve (red). We also +plot the original light curve, including the oscillations with period +PA − PB, using gray points. The short vertical solid and dashed +lines along the x-axis, colored red, blue and green, denote the mid- +eclipse times of the primary and secondary eclipses of binaries A, +B and C, respectively. The residual curves are shown below the +main light curve. +shown in Figure 3. Note that Table 3 presents the absolute +physical parameters for the C binary and both its compo- +nents, despite the fact that we do not have any directly ob- +served radial velocities for this pair. However, we have the +RVs of both the A and B binaries as well as changes of the +RVs on their mutual orbit. We can consequently derive the +properties of this binary by using the Lightcurvefactory +code to combine the light curve modeling of the C binary and +the SED of the overall system. +4 THE FINAL PARAMETERS +Our thorough modeling of the system also yields the position +of each star in the H-R diagram. Due to the fact that the age +of the system was also taken as a free parameter (under the +assumption that all six stars are coeval), we can characterize +its evolutionary state. From the calculated value of the sys- +tem’s logarithmic age (presented in Table 3), the system is +rather young and therefore located close to the ZAMS. This +is in agreement with the fact that all of the orbits are slightly +eccentric, so the circularization process is still ongoing (see, +e.g., Claret & Cunha 1997). +Only a few sextuple systems have well-constrained param- +eters, including their masses and orbital elements; as a re- +sult, it is not very easy to compare V994 Her with oth- +ers. Interestingly, the recent analysis of the sextuple system +TIC 168789840 (Powell et al. 2021) revealed a vastly differ- +ent configuration, wherein all three binary pairs have similar +mass ratios. In V994 Her, our analysis suggests that all three +components have very different mass ratios. Perhaps V994 +Her is more similar to the well-known Castor system, which +has a similar architecture, with its component binaries having +very different mass ratios (see, e.g., Tokovinin 2018). +The whole system is plausibly close to a co-planar config- +uration, given the inclination angles in Table 3. However, to +MNRAS 000, 1–?? (2023) + +8 +Zasche et al. +Table 3. Median values of the parameters from the joint spectro-photodynamical analysis of (i) all three EB light curves, (ii) both sets +of radial velocities from the SB2 (i.e., the quadruple consisting of binaries A and B), (iii) all three sets of ETVs, and (iv) joint SED and +PARSEC evolutionary tracks. +Parameter +Binary A +Binary B +Binary C +A–B orbit +Pa [days] +2.0832039+0.0000042 +−0.0000039 +1.4200981+0.0000033 +−0.00000040 +1.9601064+0.0000018 +−0.0000018 +1062.3+2.8 +−2.4 +a [R⊙] +11.85+0.17 +−0.11 +8.30+0.07 +−0.05 +9.45+0.17 +−0.23 +910+12 +−8 +ia [deg] +84.66+0.20 +−0.33 +89.19+0.63 +−0.62 +80.42+2.39 +−0.99 +83+4 +−5 +e +0.0276+0.0010 +−0.0010 +0.1186+0.0007 +−0.0007 +0.1893+0.0052 +−0.0046 +0.687+0.050 +−0.037 +ω [deg] +208.4+3.4 +−4.3 +174.7+2.4 +−2.7 +314.0+3.4 +−2.3 +59.6+3.6 +−3.9 +˙ω [deg/yr] +1.86+0.42 +−0.55 +3.66+0.17 +−0.16 +1.68+0.24 +−0.24 +− +τ [BJD - 2 400 000] +59 010.855+0.020 +−0.025 +59 010.393+0.009 +−0.011 +59 009.306+0.007 +−0.007 +58 166.8+13.6 +−9.3 +tprim eclipse [BJD - 2 400 000] +59 011.1821+0.0014 +−0.0019 +59 010.7324+0.0028 +−0.0021 +59 011.1246+0.0004 +−0.0004 +− +q (= m2/m1) +0.757+0.008 +−0.009 +1.009+0.009 +−0.009 +0.583+0.104 +−0.074 +0.738+0.017 +−0.013 +Kpri [km s−1] +124+2 +−2 +150+1 +−1 +90+11 +−8 +25+2 +−1 +Ksec [km s−1] +163+2 +−2 +148+1 +−1 +154+8 +−7 +34+3 +−2 +γ [km/s] +− +− +− +−38.7+0.3 +−0.3 +individual stars +Aa +Ab +Ba +Bb +Ca +Cb +Relative Quantities: +fractional radiusb [R/a] +0.1785+0.0022 +−0.0026 +0.1452+0.0013 +−0.0015 +0.1895+0.0019 +−0.0019 +0.1903+0.0018 +−0.0017 +0.1648+0.0038 +−0.0039 +0.1016+0.0198 +−0.0106 +fractional luminosity in TESS-band +0.3822+0.0099 +−0.0143 +0.1847+0.0075 +−0.0072 +0.1227+0.0025 +−0.0022 +0.1258+0.0029 +−0.0025 +0.1086+0.0199 +−0.0107 +0.0145+0.0113 +−0.0055 +fractional luminosity in V -band +0.4115+0.0169 +−0.0175 +0.1909+0.0098 +−0.0115 +0.1184+0.0041 +−0.0054 +0.1220+0.0047 +−0.0056 +0.1021+0.0227 +−0.0163 +0.0085+0.0074 +−0.0035 +extra light [ℓS26] +0.0576+0.0155 +−0.0221 +[ℓS40,53] +0.1808+0.0141 +−0.0215 +[ℓV ] +0.0366+0.0334 +−0.0256 +Physical Quantities: +T c +eff [K] +11890+310 +−264 +9915+256 +−201 +8832+158 +−156 +8895+175 +−170 +8514+514 +−310 +5893+451 +−384 +mass [M⊙] +2.929+0.124 +−0.093 +2.216+0.106 +−0.070 +1.913+0.050 +−0.040 +1.889+0.055 +−0.049 +1.810+0.169 +−0.072 +1.077+0.162 +−0.109 +radiusc [R⊙] +2.118+0.026 +−0.030 +1.721+0.038 +−0.031 +1.572+0.028 +−0.019 +1.579+0.027 +−0.018 +1.544+0.071 +−0.035 +0.961+0.199 +−0.115 +luminosityc [L⊙] +80.2+10.4 +−7.7 +25.7+3.9 +−2.6 +13.5+1.1 +−1.0 +14.1+1.3 +−1.2 +11.2+4.4 +−1.8 +1.00+0.94 +−0.41 +[Mbol] +0.01+0.11 +−0.13 +1.25+0.12 +−0.16 +1.94+0.09 +−0.09 +1.90+0.10 +−0.09 +2.15+0.19 +−0.36 +4.77+0.57 +−0.72 +log gc [cgs] +4.252+0.017 +−0.013 +4.312+0.005 +−0.005 +4.321+0.007 +−0.007 +4.321+0.007 +−0.007 +4.317+0.009 +−0.007 +4.503+0.065 +−0.103 +log(age) [dex] +7.92+0.12 +−0.23 +[M/H] [dex] +0.073+0.042 +−0.056 +E(B − V ) [mag] +0.050+0.011 +−0.014 +(MV )c +tot +−0.30+0.07 +−0.07 +distance [pc] +274+6 +−6 +Notes: (a) Calculated only from the sin i terms; (b) Polar radii; (c) Interpolated from the PARSEC isochrones +derive its true orbital architecture we would also need to cal- +culate the values for the longitude of the ascending node Ω. +To do so, one would need to derive a precise interferometric +orbit, which is not available to us currently. Once this in- +formation is obtained, we can speculate whether or not the +system can exhibit Kozai-Lidov cycles (Kozai 1962; Lidov +1962); however, these may be halted anyway by rapid pre- +cession of the pericenters of the component binaries (Table 3 +and Vokrouhlický 2016, for an example). +4.1 Apsidal motion +Given that we have multiple high-precision sets of eclipse +times (see Tables A1, A2, A3, and A4), and that each binary +has an eccentric orbit, we are able to derive apsidal motion +rates for all three pairs in the system. We find that this rate +is approximately a few degrees per year, suggesting that their +apsidal advance is not at all negligible. In order to properly +interpret these empirically fitted rates, we first have to sub- +tract any contributions from the apsidal advance that can be +accurately computed. +First, we determine the relativistic contribution to the ob- +served apsidal motion (see, e.g., Claret & Giménez 2010). +Given the orbital and physical parameters in Table 3 we find +this effect represents about 10%, 8%, and 6% of the total for +the binaries A, B, and C, respectively. Because these percent- +ages are rather small, we can consider all three sub-systems of +50000 +52000 +54000 +56000 +58000 +195 +200 +205 +ωA [deg] +50000 +52000 +54000 +56000 +58000 +165 +170 +175 +ωB [deg] +50000 +52000 +54000 +56000 +58000 +58.5 +59 +HJD−2400000 +ωAB [deg] +Figure 6. Results of the numerical modeling of the orbits of bina- +ries A and B and the quadruple AB. Here, we show the long-term +evolution of the arguments of periastron (without the tidal term). +Pair A is plotted in black; pair B, in red; and their mutual orbit +(A-B), in blue. See section 4.1 for details. +MNRAS 000, 1–?? (2023) + +V994 Herculis: A Triply Eclipsing Sextuple +9 +V994 Her as classical apsidal rotators, rather than relativistic +ones. +Because the inner 2+2 component of binaries A and B is +not too wide, there also exists a classical (Newtonian) contri- +bution to their apsidal motion, arising from mutual dynami- +cal perturbations between A and B. In order to estimate this +effect, we ran a simulation using the N-body code developed +by Brož (2017) and Brož et al. (2022). For the sake of defi- +niteness, we assumed a nearly-coplanar system configuration +by imposing identical initial values of the nodal longitude for +both the A and B orbits, with other orbital parameters taken +from Table 3. We found that such mutual perturbations in +the A and B system account for another 12% and 5% of the +total apsidal motion in the respective component (see Figure +6, which shows these contributions). The binary C is deemed +to be distant enough from the A and B binaries so that we +do not provide the classical apsidal contribution in this case. +With those two effects estimated and subtracted from the +total values of the observed apsidal motions in the A, B and +C systems, we can assume that the remainder is attributable +to the effect of the stellar tidal interactions. From these rates +of ˙ωtidal, one can usually derive the internal structure con- +stants and compare them with theoretical ones from stellar +evolution models (e.g. Claret 2004). However, when compar- +ing the results for pair B (which has the best coverage of +its apsidal period, since it has the fastest rate), our resulting +values deviate slightly from the predicted theoretical ones. +The tidal contribution to the apsidal rate was found to be +˙ωtidal,B = (3.14 ± 0.20) deg/yr, yielding an internal struc- +ture constant of log k2 = −2.44 ± 0.05, while the theoreti- +cal models of Claret (2004) suggest that its value should be +−2.36 ± 0.02. In order for the respective error intervals to +overlap, one needs to have either larger uncertainties in the +derived parameters, a slightly faster apsidal motion (of about +5%), subsynchronous rotation of the component stars (about +20% slower), or a combination of all three of these effects. An- +other way to account for this discrepancy is by using the fact +that these stars, found to be very young, were likely born in +a metal-rich environment. Using the stellar evolution models +of Claret (2007) with a higher metallicity (Z=0.04), we find +log k2 = −2.40 ± 0.02, which is in much better agreement +with the observed value of log k2. +The N-body modelling also allows us to estimate the ap- +sidal advance of the quadruple orbit A-B (shown in Figure +6). This motion – accumulating to ∼ 0.6◦ over the inter- +val of available observations – is orders of magnitude slower +when compared to the values for the A and B systems. How- +ever, over the next few decades, when the change will have +cumulatively added up to a few degrees, one can readily de- +tect such movement with newly obtained data. On the other +hand, other effects such as the change in orbital inclination +and eclipse depth would still be negligible on such a timescale. +We also note that the eccentric orbits of the inner binaries +are subject to the circularization effect. From the theory of +circularization by Zahn (1977) and equations by Claret & +Cunha (1997) the appropriate circularization time scales are +of the order of magnitude longer than the estimated age of +the system as resulted from our modelling. +5 DISCUSSION +5.1 V994 Her and its visual companion +In the prior sections, we conclusively demonstrated the pres- +ence of a third eclipsing binary in the V994 Her system. Here, +we discuss the likelihood that binary C is hosted by Image 2 +(fainter object to the North, as seen in Fig. 4), as well as the +probability that Image 2 is physically bound to Image 1. If +so, this would give the system a (2+2)+2 configuration. +According to the photodynamical fit for the system pa- +rameters presented in Table 3, binary C has 14% the light +of binaries A+B in the TESS band, and 13% in V band. +That corresponds to magnitude differences of 2.1 and 2.2, re- +spectively, in the TESS and V bands. The bottom panel of +Fig. 4 suggests that these contrasts correspond to being able +to resolve two objects within Image 1 (the brighter southerly +object) that are separated by ≳ 0.06′′. Since both Images 1 +and 2 are 290 pc away, the resolvable physical separation at +this magnitude contrast would correspond to 18 au. The ac- +tual semimajor axis of the binary A and B quadruple, which +resides in Image 1, is 4.2 au (see Table 3). It is always pos- +sible, of course, that at the time of the speckle observations, +the projected distance between the center of light of bina- +ries A+B and a putative close orbiting neighbor (i.e., binary +C) might inadvertently be very small due to unlucky orbital +phasing. Let us assume, however, for the sake of argument, +that A+B and C are at some nominal separation on the sky at +this particular outer orbital phase when the speckle measure- +ments were made. In that case, the outer orbit of C around +A+B, within Image 1, would have to have a semimajor axis +of not much more than ∼ 20 au before it is resolvable. +For a stable triple system (i.e., C stably orbiting A+B) the +ratio of semi-major axes must satisfy +aout ≳ 2.8 +�MABC +MAB +�2/5 (1 + eout)2/5 +(1 − eout)6/5 ain , +(1) +where equation (1) is from Rappaport et al. (2013), which +in turn is based on the work of Mardling & Aarseth (2001) +and Mikkola (2008). If we take as a very rough estimate that +aout ≲ 20 au, and we know that ain ∼ 4.2 au, then we find a +constraint on eout such that +(1 + eout)2/5 +(1 − eout)6/5 ≲ 1.5 . +(2) +In turn, this requires that eout ≲ 0.25. Thus, while this is not +an unreasonably small value for an outer orbital eccentricity, +we can see from this exercise, that there is “not much room +to spare" in trying to fit binary C into an orbit about bina- +ries A+B, all within Image 1. Furthermore, recall that the +contrast limits shown in Fig. 4 are 5-σ limits. Thus, we ten- +tatively conclude that binary C, in fact, is hosted by Image +2 (the fainter one to the North). +We next look at the question of whether Image 2 (likely +containing binary C) is physically bound to Image 1 (hosting +binaries A and B). For this analysis, we have two pieces of +kinematic evidence: (i) the proper motions of Images 1 and 2 +from Gaia DR3 (Gaia Collaboration et al. 2022), and (ii) the +historical astrometric data, spanning 200 years, of the WDS +catalog (Mason et al. 2001). This information is summarized +in Table 4, and the WDS astrometric data are plotted in +Fig. 7. +The two proper motion results (Gaia and WDS) evaluated +MNRAS 000, 1–?? (2023) + +10 +Zasche et al. +Table 4. Observational Kinematics Between Image 1 and Image +2 +Cartesian motiona +mas yr−1 +km s−1 +Gaia PM RA +−0.70 ± 0.14 +−0.96 ± 0.20 +WDSb PM RA +−1.22 ± 0.10 +−1.68 ± 0.14 +Gaia PM Dec +−4.11 ± 0.11 +−5.66 ± 0.16 +WDSb PM Dec +−3.84 ± 0.35 +−5.29 ± 0.48 +angular motion +— +— +WDSb,c ˙r [mas yr−1] +−3.86 ± 0.35 +— +WDSb,c ˙θ [mrad yr−1] +−1.10 ± 0.09 +— +Notes: (a) Image 2 value - Image 1. (b) Washington Double Star +catalog (Mason et al. 2001). (c) These refer to the rate of change +in the separation and the position angle, respectively, and are +inferred from the fits shown in Fig. 7. +at the Gaia epoch are in agreement on the proper motion of +the declination (PM Dec) to better than 1 σ, while the proper +motions of the right ascension (PM RA) differ by 2.9 σ. We +attribute this discrepancy to fitting a linear function to ˙θ2 +over a 200 year interval. The total relative velocity between +Image 1 and Image 2 on the plane of the sky is in the range +5.55–5.74 km s−1 depending on whether we choose to use the +WDS or Gaia results, respectively. +To check whether Image 2 is physically bound to Image 1, +we take the escape speed to be +vesc ≃ +� +2GMABC +s +, +(3) +where Mtot is the total mass contained in Image 1 plus Im- +age 2, and s is the instantaneous (3D) separation of Im- +age 1 and Image 2. As representative values, we estimate +MABC = MA + MB + MC = 11.5 M⊙ from Table 3, and +s ≳ 307 au, where the latter is the physical separation on the +plane of the sky between Image 1 and Image 2. This leads +to an estimate for vesc ≲ 8.2 km s−1. Since this value is sub- +stantially larger than the relative speed of Image 1 vs. Image +2 (at least in the plane of the sky), we tentatively take this +as strong evidence that the Image 1 plus Image 2 system is +physically bound. However, we remain unsure about the rel- +ative speed and separation in the direction along our line of +sight. +Finally, we make another independent argument which also +strongly suggests that Image 1 and Image 2 are physically +bound. This argument relies on the fact that Image 2 is found +so close in the sky to Image 1, with similar proper motions +and distance, and the two are not too dissimilar in magnitude. +We seek to quantify the relative occurrence rate of such a +pair of stars. Using Gaia data, we searched for other stars +with similar properties to those of Image 2. In particular, +we looked for stars that have (i) PM RA within an absolute +value of 1 mas yr−1 of image 1; (ii) PM Dec with absolute +value within 5 mas yr−1 of Image 1; (iii) a parallax within +absolute value of 0.2 compared to Image 1; and (iv) having +a G magnitude brighter than 9. When we search the Gaia +database for other stars that satisfy these criteria, we find 13 +such stars within 30◦ of Image 1. Given that the search area +is ∼ 1010 times larger than the area needed to include Image +2 (at 1′′ distance from Image 1), we conclude that Image 1, +2 ˙θ ≡ rate of change in the position angle. +Figure 7. A plot of position angle and separation for the two +stars in the visual binary catalogued by the Washington Double +Star catalog (Mason et al. 2001) at the position of V994 Her. +with its given properties, is not remotely likely to be found +there by accident. +The conclusion from the above argument is that either Im- +age 2 is physically bound to Image 1, or it is comoving with +it by virtue of having been born in the same stellar nursery. +The latter scenario can be ruled out rather readily. We have +seen that the relative speed between Image 1 and Image 2 (on +the plane of the sky) is 5.6 km s−1. The age of the system +(from Table 3) is ≃ 60 Myr. If the two images were unbound +and merely approximately comoving on the sky, then during +that time the two images would have drifted apart in the en- +suing 60 Myr by some 300 pc. This is much, much larger than +the current sky separation of 290 au. Thus, we conclude that +Image 2 is physically bound to Image 1. +Given the above discussion, we believe that the most likely +configuration of this sextuple is a (2+2)+2 system, with the +inner quadruple system containing binaries A and B situated +in Image 1, and the third binary (C) in Image 2. There is +also a slim chance, but not yet fully ruled out, that Image 1 +hosts all three binaries, and Image 2 represents a 7th star or +yet another binary (‘D’). The most direct ways to prove our +most likely scenario (that binary C is in Image 2) is to (i) +check for binary C eclipses in Image 2, and/or (ii) check for +RV motions in Image 2 with P = 1.96 d. +MNRAS 000, 1–?? (2023) + +20 +(degrees) +WDS Astrometry +15 +10 +Position Angle ( +5 +5 +Rotation Rate = -0.063 +/- 0.005 deg/year +-10 +1800 +1850 +1900 +1950 +2000 +2050 +Year3.0 +WDS Astrometry +2.5 +(arc sec +2.0 +1.5 +1.0 +0.5 +dr/dt = -3.86 +/- 0.35 mas/yea +0.0 +1800 +1850 +1900 +1950 +2000 +2050 +YearV994 Herculis: A Triply Eclipsing Sextuple +11 +5.2 Outer Orbital Period Distribution +Armed with only the relative velocity between Image 1 and +Image 2 projected onto the sky, vsky ≃ 5.7 km s−1, and the +projected separation on the sky, ssky ≃ 307 au, we attempt +to estimate a probability distribution for the outer orbital +period Pout via a Monte Carlo approach. Let ⃗s and ⃗v be the +full relative position and velocity vectors between Image 1 +and Image 2. In that case +ssky +s += +sin β1 , +(4) +vsky +v += +sin β2 , +(5) +where β1 is the angle between the observer’s view direction +and ⃗s and β2 is the angle between the view direction and ⃗v. +If we know nothing about the orientation of the orbit on the +sky, then samples of sin β can be drawn randomly from: +sin β = +� +1 − R2 , +(6) +where R is a uniformly distributed random number between +0 and 1. Here, as an approximation, we treat ˆs and ˆv as +independently and randomly directed with respect to the ob- +server’s view direction. So, we randomly draw β1 and β2. +The energy of the outer orbit can now be written as +E = −GM1M2 +ssky +sin β1 + 1 +2 +M1M2 +MABC +� vsky +sin β2 +�2 += −GM1M2 +2aout +, +(7) +where M1 ≡ MA + MB and M2 ≡ MC. This reduces to a +simple expression for the semi-major axis of the outer orbit: +1 +aout = 2 sin β1 +ssky +− +� +vsky +GMABC sin β2 +�2 +. +(8) +Finally, we make a large number of random draws for β1 +and β2 and, for each combination, store the realization for +the semi-major axis and corresponding Pout. The resultant +distribution for Pout is shown in Fig. 8. The distribution has +a sharp cut at ≃ 1500 years, reflecting the fact that the min- +imum orbital separation is attained when β1 = β2 = 90◦, +i.e., when the outer orbit is in the plane of the sky. However, +since we do not know the orbital parameters in the line of +sight, long orbital periods are possible, as evidenced by the +long tail of the period distribution in Figure 8. The median +of the orbital period distribution is close to 3000 years. +We note that our uncertain knowledge of the outer orbit +could be greatly improved with a radial velocity study of +Image 2 (which presumably hosts binary C). +5.3 Possible role of the 3:2 mean motion resonance +Zasche et al. (2019) presented a thorough analysis of stel- +lar quadruple systems with a 2+2 architecture that exhibit +eclipses of both components, with binary periods less than +15 days. One of the interesting population results from this +study was the identification of a statistically significant group +of systems having a period ratio close to 3/2. Zasche et al. +(2019) speculated that these systems are either captured in +the 3:2 mean motion resonance of the binary periods, or in- +teracted with this resonance in the recent past and still reside +close to it. One of the consequences for this class of systems +would be a possible excitation of the orbital eccentricity of +Figure 8. Probability distribution for the outer orbit of the V994 +Her system. Results have been logarithmically binned. The black +histogram is the differential probability distribution, while the red +curve is the cumulative distribution. +the binary with the longer period. The V994 Her system was +considered in this class by Zasche et al. (2019). Now, with +much more detailed information about V994 Her, we revise +its status with respect to the group of objects that reside or +interacted with the 3:2 resonance in the past. +We use an analytical description of the low-order mean mo- +tion resonances in 2+2 quadruples by Tremaine (2020) (for +completeness, see also Breiter & Vokrouhlický 2018, who dis- +cuss the 1:1 mean motion resonant states in the 2+2 quadru- +ples). First, it is trivial to check that V994 Her is not currently +located in the resonance since 1−(2PA/3PB) ≃ 0.02204 is too +large (it would need to be three orders of magnitude smaller +to be considered to possess this resonance). Tremaine (2020) +also discusses sidebands of the pure 3:2 mean motion res- +onance between PA and PB generated by multiplets of the +mean motion frequency nAB of the mutual orbit. Their im- +portance is typically very small, because the sideband width +at frequency k nAB has a multiplicative factor ∝ e|k| +AB (k is an +integer). Here, eAB ≃ 0.7 is a rather large value. However, +to account for the three orders of magnitude in separation +between the observed eccentricity and the requirement for +resonance, |k| would have to be greater than 30, which is +much too large. The system, however, may have crossed the +resonance in the past; this could have contributed to an ex- +citation of the eA value. +In order to place the system into the exact 3:2 mean mo- +tion resonance, one would need to (i) increase PA by ∆PA ≃ +0.046943 day, (ii) decrease PB by ∆PB ≃ 0.031296 day, or +(iii) perform some combination of the two operations. Addi- +tionally, in order to temporarily capture V994 Her in the 3:2 +resonance in the past, PA and PB should have been converg- +ing towards each other. In what follows, we shall discuss an +end-member possibility (i) that PB was constant, and PA was +evolving from an initially larger value beyond the resonance +condition toward the current value. However, identical con- +clusions are obtained for other options, such as keeping PA +constant and PB increasing as in (ii), or their combination. +Using the results from Appendix C of Tremaine (2020), +we note that the putative past capture in the 3:2 resonance +puts a severe constraint on the speed by which the period PA +MNRAS 000, 1–?? (2023) + +1.0 +Probability +0.8 +0.6 +Relative F +0.4 +Median Period = 2930 year +0.2 +0.0 +1000 +10000 +100000 +Outer Orbital Period (years)12 +Zasche et al. +decreased. In particular, denoting the corresponding charac- +teristic timescale τA = PA/ ˙PA (with ˙PA = −dPA/dt), we find +that +τA ≥ K (MB/µB)4/3 (aAB/aB)20/3 PA , +(9) +where K ≃ 1.74 × 10−2, MB and µB represent the total and +reduced masses of the shorter period binary component, aB +and aAB are the semimajor axes of the B and A-B orbits, +and PA is the orbital period of the A binary. Substituting +the values from Table 3, we have τA ≥ 26 Gyr. Assuming an +approximately steady decay of the A orbit, we then estimate +a minimum time needed to accumulate the difference ∆PA +between the resonance and the current state +∆TA ≃ ∆PA +PA τA ≥ 580 Myr . +(10) +This is nearly an order of magnitude longer than the esti- +mated age of the V994 Her system (Table 3). Since other +possibilities outlined above lead to the same result, such as +PB drifting toward its current value from an initially smaller +value, we may conclude that the V994 Her system in all like- +lihood never interacted with the 3:2 mean motion resonance. +Its location near to it might therefore be just coincidental. As +a consequence, the eA value is fully a relic of the initial state, +with possible tidal damping. Indeed, the interaction with the +3:2 resonance would likely not be capable of explaining the +significantly larger eB value. The latter might be excited by +interaction with the 2:1 mean motion resonance between PA +and PB values; the location of this resonance, however, is +much too distant from the current system parameters. +6 SUMMARY AND CONCLUSIONS +In this paper, we have demonstrated that the first-known +doubly eclipsing system V994 Herculis is in fact at least a +sextuple system that unambiguously demonstrates three sets +of eclipses. Using TESS and archival data, we have disentan- +gled the light curves of all three binaries in the system using +three different techniques and added new measurements to +the O–C diagrams of binaries A and B. We have also identi- +fied the period of the newly-discovered binary C to be 1.9601 +days, based on TESS and older ground-based data. Finally, +we used archival data from the Washington Double Star Cat- +alog (spanning over 190 years) alongside parameters from +Gaia DR3 in order to prove that the fainter visual companion +on the night sky (1.1′′ distant) is likely gravitationally bound +to this system and may harbor binary C. +Depending on the nature of the companion star, this could +be either a rare (2+2)+2 sextuple star system—similar to +the well-known system Castor, with the same architecture +(see, e.g., Stelzer & Burwitz 2003, and Tokovinin 2018). An- +other possibility is that the brighter star has six unresolvable +stars, and the nearby visual companion is another bound +member of the system, making it even more interesting—a +potential septuple (or even higher-multiplicity) star system. +Using additional data, we can more precisely derive the outer +orbit; moreover, updated higher angular-resolution photom- +etry would be able to firmly prove whether or not the C pair +resides in the fainter nearby component. We urge the com- +munity to observe these interesting stars using the tools at +their disposal. The high-angular separation techniques (both +in photometry as well as spectroscopy) would be able to prove +the true structure of the system. As the separation of the vi- +sual pair on the night sky is slowly decreasing, it may become +increasingly difficult to carry out these observations as time +goes on. +DATA AVAILABILITY +The TESS data underlying this article were accessed using +the MAST (Barbara A. Mikulski Archive for Space Tele- +scopes) Portal (https://mast.stsci.edu/portal/Mashup/ +Clients/Mast/Portal.html). Some of the data were derived +from sources in the public domain; their URLs are provided +as footnotes. The derived data generated in this paper and +the code used for the photodynamical analysis will be shared +upon reasonable request to the corresponding author P.Z. +ACKNOWLEDGMENTS +This paper includes data collected by the TESS mission, +specifically as part of GI program G022062 (PI: A. Prša). +Funding for the TESS mission is provided by the NASA Sci- +ence Mission directorate. Resources used in this work were +provided by the NASA High End Computing (HEC) Pro- +gram through the NASA Advanced Supercomputing (NAS) +Division at Ames Research Center for the production of the +SPOC data products. Some of the data presented in this +paper were obtained from the Mikulski Archive for Space +Telescopes (MAST). STScI is operated by the Association of +Universities for Research in Astronomy, Inc., under NASA +contract NAS5-26555. Support for MAST for non-HST data +is provided by the NASA Office of Space Science via grant +NNX09AF08G and by other grants and contracts. +This research has made use of the Washington Double Star +Catalog maintained at the U.S. Naval Observatory, and we +thank Rachel Matson for providing archival data on the V994 +Her visual double. +This work has used data from the European Space Agency +(ESA) mission Gaia3, processed by the Gaia Data Processing +and Analysis Consortium (DPAC)4. Funding for the DPAC +is provided by national institutions, in particular those par- +ticipating in the Gaia Multilateral Agreement. +Some of the observations in the paper made use of the +High-Resolution Imaging instrument ‘Alopeke, obtained un- +der Gemini LLP Proposal Number: GN/S-2021A-LP-105. +‘Alopeke was funded by the NASA Exoplanet Exploration +Program and built at the NASA Ames Research Center by +Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett +Quigley. ‘Alopeke was mounted on the Gemini North (and/or +South) telescope of the international Gemini Observatory, a +program of NSF’s NOIR Lab, which is managed by the Asso- +ciation of Universities for Research in Astronomy (AURA) +under a cooperative agreement with the National Science +Foundation, on behalf of the Gemini partnership: the Na- +tional Science Foundation (United States), National Research +3 https://www.cosmos.esa.int/gaia +4 https://www.cosmos.esa.int/web/gaia/dpac/consortium +MNRAS 000, 1–?? (2023) + +V994 Herculis: A Triply Eclipsing Sextuple +13 +Council (Canada), Agencia Nacional de Investigación y De- +sarrollo (Chile), Ministerio de Ciencia, Tecnología e Inno- +vación (Argentina), Ministério da Ciencia, Tecnologia, Ino- +vações e Comunicações (Brazil), and Korea Astronomy and +Space Science Institute (Republic of Korea). +This +work +is +supported +by +MEYS +(Czech +Repub- +lic) +under +the +projects +MEYS +LM2010105, +LTT17006 +and EU/MEYS CZ.02.1.01/0.0/0.0/16_013/0001403 and +CZ.02.1.01/0.0/0.0/18_046/0016007. +M.B. and D.V. were supported by the Czech Science Foun- +dation, grant GA21-11058S. +We also used the Simbad service operated by the Centre +des Données Stellaires (Strasbourg, France) and the ESO Sci- +ence Archive Facility services (data obtained under request +number 396301). +REFERENCES +Aarseth S. J., Mardling R. A., 2001, in Podsiadlowski P., Rap- +paport S., King A. 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TESS observed times of minima of V994 Her A +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +59011.208388 +0.0 +0.000068 +59393.510297 +183.5 +0.000244 +59744.520841 +352.0 +0.000102 +59012.282173 +0.5 +0.000038 +59394.519437 +184.0 +0.000103 +59745.592606 +352.5 +0.000108 +59013.291644 +1.0 +0.000027 +59395.593429 +184.5 +0.000134 +59746.604119 +353.0 +0.000102 +59014.365223 +1.5 +0.000039 +59396.602611 +185.0 +0.000128 +59747.675710 +353.5 +0.000085 +59015.375104 +2.0 +0.000028 +59397.676809 +185.5 +0.000113 +59748.686773 +354.0 +0.000100 +59016.448767 +2.5 +0.000038 +59398.686003 +186.0 +0.000110 +59749.759375 +354.5 +0.000097 +59017.458488 +3.0 +0.000028 +59399.760013 +186.5 +0.000116 +59750.770102 +355.0 +0.000093 +59018.532198 +3.5 +0.000044 +59400.769258 +187.0 +0.000105 +59751.842493 +355.5 +0.000101 +59019.541632 +4.0 +0.000028 +59401.843411 +187.5 +0.000115 +59752.853041 +356.0 +0.000097 +59020.615100 +4.5 +0.000036 +59402.852513 +188.0 +0.000098 +59753.926086 +356.5 +0.000109 +59021.624954 +5.0 +0.000030 +59403.926596 +188.5 +0.000173 +59754.937116 +357.0 +0.000153 +59023.708034 +6.0 +0.000040 +59407.018862 +190.0 +0.000150 +59757.020019 +358.0 +0.000086 +59025.791322 +7.0 +0.000031 +59408.093094 +190.5 +0.000142 +59759.103225 +359.0 +0.000084 +59027.874755 +8.0 +0.000029 +59410.176195 +191.5 +0.000125 +59760.175520 +359.5 +0.000103 +59028.948286 +8.5 +0.000040 +59411.185709 +192.0 +0.000125 +59761.186560 +360.0 +0.000087 +59029.957752 +9.0 +0.000028 +59412.259529 +192.5 +0.000132 +59762.259204 +360.5 +0.000102 +59031.031620 +9.5 +0.000041 +59413.268755 +193.0 +0.000107 +59763.269832 +361.0 +0.000082 +59032.041094 +10.0 +0.000030 +59414.342947 +193.5 +0.000129 +59764.342122 +361.5 +0.000094 +59033.114938 +10.5 +0.000039 +59415.352144 +194.0 +0.000113 +59765.353549 +362.0 +0.000092 +59034.124636 +11.0 +0.000032 +59416.426295 +194.5 +0.000120 +59766.426020 +362.5 +0.000132 +59391.426417 +182.5 +0.004700 +59417.435270 +195.0 +0.000112 +59767.436725 +363.0 +0.000098 +59392.436550 +183.0 +0.000110 +59418.509214 +195.5 +0.000165 +59768.508944 +363.5 +0.000190 +Table A2. TESS observed times of minima of V994 Her B +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +59010.698099 +0.0 +0.000046 +59392.097175 +268.5 +0.000104 +59418.259993 +287.0 +0.000104 +59011.515257 +0.5 +0.000044 +59392.700110 +269.0 +0.000111 +59744.251213 +516.5 +0.000111 +59012.118117 +1.0 +0.000041 +59393.517972 +269.5 +0.000156 +59744.855372 +517.0 +0.000121 +59012.934955 +1.5 +0.000038 +59394.119951 +270.0 +0.000103 +59745.671247 +517.5 +0.000131 +59013.537998 +2.0 +0.000042 +59394.936970 +270.5 +0.000111 +59746.274246 +518.0 +0.000112 +59014.355223 +2.5 +0.000040 +59395.539982 +271.0 +0.000149 +59747.092093 +518.5 +0.000150 +59014.957744 +3.0 +0.000046 +59396.357565 +271.5 +0.000083 +59747.695762 +519.0 +0.000105 +59015.774811 +3.5 +0.000043 +59396.960021 +272.0 +0.000105 +59748.511739 +519.5 +0.000095 +59016.377966 +4.0 +0.000042 +59397.776849 +272.5 +0.000149 +59749.114405 +520.0 +0.000130 +59017.194853 +4.5 +0.000039 +59398.379989 +273.0 +0.000117 +59749.931728 +520.5 +0.000125 +59017.798052 +5.0 +0.000041 +59399.197749 +273.5 +0.000093 +59750.535692 +521.0 +0.000125 +59018.614518 +5.5 +0.000035 +59399.800066 +274.0 +0.000158 +59751.351793 +521.5 +0.000074 +59019.218026 +6.0 +0.000041 +59400.617105 +274.5 +0.000128 +59751.954686 +522.0 +0.000141 +59020.035181 +6.5 +0.000040 +59401.219664 +275.0 +0.000102 +59752.771888 +522.5 +0.000125 +59020.638250 +7.0 +0.000041 +59402.037976 +275.5 +0.000125 +59753.376286 +523.0 +0.000101 +59021.455270 +7.5 +0.000042 +59402.640244 +276.0 +0.000103 +59754.192150 +523.5 +0.000128 +59022.058251 +8.0 +0.000047 +59403.457233 +276.5 +0.000109 +59754.794952 +524.0 +0.000153 +59023.478273 +9.0 +0.000047 +59404.059974 +277.0 +0.000109 +59755.612006 +524.5 +0.000110 +59024.295584 +9.5 +0.000043 +59405.480590 +278.0 +0.000133 +59757.032126 +525.5 +0.000142 +59024.898039 +10.0 +0.000047 +59406.297671 +278.5 +0.000102 +59757.634921 +526.0 +0.000116 +59025.715847 +10.5 +0.000044 +59406.900386 +279.0 +0.000163 +59758.452032 +526.5 +0.000112 +59026.318513 +11.0 +0.000048 +59407.718358 +279.5 +0.000116 +59759.055882 +527.0 +0.000183 +59027.136039 +11.5 +0.000051 +59408.320651 +280.0 +0.000110 +59759.872291 +527.5 +0.000123 +59027.738752 +12.0 +0.000044 +59409.137946 +280.5 +0.000172 +59760.475192 +528.0 +0.000116 +59028.555943 +12.5 +0.000043 +59409.740282 +281.0 +0.000106 +59761.291883 +528.5 +0.000129 +59029.158935 +13.0 +0.000039 +59410.558443 +281.5 +0.000107 +59761.895816 +529.0 +0.000116 +59029.975931 +13.5 +0.000043 +59411.160938 +282.0 +0.000165 +59762.712201 +529.5 +0.000123 +59030.578838 +14.0 +0.000042 +59411.977957 +282.5 +0.000112 +59763.315197 +530.0 +0.000126 +59031.395642 +14.5 +0.000044 +59412.580171 +283.0 +0.000096 +59764.132083 +530.5 +0.000106 +59031.998690 +15.0 +0.000045 +59413.397720 +283.5 +0.000171 +59764.735789 +531.0 +0.000140 +59032.815480 +15.5 +0.000042 +59414.000752 +284.0 +0.000117 +59765.552266 +531.5 +0.000117 +59033.418734 +16.0 +0.000045 +59414.817875 +284.5 +0.000108 +59766.155117 +532.0 +0.000126 +59034.235331 +16.5 +0.000040 +59415.420146 +285.0 +0.000182 +59766.972072 +532.5 +0.000101 +59034.838698 +17.0 +0.000044 +59416.238006 +285.5 +0.000118 +59767.575534 +533.0 +0.000120 +59390.677560 +267.5 +0.000177 +59416.840945 +286.0 +0.000127 +59768.392581 +533.5 +0.000089 +59391.279999 +268.0 +0.000108 +59417.658130 +286.5 +0.000136 +MNRAS 000, 1–?? (2023) + +V994 Herculis: A Triply Eclipsing Sextuple +15 +Table A3. TESS observed times of minima of V994 Her C +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +Eclipse Time +Cycle +std. dev. +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +BJD-2400000 +no. +(d) +59011.124177 +0.0 +0.000157 +59392.220634 +194.5 +0.002012 +59418.829817 +208.0 +0.000596 +59011.953279 +0.5 +0.000895 +59393.348483 +195.0 +0.000439 +59744.209501 +374.0 +0.001378 +59013.083475 +1.0 +0.000176 +59394.181279 +195.5 +0.001782 +59746.169255 +375.0 +0.000467 +59013.916603 +1.5 +0.000925 +59395.308080 +196.0 +0.000453 +59746.993487 +375.5 +0.002726 +59015.045185 +2.0 +0.000170 +59396.130606 +196.5 +0.002677 +59748.129163 +376.0 +0.000505 +59015.878338 +2.5 +0.000773 +59397.270147 +197.0 +0.000509 +59748.960592 +376.5 +0.002910 +59017.005797 +3.0 +0.000173 +59398.082565 +197.5 +0.007348 +59750.091678 +377.0 +0.000459 +59017.835692 +3.5 +0.000770 +59399.229029 +198.0 +0.000449 +59750.911553 +377.5 +0.016446 +59018.965311 +4.0 +0.000174 +59401.187681 +199.0 +0.000375 +59752.050606 +378.0 +0.000432 +59019.796083 +4.5 +0.001014 +59402.019831 +199.5 +0.007591 +59752.873428 +378.5 +0.057992 +59020.925121 +5.0 +0.000169 +59403.148862 +200.0 +0.000397 +59754.009831 +379.0 +0.000405 +59021.753129 +5.5 +0.001266 +59403.972393 +200.5 +0.003418 +59754.822117 +379.5 +0.003540 +59023.720603 +6.5 +0.001604 +59405.927779 +201.5 +0.004661 +59755.972515 +380.0 +0.000694 +59024.845884 +7.0 +0.000172 +59407.069016 +202.0 +0.000706 +59757.931960 +381.0 +0.000433 +59025.682738 +7.5 +0.001401 +59407.894095 +202.5 +0.057096 +59758.756173 +381.5 +0.005398 +59026.806343 +8.0 +0.000171 +59409.028342 +203.0 +0.000656 +59759.890913 +382.0 +0.000483 +59027.640483 +8.5 +0.000943 +59409.856410 +203.5 +0.002463 +59760.706877 +382.5 +0.010731 +59028.767693 +9.0 +0.000186 +59410.989522 +204.0 +0.000463 +59761.850436 +383.0 +0.000442 +59029.597642 +9.5 +0.001409 +59411.812507 +204.5 +0.003274 +59762.669061 +383.5 +0.003683 +59030.727133 +10.0 +0.000175 +59412.949612 +205.0 +0.000503 +59763.811861 +384.0 +0.000536 +59031.559275 +10.5 +0.000946 +59414.908974 +206.0 +0.000444 +59764.636994 +384.5 +0.001506 +59032.688509 +11.0 +0.000180 +59415.758940 +206.5 +0.002578 +59765.771756 +385.0 +0.000524 +59033.522849 +11.5 +0.001363 +59416.870114 +207.0 +0.000460 +59767.732579 +386.0 +0.000409 +59034.648313 +12.0 +0.000173 +59417.696368 +207.5 +0.001980 +59768.560018 +386.5 +0.007387 +59391.388607 +194.0 +0.000580 +MNRAS 000, 1–?? (2023) + +16 +Zasche et al. +Table A4. New, unpublished eclipse times of V994 Her for binaries A, B, and C. +Eclipse Time +std. dev. +Pair +Type +Reference/ +Eclipse Time +std. dev. +Pair +Type +Reference/ +BJD-2400000 +(d) +[A/B/C] +[P/S] +Observer +BJD-2400000 +(d) +[A/B/C] +[P/S] +Observer +57843.57013 +0.00045 +A +S +R.U. +59802.47556 +0.00045 +B +S +R.U. +57855.59639 +0.00073 +B +S +R.U. +59804.49687 +0.00048 +B +P +R.U. +57902.45775 +0.00069 +B +S +R.U. +59815.34773 +0.00192 +A +P +R.U. +57916.48823 +0.00185 +A +S +R.U. +59816.42734 +0.00148 +A +S +R.U. +57917.49528 +0.00029 +A +P +R.U. +59817.43375 +0.00030 +A +P +R.U. +57940.40624 +0.00014 +A +P +R.U. +59817.43433 +0.00036 +A +P +FRAM +57946.48532 +0.00097 +B +S +R.U. +59818.50532 +0.00052 +A +S +FRAM +57968.38272 +0.00099 +B +P +R.U. +59121.62037 +0.00054 +A +P +G.P. +58232.53401 +0.00037 +B +P +R.U. +52509.78153 +0.00211 +B +P +S.D. & U.M. +58257.48416 +0.00075 +B +S +R.U. +52510.52766 +0.00651 +B +S +S.D. & U.M. +58290.38397 +0.00021 +A +P +R.U. +52702.91006 +0.00275 +B +P +S.D. & U.M. +58343.53778 +0.00079 +A +S +R.U. +52703.66135 +0.00066 +B +S +S.D. & U.M. +58387.30392 +0.00032 +B +P +R.U. +52810.86153 +0.00098 +B +P +S.D. & U.M. +58565.62065 +0.00055 +B +S +R.U. +52811.60611 +0.00190 +B +S +S.D. & U.M. +58570.62894 +0.00082 +A +S +R.U. +53096.26742 +0.00274 +B +P +S.D. & U.M. +58571.63882 +0.00069 +A +P +R.U. +53097.02123 +0.00193 +B +S +S.D. & U.M. +58593.55169 +0.00068 +A +S +R.U. +53171.53199 +0.00181 +B +P +S.D. & U.M. +58614.50216 +0.00135 +B +P +R.U. +53172.28521 +0.00092 +B +S +S.D. & U.M. +58667.47186 +0.00029 +A +P +R.U. +52692.65792 +0.00048 +A +P +S.D. & U.M. +58689.37993 +0.00062 +A +S +R.U. +52715.57549 +0.00129 +A +P +S.D. & U.M. +58957.55352 +0.00149 +B +S +R.U. +53139.55537 +0.00089 +A +S +S.D. & U.M. +58991.45731 +0.00075 +A +S +R.U. +52509.33511 +0.00026 +A +P +S.D. & U.M. +59023.47973 +0.00039 +B +P +R.U. +52510.40944 +0.00040 +A +S +S.D. & U.M. +59043.52943 +0.00062 +A +S +R.U. +52735.40298 +0.00091 +A +S +S.D. & U.M. +59040.37413 +0.00091 +A +P +R.U. +52825.97834 +0.00148 +A +P +S.D. & U.M. +59040.52213 +0.00085 +B +P +R.U. +52827.05749 +0.00202 +A +S +S.D. & U.M. +59089.36019 +0.00045 +A +S +R.U. +53102.05902 +0.00052 +A +S +S.D. & U.M. +59102.40213 +0.00043 +B +S +R.U. +53171.80904 +0.00057 +A +P +S.D. & U.M. +59343.51031 +0.00033 +A +S +R.U. +52687.49036 +0.00950 +A +S +S.D. & U.M. +59349.50020 +0.00025 +B +S +R.U. +59831.47733 +0.00027 +B +P +FRAM +59361.46286 +0.00064 +B +P +FRAM +59840.35162 +0.00046 +A +P +R.U. +59367.43420 +0.00017 +A +P +R.U. +54669.45443 +0.00100 +C +P +MR13 +59369.51977 +0.00030 +A +P +M.M. +59101.29077 +0.00531 +C +P +R.U. +59392.43409 +0.00019 +A +P +R.U. +59150.29198 +0.00134 +C +P +R.U. +59416.42568 +0.00040 +A +S +FRAM +59152.25463 +0.00389 +C +P +R.U. +59419.52306 +0.00038 +A +P +R.U. +59279.67396 +0.00770 +C +P +R.U. +59425.36003 +0.00070 +B +P +R.U. +59332.58531 +0.00473 +C +P +R.U. +59463.27078 +0.00245 +A +P +R.U. +59338.46323 +0.00633 +C +P +R.U. +59465.35947 +0.00037 +A +P +R.U. +59436.47808 +0.00354 +C +P +R.U. +59677.51573 +0.00055 +B +S +R.U. +59438.44039 +0.00257 +C +P +R.U. +59679.53400 +0.00044 +B +P +R.U. +59497.24210 +0.01257 +C +P +R.U. +59717.43638 +0.00055 +A +P +R.U. +59675.60538 +0.00101 +C +P +R.U. +59718.50933 +0.00034 +A +S +R.U. +59681.49018 +0.00107 +C +P +R.U. +59742.43700 +0.00079 +A +P +R.U. +59779.49300 +0.00100 +C +P +R.U. +59767.43486 +0.00021 +A +P +R.U. +52819.10680 +0.00232 +C +P +S.D. & U.M. +59767.57676 +0.00069 +B +P +R.U. +52693.65303 +0.00273 +C +P +S.D. & U.M. +59775.49098 +0.00054 +B +S +R.U. +59828.49557 +0.00325 +C +P +FRAM +59787.45377 +0.00079 +B +P +R.U. +59830.45265 +0.00304 +C +P +FRAM +59791.42428 +0.00071 +A +S +R.U. +Notes: G.P. = Gerald Persha, see http://var2.astro.cz ; S.D. & U.M. = S. Dallaporta & U.Munari; R.U. = R.Uhlař; M.M. = M.Mašek; +MR13 = Martín-Ruiz et al. (2013) +MNRAS 000, 1–?? (2023) + diff --git a/aNFRT4oBgHgl3EQfQTdr/content/tmp_files/load_file.txt b/aNFRT4oBgHgl3EQfQTdr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fb4198f478f2b4fa8defae78a66259a75fde731 --- /dev/null +++ b/aNFRT4oBgHgl3EQfQTdr/content/tmp_files/load_file.txt @@ -0,0 +1,2662 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf,len=2661 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 V994 Her: A Unique Triply Eclipsing Sextuple Star System P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Zasche1 ⋆, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Borkovits2,3,4,5,6, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Jayaraman7 , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Rappaport7, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Brož1, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Schwengeler9, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Pál4, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Mašek10, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Howell11, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Dallaporta12, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Munari13, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Gagliano14, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Jacobs15, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Kristiansen16,17, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' LaCourse18, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Omohundro19, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Terentev20, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Vanderburg21, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Henzl22,23, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Powell24, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Kostov24,25,26 1 Charles University, Faculty of Mathematics and Physics, Astronomical Institute, V Holešovičkách 2, Praha 8, 180 00, Czech Republic 2 Baja Astronomical Observatory of Szeged University, H-6500 Baja, Szegedi út, Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 766, Hungary 3 ELKH-SZTE Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út, Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 766, Hungary 4 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, H-1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary 5 ELTE Gothard Astrophysical Observatory, H-9700 Szombathely, Szent Imre h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 112, Hungary 6 MTA-ELTE Exoplanet Research Group, H-9700 Szombathely, Szent Imre h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 112,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Hungary 7 MIT Department of Physics and and MIT Kavli Institute for Astrophysics and Space Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' USA 8 Private Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Pohoří 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 254 01 Jílové u Prahy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Czech Republic 9 Citizen Scientist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Planet Hunter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Bottmingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Switzerland 10 FZU - Institute of Physics of the Czech Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Na Slovance 1999/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' CZ-182 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Praha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Czech Republic 11 NASA Ames Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Moffett Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' CA 94035,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' USA 12 ANS Collaboration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' c/o Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 36012 Asiago (VI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Italy 13 INAF Astronomical Observatory of Padova,' metadata={'source': 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+page_content=' Czech Republic 23 Variable Star and Exoplanet Section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Czech Astronomical Society,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Fričova 298,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 251 65 Ondřejov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Czech Republic 24 NASA Goddard Space Flight Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 8800 Greenbelt Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MD 20771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' USA 25 SETI Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 189 Bernardo Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Suite 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Mountain View,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' CA 94043,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' USA 26 GSFC Sellers Exoplanet Environments Collaboration Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We report the discovery with TESS of a third set of eclipses from V994 Herculis (TIC 424508303), previously only known as a doubly-eclipsing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The key implication of this discovery and our analyses is that V994 Her is the second fully-characterized (2+2) + 2 sextuple system, in which all three binaries eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In this work, we use a combination of ground-based observations and TESS data to analyze the eclipses of binaries A and B in order to update the parameters of the inner quadruple’s orbit (with a derived period of 1062 ± 2 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The eclipses of binary C that were detected in the TESS data were also found in older ground-based observations, as well as in more recently obtained observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The eclipse timing variations of all three pairs were studied in order to detect the mutual perturbations of their constituent stars, as well as those of the inner pairs in the (2+2) core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' At the longest periods they arise from apsidal motion, which may help constraining parameters of the component stars’ internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We also discuss the relative proximity of the periods of binaries A and B to a 3:2 mean motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This work represents a step forward in the development of techniques to better understand and characterize multiple star systems, especially those with multiple eclipsing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Key words: binaries: eclipsing – binaries: close – stars: individual: (TIC 424508303, V994 Her), sextuple system ⋆ E-mail: zasche@sirrah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='troja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='cz © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='13521v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='SR] 31 Jan 2023 2 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 1 INTRODUCTION Multiple star systems consisting of three or more stars are estimated to make up at least 30% of binaries, based on a statistical analysis of Kepler data in Borkovits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, only two thousand have been observed in detail, and the number of systems known with multiplicities higher than 5 is ≲ 50 (Tokovinin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Understanding these high- multiplicity systems is important, as they can shed new light on many of the open questions that remain in currently- accepted models of stellar formation and provide insight into the dynamical interactions of multiple stars (Aarseth & Mardling 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' V994 Herculis (V994 Her, TIC 424508303) is a bright, well- studied quadruple system, with two known eclipsing binary components (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' At the time, this was the first known doubly-eclipsing quadruple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The component stars are bright and massive, and the periods of the two eclips- ing binary components have been well-constrained: binary A consists of a B8V and an A0V star, with a period of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='083 days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' binary B consists of an A2V and an A4V star, with a period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='420 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These stars are young and occupy a position near the zero-age main sequence (ZAMS) on the Hertzsprung-Russell (H-R) diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Martín-Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2013) initially postulated, based on analyses of photometric data, that this system could har- bor another eclipsing binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, because the quality of their data was rather poor, with relatively low photomet- ric precision, their results were not conclusive enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' With the advent of high-precision space-based survey missions such as the Transiting Exoplanet Survey Satellite (TESS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2015), we are able to conclusively confirm the presence of a third set of eclipses using the TESS light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Table 1 contains basic information about V994 Her and a nearby visual companion (separated by ≈ 1′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Zasche & Uhlař (2016) were the first to accurately con- strain the period of the inner binaries’ (A and B) revolution about their common center of mass (≃ 1060 days in their study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Additionally, they argued that TIC 1685970000, a faint (mV ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8) neighbor some 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1′′ away from V994 Her, is also gravitationally bound to the main quadruple, making it one of the few known quintuple star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This putative close companion has been observed many times since its dis- covery as a visual double in 1831, and these measurements have been catalogued in the Washington Double Star Catalog (WDS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The WDS calls the known quadru- ple a “primary star,” and the fainter companion a “secondary star.” However, any physical connection between these two visually-close objects has not yet been conclusively proven;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' further follow-up and analyses (such as those in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1) can resolve this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In this paper, we introduce V994 Her as a bona fide triply eclipsing sextuple star system, which we identified using TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In addition to the known set of two binaries, the system also consists of a third binary of period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='96 days, and we demonstrate that the visual companion listed in the WDS catalog is likely gravitationally bound to the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In Section 2, we describe all available observational data and how they were prepared and used for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Then, Section 3 provides detailed modelling of the available data, while 4 discusses the results of our modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Finally, in Sec- tion 5 we discuss the possible architecture of the whole system Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Archival properties of the V994 Her visual double star Name V994 Her TYC 2110-1170-2 TIC 424508303 TIC 1685970000 RA (J2000, deg) 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='941222 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='941246 Dec (J2000, deg) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='697407 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='697757 TESS a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='017 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='3949 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 Ba 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='136 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='024 V a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='023 Ja 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='948 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='019 Ha 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0036 Ka 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='989 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='023 W1b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='07 W2b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='838 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 W3b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='903 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='018 W4b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='067 Gc 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0966 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8761 Gc Bp 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9898 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9842 Gc Rp 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9358 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9869 Parallaxc (mas) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='43639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='08394 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='48065 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='09849 PMc (RA, mas/yr) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5770 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0701 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8802 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1275 PMc (Dec, mas/yr) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0765 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1568 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0834 Notes: Magnitudes are from (a) TIC-8 catalog (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (b) WISE point source catalog (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2021), (c) Gaia DR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' PM stands for proper motion (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Some parameters for the visual companion TIC 1685970000 are difficult to come by, as the brighter primary star is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1” away from it, making measurements difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' that we infer from our findings and comment on the proxim- ity of the inner 2+2 component (binaries A and B) to their mutual 3:2 mean motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2 OBSERVATIONS OF V994 HER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 TESS Observations V994 Her was observed by TESS during Year 2 in Sector 26 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', June 2020), and during Year 4 in Sectors 40 and 53 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' July 2021 & June 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In Sector 26, this star was observed at 2-minute cadence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' this light curve was prepro- cessed and detrended by the Science Processing Operations Center (SPOC) pipeline (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2016), which is par- tially based on that used for Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The detrended SPOC light curve from Sector 26 is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For the Year 4 observations, however, only the full-frame image (FFI) data (at 10-minute cadence) are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These data were processed using the convolution-based differential im- age analysis methods of the fitsh package (Pál 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' V994 Her’s triply eclipsing nature was identified both algorithmi- cally and through a visual survey1 of all stars brighter than 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 mag in the TESS FFIs (for more information on the latter initiative, see Kristiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 Three methods for disentangling Using the TESS data, we applied three different methods to disentangle the combined light curve into the three compo- nent eclipsing signals: the time-domain iterative disentangle- 1 This search makes use of the LcTools desktop application (Schmitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2019) to view and study light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) V994 Herculis: A Triply Eclipsing Sextuple 3 2010 2014 2018 2022 2026 2030 2034 Time (BTJD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 Normalized Flux 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='05 Secondary Eclipse (C) Primary Eclipse (C) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The TESS Sector 26 light curve of TIC 424508303, aka V994 Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The x-axis is plotted in Barycentric TESS Julian Day (BTJD), which corresponds to BJD–2457000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The main plot shows the full 25-day light curve, which includes multiple eclipses from the previously known eclipsing binaries A and B (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Zasche & Uhlař 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' It also contains relatively shallow eclipses from the new binary C, discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The inset panel shows a zoom-in on a roughly 9-d segment of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Three clearly visible primary eclipses of the C binary are overplotted in blue, while the eclipse lost in a deeper eclipse from the A binary is indicated with a blue line above its expected location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The (shallow) secondary eclipses of the C binary are overplotted in gold, with the gold line at BTJD ∼ 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 indicating a secondary eclipse that is lost in one of the deeper eclipses from the “main” quadruple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' ment method, the Fourier-decomposition method, and the it- erative phenomenological model method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Results for all three methods are plotted side-by-side in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' First, we used the method of time-domain iterative dis- entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This technique is a powerful tool for separat- ing the light curves of strongly blended targets, and was de- scribed in detail in Section 3 of Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2021), where it was applied for the first time to disentangle the blended light curves of three eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' To verify our results and compare the three methods, we used two other methods to disentangle the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The second one is the Fourier-based iterative method, which was also described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 of Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Such a technique is suitable for these data because all the signals of interest are strictly periodic over at least one sector of TESS data, wherein movement on the longer outer orbit can be ne- glected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The third and final method is based on iteratively fitting the individual pairs with their respective phenomeno- logical models and then subtracting these from the overall light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' After a few (usually two to five) iterative steps, a shape for the eclipsing light curve of the C pair was clearly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The method itself and the code used here are de- scribed in the Appendix of Pejcha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Apart from the three eclipsing signals, the light curve also exhibits an additional pulsation-like oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Such a vari- ation shows a periodicity of (PA − PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This extra feature is apparently not present in the Fourier-disentangled light curve, as well as in the phenomenologically-disentangled one (see the bottom panels of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This may be due to the subtraction of this signal as part of the disentanglement pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Unfortunately, we have not yet been able to come up with a coherent astrophysical explanation of this signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We found the first method of time-domain iterative disen- tanglement as the most suitable for a subsequent analysis of the individual light curves, which is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This is mainly due to problematic fitting of outside-eclipse parts of the light curves by the methods 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' To derive the precise times of eclipses of each binary, we used the result of the time-domain iterative disentanglement method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These were calculated for each binary after subtrac- tion of the light curves of the other two pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Eclipse times of each binary, as observed in TESS, are presented in Tables A1, A2, and A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 Ground-based photometric Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 Baja Astronomical Observatory, Hungary (2007) V994 Her was observed with the 50-cm f/6 modified Cassegrain Baja Astronomical Robotic Telescope (BART-1), located at the Baja Astronomical Observatory in Hungary, on 40 nights between 18 June 2007 and 9 October 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The observations were carried out with a 4096×4096 Apogee Alta U16 CCD camera, using a standard Johnson V filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 4 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0 0.' metadata={'source': 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method 3, pair A (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='083 d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='95 1 Phase Flux method 1, pair B (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='420 d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01 Phase Flux method 1, pair C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='960 d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01 Phase Flux method 2, pair C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='960 d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01 Phase Flux method 3, pair C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='960 d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The disentangled and folded light curves of all three eclipsing binaries A, B, and C using different approaches: the time-domain iterative disentanglement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' method 1), the Fourier-decomposition (method 2), and the iterative phenomenological model methods (method 3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For details see the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The original goal of this photometric monitoring of V994 Her was to prove and publish for the first time the previously- unknown doubly eclipsing nature of this system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' however, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2008) independently discovered and characterized this system’s true nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, our team at the time chose to not further analyze the data and simply published the de- rived times of minima in Borkovits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, we make use of this archival photometric data set in the present work, as it is especially useful for an additional constraining of the apsidal advance rates of binaries A and B through the complete lightcurve fittings and, also for checking the con- stancy (or variability) of the eclipse depths within a one and half decade-long interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 Additional observations V994 Her was monitored over several dozens of nights by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' at his private observatory in the Czech Republic, as well as remotely from northern Italy using three different telescopes: a 34-mm refractor, a 150-mm reflector, and a 200-mm re- flector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Some of these observations were obtained using fil- tered photometry (usually with R or I filters), while others were carried out without any filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Due to the different in- strumental setups of these instruments, the comparison stars were different for each telescope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' however, they were always chosen to be adequately close to the target and of a similar spectral type in order to minimize the effect of differential ex- tinction during the nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Additionally, four more nights of data were obtained by the 250-mm F/(Ph)otometric Robotic Atmospheric Monitor (FRAM) telescope CTA-N, located on the island of La Palma, Spain (Prouza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We also have data from one night of observations by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' at his private observatory in the Czech Republic, using a 200-mm reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' From the combination of these datasets, more than 40 new times of eclipses for pair A were derived, and more than 30 for pair B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Several new estimates for pair C were also calculated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' however, these are of lower quality due to the sig- nificantly lower photometric amplitude of its variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Between 2002 June 10 and 2004 July 14, a total of 1170 measurements in V -band and 653 in B-band were collected for V994 Her by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', using a 28-cm telescope lo- cated in Cembra (Trento, Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This telescope was equipped with an Optec SSP-5 photoelectric photometer and Johnson B and V filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The comparison and check stars were, respec- tively, HIP 89975 (V = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='978 mag, B − V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='095 mag) and HIP 90637 (V = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='862 mag, B − V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='099 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These stars are nearly identical in B − V color to V994 Her and are located nearby on the sky (≤2◦ angular separation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' From these data, we were also able to derive several times of eclipses for both the A & B pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Moreover, we were able to derive a rough value for the times of eclipse for pair C, which allowed us to significantly improve our estimate of its orbital period due to the increased time coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' All the previously-unpublished eclipse times are given in Table A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The minima presented in this work for the first time, as well as the previously-published ones, were used for a final fit of the data over the whole interval (covering more than 30 years now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This is shown in Figure 3 for all three pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='3 Other Catalogs We queried the WDS Catalog for archival data on V994 Her and its nearby visual companion (TIC 1685970000), with MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) V994 Herculis: A Triply Eclipsing Sextuple 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='04 1990 1995 2000 2005 2010 2015 2020 2025 V994 Her A T0=2459011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2085 P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0832649d ETV [in days] Calendar Year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 50000 55000 60000 Residual BJD - 2400000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='14 1990 1995 2000 2005 2010 2015 2020 2025 V994 Her B T0=2459010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6980 P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4200401d ETV [in days] Calendar Year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 50000 55000 60000 Residual BJD - 2400000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 1990 1995 2000 2005 2010 2015 2020 2025 V994 Her C T0=2459011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1245 P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='960110d ETV [in days] Calendar Year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 50000 55000 60000 Residual BJD - 2400000 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Eclipse timing variations (ETVs) of V994 Her collected over the past three decades, with the TESS eclipses included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The top, middle and bottom panels show the ETVs for binary A, B and C, in that order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' the red and blue points denote the primary and secondary eclipses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The red and blue curves are the photodynamical fitting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The “divergence” of the ETVs for the primary and secondary eclipses are due to apsidal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' measurements spanning from 1831 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This data con- sisted of position angles and separations for the system be- tween these years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Additionally, we calculated the position angle and separation for the visual double using data from Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These data were used in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 to investigate whether or not the visual double star is gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' System parameters derived from speckle imaginga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Parameter Value obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' date [JD] 2459710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='018 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='001 position angle [deg] 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 separation [arc sec] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01 δV [mag]b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 δI [mag]b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 Notes: (a) Observations made at 562 nm and 832 nm with the ‘Alopeke speckle interferometric imager mounted on the Gemini North 8-m telescope (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (b) Difference in magnitude between Image 1 (containing binaries A and B) and Image 2 (likely hosting binary C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 Speckle observation V994 Her was observed on 10 May 2022 using the, ‘Alopeke speckle interferometric imager mounted on the Gemini North 8-m telescope (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' ‘Alopeke provides simulta- neous speckle imaging in two bands (562 nm and 832 nm), with output data products including a reconstructed image and derived parameters for any detected close companion stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Three sets of 1000 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06 sec exposures were collected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' these underwent Fourier analysis in the standard reduction pipeline (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Figure 4 shows the image around V994 Her, with a bright component to the South (hereafter ‘Image 1’) which hosts binaries A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The fainter image to the North (hereafter ‘Image 2’) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06′′ away, and we believe that this image hosts binary C, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' At the 290 pc distance to V994 Her (derived from the Gaia DR3 parallax), this cor- responds to a spatial separation of ∼ 307 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The middle panel shows a zoom-in around Image 1, revealing that there are no resolved components within it, down to a limiting res- olution of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Both the image of the quadruple system to the South (Image 1), and binary C (likely residing in Image 2) remain unresolved into their component parts—respectively, either the A and B binaries, and the primary and secondary star in binary C—as these components are separated on the sky by less than our 20 mas nominal angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This value is the Gemini optical diffraction limit when Nyquist sampled with 2×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01′′ pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Our derived 5-σ contrast curves for this observation, for both the 562 nm and 832 nm images, are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' These curves will be further discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 as we attempt to rule out the possibility that binary C might actually be located in Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The system properties gleaned from the speckle observa- tions are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 3 PHOTODYNAMICAL MODELING We carried out a joint photodynamical modeling in which we combined the three sectors of TESS data alongside the 2007 V band Baja light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As part of this, we also mod- eled the eclipse timing variation (ETV) curves of all three binaries, the radial velocity (RV) points of binaries A and B obtained by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2008), and the net stellar spectral energy distributions (SEDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' To prepare for this analysis, we improved the Lightcurvefactory software package to al- low it to handle hierarchical configurations of (2+2)+2 stars in their entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Specifically, the updated code calculates the MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 6 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Speckle imaging of V994 Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' North is up and East is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Top panel: 832 nm image of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5′′ ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5′′ region near V994 Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We define the brighter feature to the South as ‘Image 1,’ which contains binaries A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We label the ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='7 magnitude fainter object to the North as ‘Image 2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Middle panel: Same as top panel but zoomed in around Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Each pixel is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01′′ in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Bottom panel: 5-σ confidence level contrast curves (obtained at 562 nm and 832 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The image spans angular scales from the diffraction limit, near 20 mas, out to ∼ 1′′, the approximate end of speckle coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The dotted black lines mark the detectable separation distance of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06′′ of a source that is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 magnitudes fainter than Image 1 itself (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', the approximate brightness of binary C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' revolutions of the six bodies on their three inner orbits, the middle orbit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', of the quadruple), and the outer orbit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', of the sextuple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' All five orbits may be considered either to be purely Keplerian or, for tight systems, Lightcurvefac- tory is able to take into account the mutual perturbations of the constituent stars with numerical integration of the or- bital motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Moreover, any combinations of two-body or multiple-body eclipses are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The updated code does not require the disentangling of the three eclipsing bi- nary light curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' rather, they can be modeled in their ob- served, blended form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Apart from this improvement, the software package is functionally iden- tical to that described in previous work (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Borkovits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For the specific case of V994 Her, we find that bina- ries A and B form a relatively wide 2+2 quadruple system (PA−B/PB > PA−B/PA > 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As a result, the gravitational perturbations of the binary components in the 2+2 quadruple are small and can be described by simple Keplerian orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Therefore, we use a simple analytic Keplerian formalism in order to calculate the stellar positions at any given time, with the slight empirical modification of considering, for all three binaries, a constant apsidal advance rate ( ˙ωA,B,C) and ref- erence values for the argument of periastron (ωA,B,C) at a specific epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' A physical interpretation of the apsidal mo- tion is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In our joint photodynamical analysis, we optimized the following parameters using a Markov Chain Monte Carlo (MCMC) method: (i) Orbit-related parameters: – For all four orbits (three eclipsing pairs and the quadruple A-B): The components of the eccentricity vec- tors at epoch t0: (e sin ω)A,B,C,A−B, (e cos ω)A,B,C,A−B, and the inclinations relative to the plane of the sky: iA, iB, iC, iA−B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' – For the A–B orbit: the period PA−B and the periastron passage time τA−B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' – For the three eclipsing pairs: the (constant) apsidal advance rates: ˙ωA,B,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (ii) Stellar parameters: – Six mass-related parameters: the masses of the pri- maries (mAa,Ba,Ca), and the mass ratios of the three EBs (qA,B,C), – The metallicity of the system ([M/H]), – The (logarithmic) age of the six coeval stars (log τ), – The interstellar reddening E(B − V ), and – The “extra light” contamination (ℓ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' A couple of other parameters were constrained instead of being adjusted or held constant during our analyses: (i) Orbits: – The sidereal orbital periods of the inner binaries (PA,B,C) and their respective orbital phases (derived using the time of an arbitrary primary eclipse) were constrained internally through the ETV curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' – The systemic radial velocity of the whole sextuple sys- tem (γ) is calculated a posteriori at the end of each trial step by minimizing the value of χ2 RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Note that the (2+2)+2 mode of Lightcurvefactory MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) Image 2 V994 Her 832 nm Image 1V994 Her Image 1 832 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2"0 V994 Her Contrast Curves 7 562 nm 2 832 nm 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 1 Angular Separation (arc sec)V994 Herculis: A Triply Eclipsing Sextuple 7 requires the orbital elements of the outermost (AB-C) orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In the present case, this orbit is completely unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We do know, however, that it must be so wide that we do not expect any observable variations in the positions of the six stars arising from their motion along this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, we chose the elements of this outmost orbit arbitrarily;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' we use a circular orbit seen face-on with a period of ∼ 30 kyr with its parameters kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (ii) Stars: – The radii and temperatures of the six stars were cal- culated with the use of three linear interpolations from the precomputed 3D PARSEC grids (the dimensions were metal- licity, logarithmic age, and stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' – The distance of the system (needed for the SED fit- ting) was calculated a posteriori at the end of each trial step, by minimizing the value of χ2 SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For a detailed ex- planation of this process, see Borkovits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The atmospheric parameters of the stars were handled in a similar manner as in our previous photodynamical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We utilized a logarithmic limb-darkening law (Klinglesmith & Sobieski 1970), for which the passband-dependent linear and non-linear coefficients were interpolated in each trial step via the tables from the original version of the Phoebe software (Prša & Zwitter 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We set the (constant) gravity darken- ing exponents for five radiative stars to β = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' for the coolest, solar-like component Cb, however, we used β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='32, which is in line with the classical model of Lucy (1967) and is valid for convective stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Prior to conducting our analysis, we performed some fur- ther preparatory steps on the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' First, after disen- tangling the three eclipsing binaries as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1, we found that the residual TESS light curves contained oscil- lations with an amplitude of approximately 2% and a char- acteristic period of (PA − PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We removed this oscillation from the light curves before performing the full photodynami- cal analysis by subtracting the final residual light curve of the iterative disentanglement process (which contained this peri- odic variability) from the original TESS light curves for each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Second, for the sake of equal sampling across sectors, we binned the 2-min sector 26 TESS light curve to 10-min bins, identical to the cadence time of the sector 40 and 53 FFI light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We also binned the 2007 Baja photometry to 10-min bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Third, we noticed that the eclipse depths of all three binaries in sector 26 were deeper by a few percent than the corresponding eclipses in the sectors 40 and 53 data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' the depths in the latter two sectors were similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As a result, we assume that in sector 26, the ratio of contaminating light is somewhat lower than in the other two sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, for the sector 26 light curve, we adjusted the amount of contam- inating light independent of sectors 40 and 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The effect of slightly different flux contamination for the same star in data from different TESS sectors has been studied previously in the literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Plachy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=') The median values and their 1-σ uncertainties (derived from the MCMC calculation) for the orbital and physical parameters of the sextuple system, as well as some derived quantities, are tabulated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Furthermore, a compar- ison of the observed and model light curves are plotted in Figure 5, while a similar comparison for the ETV curves is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='05 Normalized Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02 59408 59410 59412 Residual Flux BJD - 2400000 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' A section of the TESS sector 40 light curve (blue points) after the removal of the oscillations with period (PA −PB), plotted with the photodynamically fitted model light curve (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We also plot the original light curve, including the oscillations with period PA − PB, using gray points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The short vertical solid and dashed lines along the x-axis, colored red, blue and green, denote the mid- eclipse times of the primary and secondary eclipses of binaries A, B and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The residual curves are shown below the main light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Note that Table 3 presents the absolute physical parameters for the C binary and both its compo- nents, despite the fact that we do not have any directly ob- served radial velocities for this pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, we have the RVs of both the A and B binaries as well as changes of the RVs on their mutual orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We can consequently derive the properties of this binary by using the Lightcurvefactory code to combine the light curve modeling of the C binary and the SED of the overall system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 4 THE FINAL PARAMETERS Our thorough modeling of the system also yields the position of each star in the H-R diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Due to the fact that the age of the system was also taken as a free parameter (under the assumption that all six stars are coeval), we can characterize its evolutionary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' From the calculated value of the sys- tem’s logarithmic age (presented in Table 3), the system is rather young and therefore located close to the ZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This is in agreement with the fact that all of the orbits are slightly eccentric, so the circularization process is still ongoing (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Claret & Cunha 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Only a few sextuple systems have well-constrained param- eters, including their masses and orbital elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' as a re- sult, it is not very easy to compare V994 Her with oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Interestingly, the recent analysis of the sextuple system TIC 168789840 (Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2021) revealed a vastly differ- ent configuration, wherein all three binary pairs have similar mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In V994 Her, our analysis suggests that all three components have very different mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Perhaps V994 Her is more similar to the well-known Castor system, which has a similar architecture, with its component binaries having very different mass ratios (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Tokovinin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The whole system is plausibly close to a co-planar config- uration, given the inclination angles in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, to MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 8 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Median values of the parameters from the joint spectro-photodynamical analysis of (i) all three EB light curves, (ii) both sets of radial velocities from the SB2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', the quadruple consisting of binaries A and B), (iii) all three sets of ETVs, and (iv) joint SED and PARSEC evolutionary tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Parameter Binary A Binary B Binary C A–B orbit Pa [days] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0832039+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0000042 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0000039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4200981+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0000033 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00000040 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='23 [M/H] [dex] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='073+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='042 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='056 E(B − V ) [mag] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='050+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='014 (MV )c tot −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='07 distance [pc] 274+6 −6 Notes: (a) Calculated only from the sin i terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (b) Polar radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (c) Interpolated from the PARSEC isochrones derive its true orbital architecture we would also need to cal- culate the values for the longitude of the ascending node Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' To do so, one would need to derive a precise interferometric orbit, which is not available to us currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Once this in- formation is obtained, we can speculate whether or not the system can exhibit Kozai-Lidov cycles (Kozai 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Lidov 1962);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' however, these may be halted anyway by rapid pre- cession of the pericenters of the component binaries (Table 3 and Vokrouhlický 2016, for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 Apsidal motion Given that we have multiple high-precision sets of eclipse times (see Tables A1, A2, A3, and A4), and that each binary has an eccentric orbit, we are able to derive apsidal motion rates for all three pairs in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We find that this rate is approximately a few degrees per year, suggesting that their apsidal advance is not at all negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In order to properly interpret these empirically fitted rates, we first have to sub- tract any contributions from the apsidal advance that can be accurately computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' First, we determine the relativistic contribution to the ob- served apsidal motion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Claret & Giménez 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Given the orbital and physical parameters in Table 3 we find this effect represents about 10%, 8%, and 6% of the total for the binaries A, B, and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Because these percent- ages are rather small, we can consider all three sub-systems of 50000 52000 54000 56000 58000 195 200 205 ωA [deg] 50000 52000 54000 56000 58000 165 170 175 ωB [deg] 50000 52000 54000 56000 58000 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 59 HJD−2400000 ωAB [deg] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Results of the numerical modeling of the orbits of bina- ries A and B and the quadruple AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Here, we show the long-term evolution of the arguments of periastron (without the tidal term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Pair A is plotted in black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' pair B, in red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' and their mutual orbit (A-B), in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' See section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) V994 Herculis: A Triply Eclipsing Sextuple 9 V994 Her as classical apsidal rotators, rather than relativistic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Because the inner 2+2 component of binaries A and B is not too wide, there also exists a classical (Newtonian) contri- bution to their apsidal motion, arising from mutual dynami- cal perturbations between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In order to estimate this effect, we ran a simulation using the N-body code developed by Brož (2017) and Brož et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For the sake of defi- niteness, we assumed a nearly-coplanar system configuration by imposing identical initial values of the nodal longitude for both the A and B orbits, with other orbital parameters taken from Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We found that such mutual perturbations in the A and B system account for another 12% and 5% of the total apsidal motion in the respective component (see Figure 6, which shows these contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The binary C is deemed to be distant enough from the A and B binaries so that we do not provide the classical apsidal contribution in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' With those two effects estimated and subtracted from the total values of the observed apsidal motions in the A, B and C systems, we can assume that the remainder is attributable to the effect of the stellar tidal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' From these rates of ˙ωtidal, one can usually derive the internal structure con- stants and compare them with theoretical ones from stellar evolution models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Claret 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, when compar- ing the results for pair B (which has the best coverage of its apsidal period, since it has the fastest rate), our resulting values deviate slightly from the predicted theoretical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The tidal contribution to the apsidal rate was found to be ˙ωtidal,B = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='20) deg/yr, yielding an internal struc- ture constant of log k2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='05, while the theoreti- cal models of Claret (2004) suggest that its value should be −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In order for the respective error intervals to overlap, one needs to have either larger uncertainties in the derived parameters, a slightly faster apsidal motion (of about 5%), subsynchronous rotation of the component stars (about 20% slower), or a combination of all three of these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' An- other way to account for this discrepancy is by using the fact that these stars, found to be very young, were likely born in a metal-rich environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Using the stellar evolution models of Claret (2007) with a higher metallicity (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='04), we find log k2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02, which is in much better agreement with the observed value of log k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The N-body modelling also allows us to estimate the ap- sidal advance of the quadruple orbit A-B (shown in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This motion – accumulating to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6◦ over the inter- val of available observations – is orders of magnitude slower when compared to the values for the A and B systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' How- ever, over the next few decades, when the change will have cumulatively added up to a few degrees, one can readily de- tect such movement with newly obtained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' On the other hand, other effects such as the change in orbital inclination and eclipse depth would still be negligible on such a timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We also note that the eccentric orbits of the inner binaries are subject to the circularization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' From the theory of circularization by Zahn (1977) and equations by Claret & Cunha (1997) the appropriate circularization time scales are of the order of magnitude longer than the estimated age of the system as resulted from our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 5 DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 V994 Her and its visual companion In the prior sections, we conclusively demonstrated the pres- ence of a third eclipsing binary in the V994 Her system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Here, we discuss the likelihood that binary C is hosted by Image 2 (fainter object to the North, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 4), as well as the probability that Image 2 is physically bound to Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' If so, this would give the system a (2+2)+2 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' According to the photodynamical fit for the system pa- rameters presented in Table 3, binary C has 14% the light of binaries A+B in the TESS band, and 13% in V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' That corresponds to magnitude differences of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2, re- spectively, in the TESS and V bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 4 suggests that these contrasts correspond to being able to resolve two objects within Image 1 (the brighter southerly object) that are separated by ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='06′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Since both Images 1 and 2 are 290 pc away, the resolvable physical separation at this magnitude contrast would correspond to 18 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The ac- tual semimajor axis of the binary A and B quadruple, which resides in Image 1, is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 au (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' It is always pos- sible, of course, that at the time of the speckle observations, the projected distance between the center of light of bina- ries A+B and a putative close orbiting neighbor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', binary C) might inadvertently be very small due to unlucky orbital phasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Let us assume, however, for the sake of argument, that A+B and C are at some nominal separation on the sky at this particular outer orbital phase when the speckle measure- ments were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In that case, the outer orbit of C around A+B, within Image 1, would have to have a semimajor axis of not much more than ∼ 20 au before it is resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For a stable triple system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', C stably orbiting A+B) the ratio of semi-major axes must satisfy aout ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 �MABC MAB �2/5 (1 + eout)2/5 (1 − eout)6/5 ain , (1) where equation (1) is from Rappaport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2013), which in turn is based on the work of Mardling & Aarseth (2001) and Mikkola (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' If we take as a very rough estimate that aout ≲ 20 au, and we know that ain ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 au, then we find a constraint on eout such that (1 + eout)2/5 (1 − eout)6/5 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2) In turn, this requires that eout ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, while this is not an unreasonably small value for an outer orbital eccentricity, we can see from this exercise, that there is “not much room to spare" in trying to fit binary C into an orbit about bina- ries A+B, all within Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Furthermore, recall that the contrast limits shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 4 are 5-σ limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, we ten- tatively conclude that binary C, in fact, is hosted by Image 2 (the fainter one to the North).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We next look at the question of whether Image 2 (likely containing binary C) is physically bound to Image 1 (hosting binaries A and B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' For this analysis, we have two pieces of kinematic evidence: (i) the proper motions of Images 1 and 2 from Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2022), and (ii) the historical astrometric data, spanning 200 years, of the WDS catalog (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This information is summarized in Table 4, and the WDS astrometric data are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The two proper motion results (Gaia and WDS) evaluated MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 10 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Observational Kinematics Between Image 1 and Image 2 Cartesian motiona mas yr−1 km s−1 Gaia PM RA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='20 WDSb PM RA −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='10 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='14 Gaia PM Dec −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='16 WDSb PM Dec −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='35 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='48 angular motion — — WDSb,c ˙r [mas yr−1] −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='35 — WDSb,c ˙θ [mrad yr−1] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='09 — Notes: (a) Image 2 value - Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (b) Washington Double Star catalog (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (c) These refer to the rate of change in the separation and the position angle, respectively, and are inferred from the fits shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' at the Gaia epoch are in agreement on the proper motion of the declination (PM Dec) to better than 1 σ, while the proper motions of the right ascension (PM RA) differ by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We attribute this discrepancy to fitting a linear function to ˙θ2 over a 200 year interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The total relative velocity between Image 1 and Image 2 on the plane of the sky is in the range 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='55–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='74 km s−1 depending on whether we choose to use the WDS or Gaia results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' To check whether Image 2 is physically bound to Image 1, we take the escape speed to be vesc ≃ � 2GMABC s , (3) where Mtot is the total mass contained in Image 1 plus Im- age 2, and s is the instantaneous (3D) separation of Im- age 1 and Image 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As representative values, we estimate MABC = MA + MB + MC = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 M⊙ from Table 3, and s ≳ 307 au, where the latter is the physical separation on the plane of the sky between Image 1 and Image 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This leads to an estimate for vesc ≲ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Since this value is sub- stantially larger than the relative speed of Image 1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Image 2 (at least in the plane of the sky), we tentatively take this as strong evidence that the Image 1 plus Image 2 system is physically bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, we remain unsure about the rel- ative speed and separation in the direction along our line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Finally, we make another independent argument which also strongly suggests that Image 1 and Image 2 are physically bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This argument relies on the fact that Image 2 is found so close in the sky to Image 1, with similar proper motions and distance, and the two are not too dissimilar in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We seek to quantify the relative occurrence rate of such a pair of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Using Gaia data, we searched for other stars with similar properties to those of Image 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In particular, we looked for stars that have (i) PM RA within an absolute value of 1 mas yr−1 of image 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (ii) PM Dec with absolute value within 5 mas yr−1 of Image 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (iii) a parallax within absolute value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 compared to Image 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' and (iv) having a G magnitude brighter than 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' When we search the Gaia database for other stars that satisfy these criteria, we find 13 such stars within 30◦ of Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Given that the search area is ∼ 1010 times larger than the area needed to include Image 2 (at 1′′ distance from Image 1), we conclude that Image 1, 2 ˙θ ≡ rate of change in the position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' A plot of position angle and separation for the two stars in the visual binary catalogued by the Washington Double Star catalog (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 2001) at the position of V994 Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' with its given properties, is not remotely likely to be found there by accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The conclusion from the above argument is that either Im- age 2 is physically bound to Image 1, or it is comoving with it by virtue of having been born in the same stellar nursery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The latter scenario can be ruled out rather readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We have seen that the relative speed between Image 1 and Image 2 (on the plane of the sky) is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The age of the system (from Table 3) is ≃ 60 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' If the two images were unbound and merely approximately comoving on the sky, then during that time the two images would have drifted apart in the en- suing 60 Myr by some 300 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This is much, much larger than the current sky separation of 290 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Thus, we conclude that Image 2 is physically bound to Image 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Given the above discussion, we believe that the most likely configuration of this sextuple is a (2+2)+2 system, with the inner quadruple system containing binaries A and B situated in Image 1, and the third binary (C) in Image 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' There is also a slim chance, but not yet fully ruled out, that Image 1 hosts all three binaries, and Image 2 represents a 7th star or yet another binary (‘D’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The most direct ways to prove our most likely scenario (that binary C is in Image 2) is to (i) check for binary C eclipses in Image 2, and/or (ii) check for RV motions in Image 2 with P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='96 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 20 (degrees) WDS Astrometry 15 10 Position Angle ( 5 5 Rotation Rate = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='063 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='005 deg/year 10 1800 1850 1900 1950 2000 2050 Year3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 WDS Astrometry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 (arc sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='5 dr/dt = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='86 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='35 mas/yea 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 1800 1850 1900 1950 2000 2050 YearV994 Herculis: A Triply Eclipsing Sextuple 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 Outer Orbital Period Distribution Armed with only the relative velocity between Image 1 and Image 2 projected onto the sky, vsky ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='7 km s−1, and the projected separation on the sky, ssky ≃ 307 au, we attempt to estimate a probability distribution for the outer orbital period Pout via a Monte Carlo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Let ⃗s and ⃗v be the full relative position and velocity vectors between Image 1 and Image 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In that case ssky s = sin β1 , (4) vsky v = sin β2 , (5) where β1 is the angle between the observer’s view direction and ⃗s and β2 is the angle between the view direction and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' If we know nothing about the orientation of the orbit on the sky, then samples of sin β can be drawn randomly from: sin β = � 1 − R2 , (6) where R is a uniformly distributed random number between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Here, as an approximation, we treat ˆs and ˆv as independently and randomly directed with respect to the ob- server’s view direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' So, we randomly draw β1 and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The energy of the outer orbit can now be written as E = −GM1M2 ssky sin β1 + 1 2 M1M2 MABC � vsky sin β2 �2 = −GM1M2 2aout , (7) where M1 ≡ MA + MB and M2 ≡ MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This reduces to a simple expression for the semi-major axis of the outer orbit: 1 aout = 2 sin β1 ssky − � vsky GMABC sin β2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (8) Finally, we make a large number of random draws for β1 and β2 and, for each combination, store the realization for the semi-major axis and corresponding Pout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The resultant distribution for Pout is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The distribution has a sharp cut at ≃ 1500 years, reflecting the fact that the min- imum orbital separation is attained when β1 = β2 = 90◦, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', when the outer orbit is in the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, since we do not know the orbital parameters in the line of sight, long orbital periods are possible, as evidenced by the long tail of the period distribution in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The median of the orbital period distribution is close to 3000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We note that our uncertain knowledge of the outer orbit could be greatly improved with a radial velocity study of Image 2 (which presumably hosts binary C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='3 Possible role of the 3:2 mean motion resonance Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2019) presented a thorough analysis of stel- lar quadruple systems with a 2+2 architecture that exhibit eclipses of both components, with binary periods less than 15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' One of the interesting population results from this study was the identification of a statistically significant group of systems having a period ratio close to 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2019) speculated that these systems are either captured in the 3:2 mean motion resonance of the binary periods, or in- teracted with this resonance in the recent past and still reside close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' One of the consequences for this class of systems would be a possible excitation of the orbital eccentricity of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Probability distribution for the outer orbit of the V994 Her system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Results have been logarithmically binned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The black histogram is the differential probability distribution, while the red curve is the cumulative distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' the binary with the longer period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The V994 Her system was considered in this class by Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Now, with much more detailed information about V994 Her, we revise its status with respect to the group of objects that reside or interacted with the 3:2 resonance in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We use an analytical description of the low-order mean mo- tion resonances in 2+2 quadruples by Tremaine (2020) (for completeness, see also Breiter & Vokrouhlický 2018, who dis- cuss the 1:1 mean motion resonant states in the 2+2 quadru- ples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' First, it is trivial to check that V994 Her is not currently located in the resonance since 1−(2PA/3PB) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02204 is too large (it would need to be three orders of magnitude smaller to be considered to possess this resonance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Tremaine (2020) also discusses sidebands of the pure 3:2 mean motion res- onance between PA and PB generated by multiplets of the mean motion frequency nAB of the mutual orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Their im- portance is typically very small, because the sideband width at frequency k nAB has a multiplicative factor ∝ e|k| AB (k is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Here, eAB ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='7 is a rather large value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, to account for the three orders of magnitude in separation between the observed eccentricity and the requirement for resonance, |k| would have to be greater than 30, which is much too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The system, however, may have crossed the resonance in the past;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' this could have contributed to an ex- citation of the eA value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In order to place the system into the exact 3:2 mean mo- tion resonance, one would need to (i) increase PA by ∆PA ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='046943 day, (ii) decrease PB by ∆PB ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='031296 day, or (iii) perform some combination of the two operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Addi- tionally, in order to temporarily capture V994 Her in the 3:2 resonance in the past, PA and PB should have been converg- ing towards each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In what follows, we shall discuss an end-member possibility (i) that PB was constant, and PA was evolving from an initially larger value beyond the resonance condition toward the current value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' However, identical con- clusions are obtained for other options, such as keeping PA constant and PB increasing as in (ii), or their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Using the results from Appendix C of Tremaine (2020), we note that the putative past capture in the 3:2 resonance puts a severe constraint on the speed by which the period PA MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='6 Relative F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='4 Median Period = 2930 year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0 1000 10000 100000 Outer Orbital Period (years)12 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' In particular, denoting the corresponding charac- teristic timescale τA = PA/ ˙PA (with ˙PA = −dPA/dt), we find that τA ≥ K (MB/µB)4/3 (aAB/aB)20/3 PA , (9) where K ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='74 × 10−2, MB and µB represent the total and reduced masses of the shorter period binary component, aB and aAB are the semimajor axes of the B and A-B orbits, and PA is the orbital period of the A binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Substituting the values from Table 3, we have τA ≥ 26 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Assuming an approximately steady decay of the A orbit, we then estimate a minimum time needed to accumulate the difference ∆PA between the resonance and the current state ∆TA ≃ ∆PA PA τA ≥ 580 Myr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (10) This is nearly an order of magnitude longer than the esti- mated age of the V994 Her system (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Since other possibilities outlined above lead to the same result, such as PB drifting toward its current value from an initially smaller value, we may conclude that the V994 Her system in all like- lihood never interacted with the 3:2 mean motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Its location near to it might therefore be just coincidental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As a consequence, the eA value is fully a relic of the initial state, with possible tidal damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Indeed, the interaction with the 3:2 resonance would likely not be capable of explaining the significantly larger eB value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The latter might be excited by interaction with the 2:1 mean motion resonance between PA and PB values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' the location of this resonance, however, is much too distant from the current system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 6 SUMMARY AND CONCLUSIONS In this paper, we have demonstrated that the first-known doubly eclipsing system V994 Herculis is in fact at least a sextuple system that unambiguously demonstrates three sets of eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Using TESS and archival data, we have disentan- gled the light curves of all three binaries in the system using three different techniques and added new measurements to the O–C diagrams of binaries A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We have also identi- fied the period of the newly-discovered binary C to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='9601 days, based on TESS and older ground-based data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Finally, we used archival data from the Washington Double Star Cat- alog (spanning over 190 years) alongside parameters from Gaia DR3 in order to prove that the fainter visual companion on the night sky (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1′′ distant) is likely gravitationally bound to this system and may harbor binary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Depending on the nature of the companion star, this could be either a rare (2+2)+2 sextuple star system—similar to the well-known system Castor, with the same architecture (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Stelzer & Burwitz 2003, and Tokovinin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' An- other possibility is that the brighter star has six unresolvable stars, and the nearby visual companion is another bound member of the system, making it even more interesting—a potential septuple (or even higher-multiplicity) star system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Using additional data, we can more precisely derive the outer orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' moreover, updated higher angular-resolution photom- etry would be able to firmly prove whether or not the C pair resides in the fainter nearby component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' We urge the com- munity to observe these interesting stars using the tools at their disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The high-angular separation techniques (both in photometry as well as spectroscopy) would be able to prove the true structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' As the separation of the vi- sual pair on the night sky is slowly decreasing, it may become increasingly difficult to carry out these observations as time goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' DATA AVAILABILITY The TESS data underlying this article were accessed using the MAST (Barbara A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Mikulski Archive for Space Tele- scopes) Portal (https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='edu/portal/Mashup/ Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Some of the data were derived from sources in the public domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' their URLs are provided as footnotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' The derived data generated in this paper and the code used for the photodynamical analysis will be shared upon reasonable request to the corresponding author P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' ACKNOWLEDGMENTS This paper includes data collected by the TESS mission, specifically as part of GI program G022062 (PI: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Prša).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Funding for the TESS mission is provided by the NASA Sci- ence Mission directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Resources used in this work were provided by the NASA High End Computing (HEC) Pro- gram through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' STScI is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This research has made use of the Washington Double Star Catalog maintained at the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Naval Observatory, and we thank Rachel Matson for providing archival data on the V994 Her visual double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' This work has used data from the European Space Agency (ESA) mission Gaia3, processed by the Gaia Data Processing and Analysis Consortium (DPAC)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Funding for the DPAC is provided by national institutions, in particular those par- ticipating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Some of the observations in the paper made use of the High-Resolution Imaging instrument ‘Alopeke, obtained un- der Gemini LLP Proposal Number: GN/S-2021A-LP-105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' ‘Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Howell, Nic Scott, Elliott P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Horch, and Emmett Quigley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' ‘Alopeke was mounted on the Gemini North (and/or South) telescope of the international Gemini Observatory, a program of NSF’s NOIR Lab, which is managed by the Asso- ciation of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation, on behalf of the Gemini partnership: the Na- tional Science Foundation (United States), National Research 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='int/gaia 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='esa.' metadata={'source': 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CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='0/16_013/0001403 and CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', 1977, A&A, 57, 383 Zasche P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', Uhlař R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', 2016, A&A, 588, A121 Zasche P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=', 2019, A&A, 630, A128 APPENDIX A: TABLE OF TIMES OF ECLIPSES MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023) 14 Zasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' TESS observed times of minima of V994 Her A Eclipse Time Cycle std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Eclipse Time Cycle std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' dev.' metadata={'source': 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unpublished eclipse times of V994 Her for binaries A, B, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Eclipse Time std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Pair Type Reference/ Eclipse Time std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Pair Type Reference/ BJD-2400000 (d) [A/B/C] [P/S] Observer BJD-2400000 (d) [A/B/C] [P/S] Observer 57843.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='49557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00325 C P FRAM 59787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='45377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00079 B P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' 59830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='45265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00304 C P FRAM 59791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='42428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='00071 A S R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Notes: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' = Gerald Persha, see http://var2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='cz ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' & U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' Dallaporta & U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='Munari;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='Uhlař;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='Mašek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' MR13 = Martín-Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2013) MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} +page_content=' (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFRT4oBgHgl3EQfQTdr/content/2301.13521v1.pdf'} diff --git a/adE5T4oBgHgl3EQfeg9M/vector_store/index.faiss b/adE5T4oBgHgl3EQfeg9M/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7415dd6ea85ed386a5af0cc2d66aae2c4f52fcbe --- /dev/null +++ b/adE5T4oBgHgl3EQfeg9M/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ea1569abc636cdea11b9f216443d53d4d3fa18673c7d03961e27cb6c6495481 +size 4718637 diff --git a/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf b/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..89852c2303f630e97c1db74dfb77e51c5907a844 --- /dev/null +++ b/btAzT4oBgHgl3EQfnf1c/content/2301.01581v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d56938fe53d14cdbe5483afd6b2b1c9163cd38c642003faa8f51e2f8af40376 +size 497811 diff --git 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+Saint Louis University +Technical Report +Abstract—Everyday, large amounts of sensitive data is dis- +tributed across mobile phones, wearable devices, and other +sensors. Traditionally, these enormous datasets have been pro- +cessed on a single system, with complex models being trained +to make valuable predictions. Distributed machine learning +techniques such as Federated and Split Learning have recently +been developed to protect user data and privacy better while +ensuring high performance. Both of these distributed learning +architectures have advantages and disadvantages. In this paper, +we examine these tradeoffs and suggest a new hybrid Federated +Split Learning architecture that combines the efficiency and +privacy benefits of both. Our evaluation demonstrates how +our hybrid Federated Split Learning approach can lower the +amount of processing power required by each client running +a distributed learning system, reduce training and inference +time while keeping a similar accuracy. We also discuss the +resiliency of our approach to deep learning privacy inference +attacks and compare our solution to other recently proposed +benchmarks. +1. Introduction +CENTRALIZED machine learning (ML) training is be- +coming unsustainable [1]. Aside from the advantages of +re-training often to optimize revenues [2], several learning +applications need to run their processes at the edge of the +network, not in the core of a datacenter, for multiple rea- +sons, including end-to-end latency minimization by running +machine learning algorithms locally on an end-device, and +privacy concerns of trusting third-party clouds [3]. Several +Machine Learning (ML) models trade user experience im- +provements on mobile devices for sensible data exploitation; +see e.g., text recommendation in keyboards [4], [5] or vocal +assistants [6]. In these and other applications, a decentralized +learning approach may be preferable to a centralized system +since sensitive data may remain locally within a client and +not transferred over a computer network. +Despite its benefits in several use cases, running machine +learning training and inference jobs within local devices +has several limitations: computing capacity is often limited, +battery drains faster with intensive processing and the mo- +bile or other end-devices have limited memory and storage +capabilities. For example, our experiments show that to +fine-tune a VGG-16 [7] Neural Network, pre-trained on +ImageNet [8] with Cifar-10 [9], tens of minutes are needed +to reach 90% accuracy on an NVDIA V100 GPU. +Different distributed neural network architectures have +been proposed to preserve privacy and guarantee timely +convergence – for example Federated Learning [1], Split +Learning [10], or hybrid approaches [11], [12], [13]. Feder- +ated Learning (FL) [14] averages the weights of the learned +Neural Network model on each edge device to create a +single model, which will update the local ones. Previous +research has shown that this strategy can achieve higher +accuracy than considering only a local model [14], [15], +[16], [17] and at the same time can preserve the privacy of +the data. Split Learning (SL) architecture splits the entire +NN into partitions of layers. Each partition is executed on +a different entity (i.e., edge and cloud), and different edge +NN partitions can be paired with the cloud partition. Thus, +this approach takes advantage of distributed datasets while +keeping user data private. +On one hand, FL is easy to scale to many devices given +that there are enough resources to meet training Service +Level Agreements (SLAs). Thus, FL is impractical in edge +training/inference settings, where resources are limited. On +the other hand, Split Learning can train with limited re- +sources, but it doesn’t scale to many devices well since +it is not parallel. Especially when we pair different edge +devices with not independent and identically distributed +(Non-IID) data with the central cloud, the training may not +converge at all [18]. Another drawback of SL is that the +intermediate data can be costly to transmit and store in client +or server nodes [19], [20], [21], [22]. Furthermore, since +the intermediate data in the forward propagation is derived +from the source data, there is a privacy concern. We further +discuss these two models in Section 2. +To cope with the inefficiencies of the existing distributed +learning models, we propose a novel distributed learning +architecture, Federated Split Learning (FSL) [23], which +combines the benefits of FL and SL while mitigating their +drawbacks. We discuss the generality of our FSL and a novel +methodology to optimize both delay and privacy guarantees. +arXiv:2301.01824v1 [cs.LG] 4 Jan 2023 + +The FSL model is characterized by multiple edge client +– server pairs. Such pairs train their copy of the NN simul- +taneously, providing the parallelism of federated learning, +while the client-server separation brings the advantages of +the split learning. Each computing pair partitions the NN. +After some pairs have completed some training epochs, the +server NN weights are averaged in a central cloud server, as +in classical FL algorithms [14]. We call this central cloud +node the “Parameter Server node” in Figure 1d. +There are other techniques, i.e., Parallel Split Learning +(PSL) [13], and Federated Reconstruction (FRC) [24], close +to our proposal. But they have some disadvantages com- +paring to FSL. While the PSL architecture has only one +cloud server node, FSL allows parallel server computations. +FSL and FRC have similar privacy levels (Sec 4.1), but +our evaluations shows that FRC doubles the training time +compared to FSL, considering no dominant transmission +delays. +We evaluate the benefits of our FSL architecture by +testing it with different NN models and tasks: from im- +age classification to an Internet traffic classification [25]. +Aside from testing training performance of our proposed +hybrid federated-split architecture, we evaluated the privacy- +performance tradeoff of FSL and SL, and give ideas on how +to enhance the privacy guarantee of these schemes using +Client-based Privacy Approach (CPA) and novel neural net- +work partitioning approaches. Furthermore, we realized that +certain ways of partitioning NN could reduce transmission +delay and enhance privacy together. Our experiment results +show that by combining different privacy approaches and +NN partitioning methods, our FSL may achieve both high +efficiency with respect to training time, privacy guarantees +and accuracy. +The rest of the paper is organized as follows: in Sec- +tion 2, we introduce the background and the related works to +discuss FSL. Then, we describe the proposed FSL system in +Section 3. In Section 4 we discuss the Client-based Privacy +Approach (CPA) and motivations to partition NN at edge +that can be applied in any split learning based architec- +ture. Our evaluation with training and inference metrics is +presented in Section 5.2, while the privacy evaluation is +presented in Section 5.3. +2. Distributed Learning Architectures: +Background and Related Work +Federated Learning [1], [15], [16], [17], [26] is a de- +centralized machine learning technique that trains neural +network models using data sources “owned” by multiple +clients (Figure 1a). A logically centralized parameter server +holds the latest neural network model, and orchestrates the +sharing of its weights between all clients. +At the beginning of the training phase of a federated +learning process, the parameter server sends the same ran- +domly initialized set of neural network weights to each +client. Each client then trains a local model for multiple +epochs using its local dataset. Until the client models have +extracted enough features, that is, a given accuracy threshold +is reached, the parameter server keeps retrieving, averag- +ing, and overwriting the weights (Figure 1a – steps 1 to +3). Thus, the global model could take advantage of the +privately-owned datasets which would not transmit through +the network. Moreover, FL has parameters to specify the +frequency for the parameter server to average the weights +of a certain group of clients. In this way, system architects +can balance the network traffic and model accuracy. Thus, +FL is considered scalable in terms of the number of clients, +as long as such clients have enough computational power +and storage resources to meet the training constraints, or +Service Level Objectives (SLOs). +Split Learning [10] is a distributed machine learning tech- +nique that is characterized by a computational split of the +neural network model into two partitions. Each partition +could run on a separate computing node, hence splitting +the computational resource demand. To train a NN model +with split learning, the NN must first be partitioned on +different nodes. Then, the forward propagation phase starts. +When the end of the NN partition at the first node is +reached, the outputs of the last layer of activation functions +are sent to the server node — step 1 (Figure 1b). Then +those outputs are used as inputs to the second NN partition +and continue the forward propagation. After calculating the +loss, the backward propagation in the server node is started. +Once the backward propagation reaches the input layer of +the server’s partition, the gradients of the inputs are sent +to the last activation in the client to finish the backward +propagation for client NN — step 2. Finally, the complete +NN weights are updated with the gradients. However, the +server’s NN partition can also pair with other clients’ NN +partition with the same client NN structure. First, the trained +weights on the last client are moved to the new client — step +3. Then the new server and client pair trains as mentioned +above — steps 4 and 5. +The SL algorithm preserves data privacy but suffers from +a long convergence time with Non-Independent and Identi- +cally Distributed (non-IID) data sources (Figure 2), and large +intermediate data to transmit. Moreover, training with more +than one client is sequential, hence poorly scalable. Some +solutions attempted to mitigate the transmission delay of +such intermediate data. For example, in BottleNet [20] and +BBNet [19], the authors aim at compressing the intermediate +data with a particular NN design. In Early-Exit [21] instead +the idea is to add classifiers at the early layers to avoid +computing the complete model. Previous work [22] also +used knowledge distillation to reduce the complexity of +client models and the data to transmit. +In this paper, we consider hybrid split-federated learning +systems, to combine their benefits while minimizing their +drawbacks. For example, we show that hybrid Federated +Split Learning and Parallel Split Learning can converge with +non-IID sources faster than Split Learning. +Combining Split and Federated Learning. Other re- +searchers have proposed combining the advantages of split +and federated learning. In Parallel Split Learning (PSL) [13] +(Figure 1c), they train the client NN partition on multiple + +AVG = +! +" ∑#$% +" +𝑤# +1 +3 +1 +3 +2 +SERVER +CLIENT A +CLIENT B +data +data +WEIGHTS +INTERMEDIATE RESULTS +GRADIENTS +(a) Federated Learning +1 +2 +4 +5 +3 +SERVER +CLIENT A +CLIENT B +data +data +WEIGHTS +INTERMEDIATE RESULTS +GRADIENTS +(b) Split Learning +1 +2 +1 +2 +SERVER +CLIENT A +CLIENT B +data +data +WEIGHTS +INTERMEDIATE RESULTS +GRADIENTS +(c) Parallel Split Learning +(d) Federated Split Learning +Figure 1: Distributed NN Training Architectures: (a) Federated Learning: The NN is in the client. The parameter server +calculates the average weights among clients and overrides the local weights. (b) Split Learning: The server partition +sequentially trains with each of the clients. Client weights are shared with the next training client. (c) Parallel Split Learning: +The server trains clients’ output in batches in parallel, but the client’s weights are kept private. (d) Federated Split Learning: +Multiple Edge Server and Client pairs train simultaneously. The Edge Servers’ weights are averaged by a Parameter Server. +The clients weights are kept private. +edge nodes in parallel. During the Forward Propagation +phase, the activation function results from different clients +are sent to a single remote server. Such server then backward +propagates the gradient from the loss function to the clients. +Model specific approaches. FedSL [11] is one example of +Recurrent Neural Networks (RNN). The idea is to unroll the +RNN’s feedback loop and split the recurrent NN partition +to different nodes with the sequence data segments. After +each epoch, devices average and overwrite their weights. +This approach combines the benefit of SL and FL in the +RNN training setting without introducing much overhead. +FL Extensions. Some works extend the FL model instead +of designing a hybrid architecture. The authors of Federated +Construction (FRC) [24], prioritize user data privacy trading +off the efficiency of the training process. Their model is +partitioned into global and local shards and both deployed +in each edge device, and the two partitions are trained +alternately. A parameter server then sends the global shard +and retrieves the corresponding updated weights. This de- +sign makes the training stateless and, consequently, highly +scalable for storage. +Source Data Privacy. One of the advantages of these edge +training systems is the high perceived level of source data +privacy: user’s data doesn’t leave the edge device. However, +the adversaries can also learn the source data from the NN +weights. All systems mentioned except for FRC and PSL +cannot maintain the privacy for NN weights [27], [28]. They +maintain this privacy since they do not share the complete +NN weights or the activation results. +Another threat discussed in NoPeek [29] targets the +results of the activation function, which can be used to +reconstruct the source data with an autoencoder NN. Certain +edge intelligence systems that partition Neural Networks +and transmit intermediate data in between partitions are +Figure 2: Accuracy degradation when training with Non-IID +data. +vulnerable to this threat. Possible mitigation uses the Dis- +tance Correlation (DC) loss [29] added to the original loss +function to measure the difference between the source and +the intermediate data. This approach maximizes DC loss +and accuracy while updating the NN’s weights by solving +a Multi-Objective Optimization Problem. +Client-Based Privacy Approach (CPA). NoPeek’s loss +function improves the resilience in reconstructing private +source data from the intermediate data. However, sharing +the loss and constructing the distributed gradient graphs +are either unsafe or need an extra management system. +To overcome this limitation, we propose an extension from +NoPeek, namely Client-based Privacy Approach. The idea +is to add noise with user specified methods to the results +of activation functions in clients, so there would be less +features to be used by the autoencoder NN (Section 4.3). +Privacy Approaches. We will mainly compare DC and +Differential Privacy (DP-SGD) under CPA framework. DP- +SGD is a widely used lightweight algorithmic approach for +data privacy [30], [31]. The idea is to add a Gaussian noise + +1.0 +0.9 +Accuracy +0.8 +0.7 +0.6 +ParallelSplit +FederatedSplit +0.5 +SplitLearning +0.5 +0.85 +0.9 +0.93 +0.95 +0.97 +0.99 +1.0 +X(percentage of unbalance)PARAMETERSERVER +- ↓Zl=o Wi +SerAVG : +3 +EDGE SERVER AEDGE SERVER B +CLIENT A +CLIENTB +data +data +INTERMEDIATERESULTS +GRADIENTS +WEIGHTSto the gradients during a training phase of a NN. Thus +the client’s output, generated by the updated weights, will +confuse the adversary. Comparing to Homomorphic Encryp- +tion (HE) or Secure Multi-Party Computation (SMPC), we +consider it fits better in our client and edge server setup, +while the privacy level of encryption has been well discussed +[31]. It is not cost efficient at edge to consume a lot of +battery for encryption and decryption steps. There are also +methods, i.e., model compression( [19], [22], [32]), can +potentially enhance privacy guarantee and consume power +lower than HE but higher than DP-SGD . But comparing +existing privacy methods is not the focus of this project and +we consider it as a future work. We mainly want to show +the proposed CPA is a general approach and it provides high +attack resilience with different methods including DC and +DP-SGD. Thus, in this paper, we evaluate the traditional +DP-SGD applied to the complete model and compare it +with DP-SGD only applied to the clients using the CPA +framework (Section 5.3.6). +3. Privacy-Oblivious FSL +In this section, we detail our proposed Federated Split +Learning (FSL) architecture, that we originally proposed +in [23]. FSL is a hybrid approach that combines the ad- +vantages of SL and FL. It avoids sending users’ source data +or sharing the complete NN parameters through the network +while being scalable. +Our FSL architecture shown in Fig. 1d has three types +of entities: (i) edge servers, (ii) clients, and (iii) param- +eter servers. To train a NN with FSL, we first setup an +authentication protocol [33], [34] among the entities to pair +each client with one edge server, and edge servers with a +parameter server. After pairs are found, the communications +are sent without encryption. Then, in each client and server +pair, we partition the complete NN into the client’s partition +and (edge) server’s partition. +FSL has three training steps. In step 1, the client forward +propagates with source data and transmits the intermediate +data to the edge server. The server then finishes the propa- +gation and calculates the loss. In step 2, the server backward +propagates to client source data. In step 3, after epochs, the +parameter server averages the weights in the edge servers. +Consequently, FSL will have multiple advantages com- +pared to the other approaches discussed. FSL clients will +have a lower resource demand compared to FL since it will +have fewer NN layers to train. Thus FSL is a more practical +scheme for edge intelligence application. Also, while FSL +only averages the weights in the edge servers, FedAVG +averages the complete weights. Therefore FSL avoids po- +tential vulnerability to model inversion attacks [27], [28]. +Moreover, FSL provides better scalability than SL, since +client and server pairs can train independently. Compared +with the FRC [24] architecture, we note that the latter is +inefficient in training time. It updates the parameters of +one partition on each forward and backward propagation +execution and runs multiple times to update the full model. +Compared with Parallel Split Learning [13], we found four +potential suboptimalities. First, since the edge devices have +to synchronize with the central server, clients may have +to wait until the server has finished with processing all +the results of the activation functions in its queue. In the +worst case, assuming that all activation function results +from n clients arrive at the same time, the lower bound of +waiting time for each client to continue on with backward +propagation is O(n). We hence conclude that PSL is not +as scalable as our FSL. Second, we also observe that the +PSL server would temporarily store multiple batches of the +results of activation functions, so it needs a sophisticated +logging and compaction storage system to recover from +failures. Consequently, PSL is less robust than our FSL, +since FSL has to maintain fewer states in each isolated pair. +Third, the PSL design may suffer from resiliency problems. +Meanwhile, in FSL, failure in one pair won’t prevent other +pairs from training or inference. Fourth, in PSL, since all +intermediate data will be transmitted and processed by the +single server, bandwidth and computation resource at the +server node may get congested. Our findings are presented +in Section 5. +4. Privacy-Aware FSL +We have discussed the efficiency and fault-tolerance +properties of FSL. In this section, we consider instead the +privacy-preserving properties of different architectures and +propose our privacy-aware FSL. In particular, we discuss +how to complement general split learning based architec- +tures to mitigate the problem of sharing the output values +of NN activation functions or weights over an honest but +curious network. We first give a formal definition of our +privacy attacker model, and then we discuss how a Client- +based Privacy Approach and certain ways of partitioning +Neural Network would help avoid such attack. +4.1. Privacy Attacker Model and Assumptions +We assume an attacker can capture the Intermediate +Data (the results of last layer activation functions trans- +mitted from client to edge server) in plaintext. Moreover, +we assume that attacker knows the client NN architecture. +Consequently, the adversary can implement an AutoEn- +coder [35] NN to reproduce the source data fed into the +client model. Moreover, to train the autoencoder NN, we +assume that some datasets with features similar to the client +source data are accessible to the adversary. +4.2. Attack Resilience +Given the attacker model, in order to compare the level +of privacy guarantee among different privacy approaches, +we define an Attack Resilience metric (τ) as: +τ = 1 − +∥correct∥ +∥reconstructed∥ +(1) +It measures the misclassification rate. ∥correct∥ counts the +number of images, reproduced by the attacker, which can +be correctly classified by a trained classifier (Section 5.3.3). +And ∥reconstructed∥ is the total number of reproduced +images. + +4.3. Client-Based Privacy Approach in Distributed +Setting via Distance Correlation (CPA-DC) +Motivated by the NoPeek approach [29], where weights +are updated based on the sum of two loss functions, i.e., +Cross-Entropy, and Distance Correlation (DC), we adopted +the idea of using two loss functions into an alternately +scheduling mechanism with two rounds: the regular round, +minimizes the cross-entropy loss function in the (edge) +server. The Distance Correlation round, maximizes the DC +loss function in the client. The idea is that we add noise +that make source and intermediate data different to client’s +weights. We define this alternating behavior with loss func- +tions in Equation 2: +L = +� +Loss(g(f(x)), label) +if e mod F == 0 +m · DC(x, f(x)) +otherwise, +(2) +where L is the measured loss value, e represents the epoch +index, F represents the DC Frequency, Loss(·) is the +loss function used to measure mis-classifications, DC is the +distance correlation loss, m is the loss multiplier, f is the +client NN, g is the server NN, x is the source data, and +label is the labels in the source data. Notice that F controls +the alternation frequency and m adds a weight to the DC +loss. +The alternating loss function is a policy. Our Client- +Based Privacy Approach (CPA) can also work with other +loss functions or methods adding random noise to the regular +round. In the evaluation, we explored several methods to +embed the noise. In particular, we explored the trade-off +between training time and the highest attack resilience. +4.4. How many layers do we assign +to each neural network partition? +In this section, we discuss the problem of selecting how +many layers need to be assigned for each NN partition, +i.e., client NN depth. This tradeoff will tune training time +(processing and transmission delay), privacy, and accuracy. +Capturing the tradeoff between all these metrics is challeng- +ing. To illustrate, consider the tradeoff between processing +delay and transmission delay. The size difference among +output layers in different partitions can be large, so a few +partitioning policies may lead to significant transmission +overhead, increasing training time and hence diminishing the +gain of the hybrid FSL compared to the original Federated +Learning architecture. In VGG-16 [7], the output size of the +first convolutional layer is two times the size of the second +convolutional layer. Thus, a system with a model cut after +the second convolutional layer can tradeoff the extra pro- +cessing delay at low-capacity clients while yielding a lower +transmission delay. We evaluate this effect in Section 5.2.2. +Analytically, this effect is captured by solving Prob- +lem 3, where α and β are developer-specified parameters +that represent positive weights for the transmission delay +of intermediate data (I) and computation delay (C), the pa- +rameter d represents the depth of the client neural network, +and finally, b represents the bandwidth, which is measured +periodically. To efficiently solve this problem, we follow the +approach in [36], where we build two regression models to +predict the delays I and C, given the available bandwidth +by profiling the model, i.e., computing the output size and +processing time for each layer in the client NN, instead of +training the full model. +min +d (αI(d | b) + βC(d)) +(3) +The solution of Problem 3 is optimal with respect to de- +lays, however, it can be sub-optimal with respect to privacy +and accuracy. As Section 5.2.2 shows, the client processing +and transmission delay of FSL reach the minimum when +the client NN depth is between 7 and 16. In Fig. 7a, +instead, the client NN needs more than 16 layers to be +above 90% attack resiliency. Thus, we conclude that an +optimal NN partitioning decision should balance different +objectives and constraints, including transmission delay, pro- +cessing time, privacy, and accuracy, as shown in our Problem +formulation 4. In such a problem, W represents the model +weight vector, (I′, C′, A, R) is the tuple representing the +observations for transmission delay, computation delay, ac- +curacy, and resilience, (γ, κ) are new user-specified positive +weights, and d is the client NN depth. +max +W,d +(−αI′(W, d | b) − βC′(W, d) ++ γA(W, d) + κR(W, d)) +(4) +To solve Problem 4, we have to train W for each d +until convergence and then find the best d. This brute force +method is inefficient. A more efficient approach would rely +on predicting the delays, accuracy, and privacy without the +full training of the model. Extending the approach in [36] to +go beyond profiling delays, is challenging. This is because +the accuracy and attack resilience for each client and edge +server pair is harder to profile and predict. Specifically, their +profiling depends on the weights trained on other pairs, the +distribution of source data among clients, number of clients, +number of layers to average in SerAVG, and training epochs +(Sec. 5.2.4 and 5.3). Another work can predict the model +accuracy [37], but it is based on the already trained model. +Therefore, for our FSL architecture with SerAVG, a predic- +tion method for partitioning remains an open question for +future work. In this paper, we experimentally demonstrate +the best model partitioning that balances requirements on +training time, accuracy, and privacy. +5. Evaluation Results +In this section, we describe the evaluation results re- +lated to our Privacy-Oblivious FSL (POFSL) and Privacy- +Aware FSL (PAFSL) architectures with our privacy-aware +approaches (CPA in Section 4.3 and Neural Network Par- +titioning in Section 4.4). Our evaluation demonstrates the +advantages of FSL over PSL and FRC in terms of training +time, memory usage, and convergence rate. Moreover, we +also show that our privacy-aware approaches can prevent + +the reconstruction of source images from intermediate data +in the Split Learning-based systems. We first discuss our +experimental setup, then present our evaluation results of +POFSL and PAFSL in Sections 5.2 and 5.3. +5.1. Experimental Setup +This experiment set studies the convergence for POFSL +and the privacy guarantee of PAFSL across different hard- +ware and applications with different NNs and datasets. +For the hardware, we used two types of nodes on +Chameleon Cloud [38]. One has an RTX6000 GPU, two +Intel Xeon Gold 6126 CPUs and 187 GB memory. The +other one has four NVIDIA V100 GPUs, two Intel Xeon +Gold 6230 CPUs and 128 GB of memory. We emulated the +computer network among our distributed learning entities +on the localhost interface on a physical machine, and each +experiment was set to use a single GPU. So that we can +ignore the network bandwidth bottlenecks. +For the applications, we considered three classification +tasks and implemented with PyTorch [39]. Then the dis- +tributed communication among entities of the systems was +handled by PySyft [40] and PyGrid [41] and no encryption +is applied on the transmission. The first application runs a +general image classification task with a VGG-16 [7] Convo- +lutional Neural Network (CNN). The model was pre-trained +using Imagenet [8] and then trained with the CIFAR-10 +dataset [9]. We run this task on 5 clients running a NVIDIA +V100 GPU. The second task uses a LENET [42] CNN to +recognize handwritten numbers in the MNIST dataset [43] +on 20 clients running a NVIDIA RTX6000 GPU. The third +task classifies traffic, not images. In particular, we decom- +posed a one-dimensional-CNN, trained with the ISCX VPN- +nonVPN (ISCX) traffic dataset [44], using 5 clients running +on a RTX6000 GPU. We partitioned the dataset and assigned +among different clients with Independent and Identically +Distributed (IID) probabilities and all our plots show 95% +confidence intervals, unless otherwise specified. Our goal +is to verify that FSL can always converge, with different +tasks, different NNs, different devices, and different data +distributions. We verified the advantages in delay or privacy +of FSL over existing solutions. +5.2. Evaluation Results for Privacy-Oblivious FSL +This section illustrates the methodology and draws ob- +servations of our experiments. Overall, our evaluations show +that POFSL has less overhead and similar accuracy compar- +ing to existing solutions. In particular, we evaluate training +time (Sec. 5.2.2), memory consumption (Sec. 5.2.3), and +learner accuracy (Sec. 5.2.4). +5.2.1. Experiment Design. Given the size of the different +datasets and number of clients, to reach at least 90% ac- +curacy, the neural networks used for image classification +needed 20 epochs. While for the traffic classification model, +80 epochs were used. +5.2.2. Training Time Evaluation. To evaluate training +time, let us consider the experiment whose results are re- +ported in Figures 3 and 4. The x-axis indicates the Cut Index, +i.e., the index of the last layer running in the client/local +part of the NN. When tested over the MNIST scenario, +we can observe from Figure 3a and 3b that FSL has the +shortest “Client Forward and Backward Propagation” (Client +F&B) time among all other distributed architectures. The +Client F&B time includes transmission time for gradients +and computing both activations and gradients in Client NN. +And Server F&B time includes transmission time for hidden +variable from client to server and computation in Server NN. +Notice that the weight update time is separately counted by +“Client Update Time“ and “Server Update Time“. Moreover, +we note that PSL is more vulnerable than FSL to limited +bandwidth across splits. PSL is consistently the slowest, due +to its inefficient server design; the server has to synchronize +the intermediate data, and it must process all batches of +intermediate data in each training epoch sequentially. +Observing FRC and FL, we see the F&B times do not +change along with the Cut Index (Figures 3a and 3b). Note +also that FRC is not training time efficient. It updates its +complete model with two almost full forward and backward +steps [24]. This can be noted in the same figures: the FRC +total F&B time for local and shared weights is almost +doubled compared to the FL training time. +We were able to obtain similar results comparing the +F&B times on another predicting scenario: the 1D CNN +implemented by [44] (Figures 3d and 3e). Due to the limited +size of this neural network (with only two convolutional +layers), we evaluated the architectures with merely two Cut +Indexes: at layer 3 and layer 6 of the NN. Even in this +experiment, we observe how our FSL still has the shortest +Client F&B time. PSL is the worst performant at each cut, +and FL keeps performing better than FRC. +FSL consistently uses less time in each training epoch +than the other analyzed architectures. We found that +PSL perform worse than FSL because of the single-server +architecture. PSL has similar results when comparing its +client F&B time with FL and FRC. FRC is not training +time efficient. Its total F&B time almost doubles compared +to FL. +When evaluating the training time on the CIFAR-10 +scenario, we found a different trend (Fig. 4a): PSL had the +longest training time, except for cut index of 30. Moreover, +FSL did not always perform the best. When most of the +layers run within the client, FL has a shorter training time. +This is because the size of intermediate data changes as the +cut moves, and with smaller data to send, the overall training +time can be shorter. Fig. 4b and 4c show the extra F&B time +during training. And existing works for SL have discussed +the similar behavior [19], [20], [21], [22]. +In particular, we show that FSL outperforms PSL as +PSL F&B time is more vulnerable to the intermediate +data transmission. In Figure 3, Client F&B time of FSL +keeps decreasing with smaller Cut Index, while that of +PSL still increases at Cut Index 6, 5 and 3, 2, although the +intermediate data in this experiment is much smaller than + +1 +2 +3 +4 +5 +6 +7 +Cut Index +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Client/Local F&B Time (s) +PSL +FSL +FL +FRC +(a) LENET+MNIST +Client Measurements +1 +2 +3 +4 +5 +6 +7 +Cut Index +0 +2 +4 +6 +8 +10 +Server/Shared F&B Time (s) +PSL +FSL +FL +FRC +(b) LENET+MNIST +Server Measurements +1 +2 +3 +4 +5 +6 +7 +Cut Index +0 +50 +100 +150 +200 +250 +300 +Intermediate Data Size (KB) +PSL +FSL +(c) LENET+MNIST +Intermediate Data +3 +6 +Cut Index +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Client F&B Time (s) +PSL +FSL +FL +FRC +(d) VPN Workload +Client Measurements +3 +6 +Cut Index +0 +1 +2 +3 +4 +5 +Server/Shared F&B Time (s) +PSL +FSL +FL +FRC +(e) VPN Workload +Server Measurements +Figure 3: LeNet+MNIST: (a) Client Time, (b) Server Time, (c) Intermediate data size. Observations: 1) Intermediate data +size is correlated with the times taken by the PSL architecture while having little correlation under FSL; 2) FRC has +almost twice the overall training time as FL; 3) Plots are obtained by averaging 20 clients’ results. Intermediate data size is +under batch size of 16 and each image was resized to (1,32,32). VPN Workload: (d) Client Time (e) Server Time. Similar +considerations are valid for the VPN dataset. Tested with 5 clients with a one dimentional NN with input size of (1,784). +Still, FSL has the shortest Client F&B time compared to the other settings. +4 +7 +16 +30 +Cut Index +0 +50 +100 +150 +200 +Training Time (s) +PSL +FSL +FL +FRC +(a) Pair Time +4 +7 +16 +30 +Cut Index +0 +10 +20 +30 +40 +Client Time (s) +PSL Client Update +FSL Client Update +PSL Client F&B +FSL Client F&B +(b) Client Time +4 +7 +16 +30 +Cut Index +0 +20 +40 +60 +80 +100 +Server Time (s) +SerAVG +PSL Server Update +FSL Server Update +PSL Server B&F +FSL Server B&F +(c) Server Time +4, +7 +16 +30 +Cut Index +0 +1 +2 +3 +4 +5 +6 +7 +8 +Intermediate Data Size (MB) +PSL +FSL +(d) Intermediate Data +Size +Figure 4: VGG + CiFar10: Plots show the effect of Cut +Index over (a) averaged overall training time, (b) time +spent in clients (5 clients average), (c) time spent in server, +(d) intermediate data size. The transmission delay caused +by intermediate data size can dominate the training time +(occurring during F&B propagation). +using VGG-16. This behavior is caused by the single server +bottleneck. Thus, FSL is more scalable in terms of training +time. Such observation also explains why both client and +server F&B times of PSL are consistently larger than FSL. +The intermediate data and gradients can cause signifi- +cant network overhead. Such overhead, however is better +mitigated by our FSL than PSL. Existing work [19] [21] for +Split Learning, as well as our partitioning strategies (Sec- +tion 4.4) can further mitigate the communication overhead. +Thus, the additional delay in FSL is not considered a severe +bottleneck compared to those systems training at the edge, +like FL. +5.2.3. Memory Consumption Evaluation. Memory usage +of each entity in the edge training and inference systems +limits the scope of devices that can join the system. To +compare which system is more flexible to deploy in terms +of memory capacity on devices, we show that each entity +has calculated memory usage in the FSL, PSL, FL, and FRC +systems. +Real-world memory utilization can be highly variable +as it depends on several implementation factors, such as +libraries used and the Remote Procedure Calls (RPCs) im- +plemented. However, the size of a model and its activation +at each layer are known. The following results show that +FSL’s clients consume less memory compared to FL and +FRC, and any of its servers occupy less memory than PSL. +The two plots in Figure 5 show the memory demands +computed at the client and the server for each architecture. +The x-axis shows the Cut Index and the y-axis represents +the corresponding expected memory usages in MB. Note +that FL and FRC do not split the NN, so their memory +demands are only shown in the left plot. +As shown in Figure 5a, the sizes of each client NN’s +weight and the results at each layer are the same in FSL +and PSL. Since FRC and FL compute the full NN in the +client during training, they require more than five times the +memory, for Cut Index 4 to 30. Figure 5b instead shows that +the server memory demand decreases with the cut index, +as expected. However, notice that PSL server need more +memory to hold intermediate data from different clients. +Memory usage of FSL compared to other systems. To +conclude, we found that FSL’s servers are lightweight com- +pared to the PSL system. Consequently, state management +would be easier in FSL. Also, FSL’s clients are lightweight +compared to FRC and FL, during training, so they are more +suitable at edge. +5.2.4. Learner Accuracy Evaluation. In this subsection, +we focus on evaluating the convergence of SerAVG, pro- + +4 +7 +16 +30 +Cut Index +0 +100 +200 +300 +400 +500 +600 +Client Memory Usage [MB] +PSL +FSL +FL +FRC +(a) +4 +7 +16 +30 +Cut Index +0 +100 +200 +300 +400 +500 +600 +Server Memory Usage [MB] +PSL +FSL +FL +FRC +(b) +Figure 5: FSL’s server has lower memory demand compared +to PSL, and FSL’s client has lower memory demand com- +pared to FRC/FL. VGG + CiFar10: (a) Client and (b) Server +memory demand (sum of model weights size and outputs at +each layer). Batch size is 32 and each image was resized to +(3,32,32). +posed and detailed in Section 3. Unlike FedAVG in Feder- +ated Learning, SerAVG averages the server NN weights. We +begin by discussing the correctness of SerAVG, comparing +the accuracy of SerAVG PSL, and FedAVG (FL). Then, we +compare the convergence rates based on different source +data sizes and Cut Indexes. +SerAVG Evaluation: In this experiment set, we evaluate if +SerAVG can enhance the accuracy of every model joining +the training process, given that the source data at clients +are non-IID distributed. Our results are reported in Fig- +ures 6a, 6b and 6c. To train the MNIST model, we split the +training set in two parts, part 0 and part 1. The two parts +represent two collections of skewed data sources at client ge- +olocations. We then let each part include data corresponding +to half of the labels in the MNIST dataset. We then split the +MNIST test set into two parts in the same way. Note that +the aforementioned way of splitting training set and test set +is an extreme Non-IID case. For example, a model trained +on part 0 of the training set has no knowledge of the labels +in part 1 of training set. To assess how well the systems +may learn and predict on Non-IID data on more realistic +data distributions, we further add 10% samples uniformly +and randomly, selected from the complete MNIST dataset +to all the four parts of datasets so that the model trained +on either part of the training set may be able to classify +labels in the other partition. Consider Figures 6a, 6b and 6c. +The x-axis represents the data partitions used in training +and validation, i.e., 0 & 1 means data part 0 was used in +training and part 1 was used during the validation phase. +The y-axis shows the validation set accuracy. From left to +right, the accuracy decreases when the server/shared NN for +FSL, PSL, and FRC is shallower. When the Cut Index is 3, +SerAVG, FedAvG, and PSL perform equally well. When the +Cut Index is greater than 3, SerAVG is worse than FedAVG +and lower but close to PSL. We conclude that SerAVG can +enhance the accuracy in the Non-IID source data setting, +while worse than FL and PSL. +Note that FedAVG averages all weights, while FSL never +shares the client’s weight in the SerAVG setting. Thus, FSL +clients cannot benefit from the gradients calculated at other +clients and that may lead to lower accuracy. Comparing +PSL and FSL, the PSL server optimizes for minimal loss +using all clients’ batch output. On the other hand, SerAVG +averages the trained weights on each server heuristically +based on FedAVG. Thus, SerAVG will have lower accuracy +compared with PSL. And each client NN trained with a +Non-IID dataset can extract little features from the other +Non-IID dataset. +Note also a similar but smoother drop in accuracy based +on Cut Indexes with Independent and Identically Distributed +(IID) partitioned CIFAR10 dataset and VGG16 NN in Fig- +ure 7a. The two experiments with MNIST and CIFAR10 +datasets suggest that SerAVG can preserve similar accuracy +patterns even when applied to different NN models and Cut +Indexes. +Tradeoff between resource demand and accuracy. In this +experiment we evaluated the accuracy of FSL with small +resource demand, i.e., when the size of the input data is +limited, and the client model runs on limited resources. We +found that with LENET and MNIST, the validation dataset +can reach an accuracy range of 87% to 93%, as long as +each client has enough data to train the machine learning +model. By varying the number of clients, we quantified +the expected drop in accuracy for both architectures. Our +results are shown in Figure 6d. The x-axis indicates the +index of epochs, and the y-axis shows the corresponding test +set accuracy after a certain number of epochs. The dataset +is IID among 20, 100, and 500 clients to study how the +data size affects convergence. In Figures 6a, 6b, 6c and 7a, +we also noted that FSL (implementing the SerAVG mech- +anism) keeps high accuracy when the Cut Index is small, +allowing deployments over resource limited device given +high performance requirement. We expect a higher accuracy +for both architectures with more effort in tuning the hyper- +parameters, given prior results in similar contexts [45], [46]. +While our accuracy results show 95% confidence intervals, +parameter tuning is out of the scope of this paper. +5.3. Evaluation Results for Privacy-Aware FSL +In this section we present the evaluation results of our +privacy-aware FSL (PAFSL) architecture and show that it +can provide certain privacy guarantees. FSL clients do not +share the source data and model weights, so adversaries +cannot directly access the source data or reconstruct them +with the model weights using model inversion attacks [27], +[28]. However, when compared with Federated Reconstruc- +tion (FRC) [24] which trains a complete model at the client, +FSL still sends the intermediate data through a network to +complete the forward and backward propagation between +clients and (edge) servers. An adversary could use such +data to reproduce the source data, e.g., through an Autoen- +coder [35] NN, trained with certain dataset, in Section 4.1. +To assess how our approach mitigates such vulnerabili- +ties, we first introduce the evaluation setup, the design and +usage of the attacker Autoencoder NN, and our experiment + +0&0 +1&0 +Train Part & Val Part +0.0 +0.2 +0.4 +0.6 +0.8 +Accuracy +FedAVG +SerAVG +NonAVG +FRC +PSL +(a) Accuracy (Cut Index = 3) +0&0 +1&0 +Train Part & Val Part +0.0 +0.2 +0.4 +0.6 +0.8 +Accuracy +FedAVG +SerAVG +NonAVG +FRC +PSL +(b) Accuracy (Cut Index = 5) +0&0 +1&0 +Train Part & Val Part +0.0 +0.2 +0.4 +0.6 +0.8 +Accuracy +FedAVG +SerAVG +NonAVG +FRC +PSL +(c) Accuracy (Cut Index = 7) +(d) Accuracy (different number +of clients) +Figure 6: In plots (a), (b) and (c), the accuracy of SerAVG is better than NonAVG but lower while mostly close to FedAVG. +SerAVG: Average the server NN’s weights; NonAVG: Each pair trains on its own; FedAVG: Average the complete NN’s +weights. In plot (d), the FSL and PSL accuracy is similar. +4 +7 +16 +30 +Cut Index +0.80 +0.82 +0.84 +0.86 +0.88 +Accuracy +PSL Accu +FSL Accu +FRC Accu +0.4 +0.6 +0.8 +Attack Resilience +PSL Resi +FSL Resi +(a) Privacy Oblivious +(VGG16+Cifar-10) +4 +7 +16 +30 +Cut Index +0.2 +0.4 +0.6 +0.8 +Accuracy +PSL Accu +FSL Accu +0.4 +0.6 +0.8 +Attack Resilience +PSL Resi +FSL Resi +(b) DC 1 & Muiltiplier=2.0 +(VGG16+Cifar-10) +1 +2 +3 +4 +5 +6 +7 +Cut Index +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy +PSL Accu +FSL Accu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Attack Resilience +PSL Resi +FSL Resi +(c) Privacy Oblivious +(Lenet+MNIST) +1 +2 +3 +4 +5 +6 +7 +Cut Index +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy +PSL Accu +FSL Accu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Attack Resilience +PSL Resi +FSL Resi +(d) DC 1 & Multiplier=0.3 +(Lenet+MNIST) +Figure 7: Accuracy and Attack Resilience for Privacy Oblivious and Privacy Aware Architectures based on Cut Index. Note: +DC 1 in captions refer to DC Frequency = 1. These figures show that to reach high accuracy and attack resilience, the +Cut Index cannot be too big or too small and loss multiplier is another way to enhance attack resilience. Furthermore, loss +multiplier can be more practical than DC Frequency, since it doesn’t introduce overhead to the training steps. +methodology. Then, we discuss the results of different pri- +vacy approaches, i.e., NoPeek [29] and the Client Based +Privacy Approach (CPA), and privacy level of different ways +of paritioning the NN. NoPeek solves a multi-objective opti- +mization problem of two loss functions, i.e., one maximizes +accuracy, and the other maximizes the differences between +source images and intermediate data. For CPA, we evaluate +CPA-DC and CPA-DP. CPA-DC (Section 4.3) optimize the +two loss function in NoPeek alternatively. CPA-DP applies +a DP-SGD [30] algorithm in the clients. Finally we evaluate +the privacy guarantee of different partitioning of client NN +and server NN motivated by Section 4.4. We conclude +this section by presenting results that demonstrate the high +resilience to privacy attacks of our proposed FSL and the +advantages in training efficiency. +5.3.1. Evaluation Settings. Our Privacy-Aware FSL and +PSL extend our Privacy-Oblivious version by adding the +CPA. Partitioning the NN was made easy by considering +only sequential NNs (e.g., LeNET and VGG16). We tested +both systems with the same image classification workloads +(e.g., MNIST and CIFAR10) on the same hardware (i.e., +NVIDIA RTX6000 and NVIDIA V100, respectively) as the +Privacy-Oblivious setting. +5.3.2. Setup the Attacker’s Auto-Encoder Neural Net- +work. In this subsection we explain how we define our pri- +vacy attack, given the assumptions mentioned in Section 4.1. +To understand how the privacy attack works, it is useful +to recall how the Autoencoder NN that we used work. In +particular, the Autoencoder is composed of two parts: an +encoder and a decoder. We need a dataset similar to the +source images to train the Autoencoder NN. +While the encoder uses convolutional layers to extract +latent variables from its input dataset, the decoder uses the +last layer’s activation function outputs from the encoder +and transposes the convolutional layers to reproduce the +input dataset of the encoder. Consequently, the encoder NN +structure is the same as the client NN structure, and the +attacker is the decoder NN. The ith layer of encoder would +be the ith last layer of decoder, with transposed convo- +lutional layer. We assume the attacker’s decoder structure +strictly mirrors the client NN structure. Thus, we expect +better attack resilience for SL-based systems in the real +world. +5.3.3. Privacy Evaluation: Methodology. In this subset of +our evaluation, we want to show both CPA and carefully +designed ways of partitioning NN provide high attack re- +silience while preserving high accuracy, based on different +datasets, NNs, for split learning based systems. +Each experiment includes trials initializing the Autoen- +coder’s weights and data-loaders with different random + +1.0 +0.6 +0.4 +FSL 20 +FSL100 +FSL500 +0.2 +PSL_20 +PSL 100 +PSL 500 +0 +3 +6 +9 +12 +15 +18 +Epochsseeds. In each trial, the attacker used a dataset containing +similar features to the learner’s source dataset. To recon- +struct MNIST, we selected EMNIST [47], and for CIFAR10 +we selected the CIFAR100 [9]. The EMNIST dataset con- +tains hand-written characters instead of numbers in MNIST, +so features like lines and curves are the same and attacker +can decode those activation function outputs. Similarly, the +CIFAR100 dataset contains 100 classes of RGB images in- +stead of the 10 classes in CIFAR10, so the common features, +e.g. classifying cat or dog, can be used to reconstruct with +the activation function outputs from the CIFAR10 dataset. +For each trial of the experiment, we first let the attacker +learn to reproduce her datasets. Based on the MSE loss, +i.e. a loss function that measures how different the original +and reproduced images are, the attacker updates her weights +in each epoch. After 20 epochs, we used the decoder to +reproduce the learner’s dataset from the intermediate data. +When an autoencoder NN is trained, we train a new +classifier with the same NN structure and source data as +the learner to classify the reproduced images for 20 epochs. +The mean and standard deviation of this classifier’s attack +resilience τ +in section 4.2. +We show the results comparing the four systems, i.e., +POFSL, POPSL, PAFSL, and PAPSL, with different CPAs +and Cut Indexes. Then, we evaluate the trade-off between +accuracy and attack resilience for FSL and PSL. +5.3.4. Privacy Evaluation using NoPeek. As we illustrated +in Section 2, NoPeek solves a multi-objective optimization +problem that takes in the source data and intermediate data +to maximize the difference with a Distance Correlation +(DC) loss function, as well as the prediction and labels to +maximize the accuracy. To solve such optimization prob- +lem, NoPeek has to share the value of the loss over an +network which may cause vulnerability or added complexity +of maintaining the gradient graph. As shown in Table 1, this +approach has both high attack resilience (i.e., 97% for PSL +and 98% for FSL) and high learner’s accuracy (i.e., 97% for +PSL and 96% for FSL) when trained for the same number +of epochs and clients as the Privacy Oblivious experiment +with the MNIST dataset and LENET NN. +cases +PSL +FSL +attack resilience(τ) +0.9733 +0.9837 +learner accuracy +0.9702 +0.9614 +TABLE 1: NoPeek stats with 20 clients training 20 epochs. +5.3.5. Evaluation Result using Client-Based Privacy Ap- +proach via Distance Correlation. To mitigate the draw- +backs of the loss value sharing, we consider a new approach +that prevents transmitting data outside clients, improving +upon NoPeek. We optimize for the similar two objectives +in NoPeek alternatively in the Client-Based Privacy Ap- +proach (CPA) via DC. As shown in Equation 2, there are +DC Frequency (F) and Loss Multiplier (m) to evaluate. +DC Frequency (F) defines how many times the DC loss +function is optimized after the loss function for accuracy is +optimized once. Loss Multiplier (m) is applied to the loss +function result but has the equivalent effect of multiplying +the learning rate by a factor m. These two parameters control +how different the intermediate data and source data will be, +by changing the frequency of optimizing the DC loss and +by changing the learning rate of gradients applied during +that optimization, respectively. +We note multiple tradeoffs in CPA-DC. First, CPA-DC +is not as training time-efficient as NoPeek. NoPeek can +optimize its two objectives simultaneously while CPA-DC +has to solve them sequentially. However, we consider that +NoPeek transfers more information than necessary over a +network. Second, there are tradeoffs for DC Frequency (F) +and Loss Multiplier (m). Increasing F adds more epochs +to optimize for DC loss, so the attack resilience would be +higher at the expense of a longer training time. Moreover, +we introduced m, so that we can keep a small F and only +increase m, which reduces training time while maintaining +attack resilience. We multiply larger m by the DC loss, +similar to increasing the gradient descent step size. Thus, +we need less DC epochs to maintain the attack resilience, +given a larger m, while the DC loss becomes less accurate. +Based on the discussion, intuitively we expect that a large +m combined with F of 1 can balance between training +time efficiency and DC loss gradients’ accuracy. These two +parameters should be carefully designed in a production +environment. +We first experimented MNIST classification with differ- +ent DC Frequencies with a constant loss multiplier of 0.1 +(equivalent to reducing the learning rate by 10%) and stud- +ied the tradeoff between accuracy and attack resilience, as +shown in Fig. 8a. The x-axis represents the DC Frequency, +the left y-axis shows the learner accuracy and the right y- +axis shows the corresponding attack resilience. +From the top plot of Figure 81, as DC Frequency is in- +creasing, for both Privacy Aware FSL (PAFSL) and Privacy +Aware PSL (PAPSL) systems, the attack resiliency increases +and the learner accuracy decreases, as expected. +Notice that PAFSL achieves better accuracy and good +resilience for most DC Frequency values. For DC Frequency +from 10 to 20, given that the attack resilience of PAFSL and +PAPSL are close within 10% difference, PAFSL achieves +more than 90% accuracy. From 25 to 35, PAPSL does +not learn any features while PAFSL still has about 80% +accuracy. When DC Frequency is five, the PAPSL has an +advantage over PAFSL, with close accuracy, and PAPSL has +around 20% more attack resilience. +The result shows that PAFSL with CPA-DC is easier to +tune for high accuracy and attack resilience. Within wider +domain of DC Frequency, PAFSL has higher accuracy and +good attack resilience compared to PAPSL. This is because +of the learning rate (step size) in SerAVG and PSL. The +server weight update rule of PSL is shown in Equation 5. +W t+1 +g += W t +g − η +NC +� +i=1 +∂g(f(xi)) +∂Wg +(5) +1. A DC Frequency of zero corresponds to POFSL and POPSL without +the privacy-aware approaches. + +The server weight update rule of FSL is shown in Equa- +tion 6. +W t+1 +g += +�NC +i=1(W t +g − η ∂g(f(xi)) +∂Wg +) +NC += W t +g − +η +NC +NC +� +i=1 +∂g(f(xi)) +∂Wg +, +(6) +where W t +g indicates the weights in the server at iteration t, +g is the server NN, f is the client NN, xi represent the i-th +batch of data, NC is the number of clients, and η is the +step size. Intuitively, since PSL has a larger step size, its +server NN can be confused quicker than FSL servers by the +intermediate data. Moreover, the confused server NN can +further confuse the client NN. It justifies our observation +that PSL’s accuracy and attack resilience become unstable +quickly when increasing the DC Frequency (F). Therefore, +we conclude that FSL is easier to tune compared to PSL. +In Figures 7b and 7d, with a fixed DC Frequency (F), we +show the accuracy (left y-axis) and attack resilience (right +y-axis) based on different Loss Multiplier (m) for different +models and datasets. Furthermore, we compared the privacy +oblivious cases (Figure 7a and Figure 7c), and the privacy +aware cases at different Cut Indexes (x-axis). As expected, +increasing the Loss Multiplier (m) enhances attack resilience +but reduces accuracy, especially when the client NN is deep. +Overall our evaluation of CPA-DC shows good attack +resilience and accuracy with a combination of small DC +Frequency (F) and big Loss Multiplier (m). And our FSL +has better accuracy and similar attack resilience to PSL. We +hence conclude that our CPA-DC can defend against our +attacker model. +5.3.6. Privacy Evaluation with Differential Privacy Ap- +proach. The previous section has discussed the CPA-DC, +but instead of DC there are other lightweight methods that +can enhance the privacy guarantee which adds noise to the +client NN while prevent depleting the client battery quickly. +In this section, we compare CPA-DP (using the popular DP- +SGD [30] algorithm inside clients) and CPA-DC. Also, we +show that na¨ıvely using DP-SGD in an FSL system would +lead to low accuracy. The implementation extends POFSL +with a DP-SGD optimizer, provided by the Opacus [48] +library. This method would add normally distributed ran- +dom noise to the gradients during backward propagation +based on noise multiplier ϵ. This parameter controls the +magnitude of the noise added. Notice that DC generates +the gradients in a specific direction to reduce correlation +between intermediate data and source data in each Distance +Correlation round. So we expect CPA-DP to have a worse +level of privacy, given the same level of learner accuracy, +compared to CPA-DC. Thus, the focus of this section is to +show that CPA can be applied with other privacy methods +like DP, despite DP’s worse privacy compared to DC. +We summarize the results of CPA-DP in Fig. 8b. This +plot shows the result when Cut Index equals 3. The x-axis +is the noise multiplier. The attack resiliency of FSL and +(a) CPA-DC (Multiplier=0.1) +(b) CPA-DP +(c) DP-SGD in Complete NN +Figure 8: LeNet+MNIST: Learner accuracy and attack re- +silience (τ) with 20 clients and Cut Index of 3 for Client- +Based Privacy Approaches (via DC (a) and via DP (b)) and +DP-SGD on the global learner model (c). Our FSL with +both client-based policies guarantee high-level of privacy +and accuracy. +PSL with noise multiplier > 0 is consistently better by +nearly 5% than FSL and PSL with noise multiplier = +0. At the same time, the accuracy decreases by less than +1% in either FSL or PSL from noise multiplier = 0 to +noise multiplier = 4. +CPA is a general approach and can be customized +with different methods to enhance attack resilience. The +evaluation shows that CPA-DP can also improve attack +resilience, while the learner accuracy does not change much. +Comparing with CPA-DC, both methods provide similar +learner accuracy, but CPA-DC’s attack resilience is higher. +We now compare CPA-DP against applying DP-SGD +in both client NN and server NN. Figure 8c shows high +attack resilience, but the learner accuracy of FSL drops +below 60% when noise multiplier ≥ 0.3. Meanwhile, +PSL shows a similar behavior as using CPA-DP. So, CPA- +DP is considered a better method for FSL to enhance its +attack resilience than with DP-SGD applied in both clients +and servers. The reason for FSL’s lower accuracy under DP- +SGD and high noise can be attributed to its SerAVG. After +applying SerAVG, the distribution of the random noise in +the server NN can be arbitrary, as shown in E.q., 7, while +the noise in the client NN stays intact. Thus, the resulting +complete NN in FSL may not converge. + +PSLAccuracy +PSL Resilience +FsLAccuracy +Client-Based Approach: DC Loss in Client +FSLResilience +1.0 +0.9 e +lience +e +0.8 +0.8' +0.6 +Re: + 0.4 +u +Atta +a +0.2 +0 +5 +10 +15 +20 +25 +30 +35 +40 +DC FrequencyPSL Accuracy +PSLResilience +FSLAccuracy +DifferentialPrivacy (Client + Server) +FSLResilience +0.75 +nce +g +0.8 +cul +Resili +0.6 +0.65 +@ 0.4 +a +0.2 +Lea +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +NoiseMultiplierPSL Accuracy +PSL Resilience +FSL Accuracy +Client-Based Approach: DP-SGD in Client +FSL Resilience +ence +0.97 +0.75 +esilier +0.96 +0.70.0 +0.95 +R +0.94 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Noise MultiplierPSL Accuracy +PSL Resilience +FSL Accuracy +Client-Based Approach: DP-SGD in Client +FSL Resilience +ence +0.97 +0.75 +esilier +0.96 +0.70.0 +0.95 +R +0.94 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Noise MultiplierW t+1 +g += W t +g − +η +NC +NC +� +i=1 +(∂g(f(xi)) +∂Wg ++ et +gi) +(7) +The variable et +gi indicates the gaussian noise added +for the i-th server NN at iteration t according to et +gi ∼ +N(0, (noise multiplier × max gradient norm)2) [48]. +5.3.7. Evaluation Using Different ways of Partitioning +NN. In Figure 7a and Figure 7c, the right y-axis shows the +attack resilience, and the x-axis indicates the Cut Indexes. +Overall, if we have a deeper client NN (i.e., moving from +smaller to bigger Cut Indexes), the attack resilience in- +creases and accuracy decreases (consistent with the SerAVG +Evaluation in Section 5.2.4). After adding more layers, the +intermediate data would have less features from the source +data, but only keeps those that can improve the classification +accuracy. Thus, less features are preserved and the attacker’s +ability to reconstruct the source’s data is hindered. +We also want to emphasize that with CIFAR10 work- +load, when there are 7 layers in the clients, the attack +resilience reaches about 80%. We reach the same result with +MNIST workload at the Cut Index of 4. No extra privacy- +aware method was used, and as we discussed earlier, the +transmission delay can also be reduced with a deeper NN +in the client due to potentially smaller intermediate data. +Cut Index is an important hyper-parameter for train- +ing delay, accuracy and privacy. Different Cut Indexes +bring the following tradeoff: the deeper client NN adds more +resource demand at the edge, but reduces the transmission +time and enhances attack resilience. On the other hand, +a shallower server NN may lead to lower accuracy with +SerAVG. +Furthermore, system architects can combine the ap- +proaches mentioned above, e.g., having a moderately deep +NN in clients and using the CPA-DC, to find a balance +between resource demand and performance. As in Fig- +ure 7d, with cut index = 4, DC Frequency = 1 and +loss multiplier = 0.3, we still get about 80% attack +resilience and more than 90% accuracy. +On the other hand, when comparing FSL and PSL, we +note that FSL has more hyper-parameters to tune. But, in +all experiments reported in Figure 7, when the Cut Index is +large, FSL has a better accuracy than PSL. So we conclude +that a carefully specified way of partitioning the NN can +benefit the most when applied in hybrid federated-split +learning systems. +6. Conclusion +Systems like Federated Learning (FL), Split Learning +(SL), and later works aim to fit specific scenarios such as +distributed model training and inference. However, they are +not flexible enough to fit some use cases with the recent +development in edge and constrained devices. +In this work, we propose and extensively evaluate Feder- +ated Split Learning (FSL), a system for efficient training and +inference with high-level privacy for clients’ source data. +We present a Client-based Privacy Approach (CPA) for split +learning-based systems to provide high attack resilience by +adding noise to the intermediate data. We also study the +training time, accuracy and privacy level of different ways +to partitioning a NN in FSL. As further works, we aim to +research prediction based NN partitioning methods. +Moreover, comparisons between FSL and existing NN +training and inference systems at the edge are carried out, +along with explanations of their pros and cons against +FSL. 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Available: https://opacus.ai/ + diff --git a/cNAzT4oBgHgl3EQf2_7N/content/tmp_files/load_file.txt b/cNAzT4oBgHgl3EQf2_7N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..717124f695c2fa32b194b576b6454d8de2505ba6 --- /dev/null +++ b/cNAzT4oBgHgl3EQf2_7N/content/tmp_files/load_file.txt @@ -0,0 +1,1130 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf,len=1129 +page_content='Privacy and Efficiency of Communications in Federated Split Learning Zongshun Zhang‡ Andrea Pinto⋆ Valeria Turina⋆ Flavio Esposito⋆ Ibrahim Matta‡ ‡Computer Science Department ⋆Computer Science Department Boston University Saint Louis University Technical Report Abstract—Everyday, large amounts of sensitive data is dis- tributed across mobile phones, wearable devices, and other sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Traditionally, these enormous datasets have been pro- cessed on a single system, with complex models being trained to make valuable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Distributed machine learning techniques such as Federated and Split Learning have recently been developed to protect user data and privacy better while ensuring high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Both of these distributed learning architectures have advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this paper, we examine these tradeoffs and suggest a new hybrid Federated Split Learning architecture that combines the efficiency and privacy benefits of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our evaluation demonstrates how our hybrid Federated Split Learning approach can lower the amount of processing power required by each client running a distributed learning system, reduce training and inference time while keeping a similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We also discuss the resiliency of our approach to deep learning privacy inference attacks and compare our solution to other recently proposed benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Introduction CENTRALIZED machine learning (ML) training is be- coming unsustainable [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Aside from the advantages of re-training often to optimize revenues [2], several learning applications need to run their processes at the edge of the network, not in the core of a datacenter, for multiple rea- sons, including end-to-end latency minimization by running machine learning algorithms locally on an end-device, and privacy concerns of trusting third-party clouds [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Several Machine Learning (ML) models trade user experience im- provements on mobile devices for sensible data exploitation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', text recommendation in keyboards [4], [5] or vocal assistants [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In these and other applications, a decentralized learning approach may be preferable to a centralized system since sensitive data may remain locally within a client and not transferred over a computer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Despite its benefits in several use cases, running machine learning training and inference jobs within local devices has several limitations: computing capacity is often limited, battery drains faster with intensive processing and the mo- bile or other end-devices have limited memory and storage capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For example, our experiments show that to fine-tune a VGG-16 [7] Neural Network, pre-trained on ImageNet [8] with Cifar-10 [9], tens of minutes are needed to reach 90% accuracy on an NVDIA V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Different distributed neural network architectures have been proposed to preserve privacy and guarantee timely convergence – for example Federated Learning [1], Split Learning [10], or hybrid approaches [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Feder- ated Learning (FL) [14] averages the weights of the learned Neural Network model on each edge device to create a single model, which will update the local ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Previous research has shown that this strategy can achieve higher accuracy than considering only a local model [14], [15], [16], [17] and at the same time can preserve the privacy of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Split Learning (SL) architecture splits the entire NN into partitions of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Each partition is executed on a different entity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', edge and cloud), and different edge NN partitions can be paired with the cloud partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, this approach takes advantage of distributed datasets while keeping user data private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' On one hand, FL is easy to scale to many devices given that there are enough resources to meet training Service Level Agreements (SLAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, FL is impractical in edge training/inference settings, where resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' On the other hand, Split Learning can train with limited re- sources, but it doesn’t scale to many devices well since it is not parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Especially when we pair different edge devices with not independent and identically distributed (Non-IID) data with the central cloud, the training may not converge at all [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Another drawback of SL is that the intermediate data can be costly to transmit and store in client or server nodes [19], [20], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Furthermore, since the intermediate data in the forward propagation is derived from the source data, there is a privacy concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We further discuss these two models in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To cope with the inefficiencies of the existing distributed learning models, we propose a novel distributed learning architecture, Federated Split Learning (FSL) [23], which combines the benefits of FL and SL while mitigating their drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We discuss the generality of our FSL and a novel methodology to optimize both delay and privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='01824v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='LG] 4 Jan 2023 The FSL model is characterized by multiple edge client – server pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Such pairs train their copy of the NN simul- taneously, providing the parallelism of federated learning, while the client-server separation brings the advantages of the split learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Each computing pair partitions the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After some pairs have completed some training epochs, the server NN weights are averaged in a central cloud server, as in classical FL algorithms [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We call this central cloud node the “Parameter Server node” in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' There are other techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', Parallel Split Learning (PSL) [13], and Federated Reconstruction (FRC) [24], close to our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' But they have some disadvantages com- paring to FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' While the PSL architecture has only one cloud server node, FSL allows parallel server computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL and FRC have similar privacy levels (Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1), but our evaluations shows that FRC doubles the training time compared to FSL, considering no dominant transmission delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We evaluate the benefits of our FSL architecture by testing it with different NN models and tasks: from im- age classification to an Internet traffic classification [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Aside from testing training performance of our proposed hybrid federated-split architecture, we evaluated the privacy- performance tradeoff of FSL and SL, and give ideas on how to enhance the privacy guarantee of these schemes using Client-based Privacy Approach (CPA) and novel neural net- work partitioning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Furthermore, we realized that certain ways of partitioning NN could reduce transmission delay and enhance privacy together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our experiment results show that by combining different privacy approaches and NN partitioning methods, our FSL may achieve both high efficiency with respect to training time, privacy guarantees and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Sec- tion 2, we introduce the background and the related works to discuss FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, we describe the proposed FSL system in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Section 4 we discuss the Client-based Privacy Approach (CPA) and motivations to partition NN at edge that can be applied in any split learning based architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our evaluation with training and inference metrics is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2, while the privacy evaluation is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Distributed Learning Architectures: Background and Related Work Federated Learning [1], [15], [16], [17], [26] is a de- centralized machine learning technique that trains neural network models using data sources “owned” by multiple clients (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' A logically centralized parameter server holds the latest neural network model, and orchestrates the sharing of its weights between all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' At the beginning of the training phase of a federated learning process, the parameter server sends the same ran- domly initialized set of neural network weights to each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Each client then trains a local model for multiple epochs using its local dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Until the client models have extracted enough features, that is, a given accuracy threshold is reached, the parameter server keeps retrieving, averag- ing, and overwriting the weights (Figure 1a – steps 1 to 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, the global model could take advantage of the privately-owned datasets which would not transmit through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, FL has parameters to specify the frequency for the parameter server to average the weights of a certain group of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this way, system architects can balance the network traffic and model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, FL is considered scalable in terms of the number of clients, as long as such clients have enough computational power and storage resources to meet the training constraints, or Service Level Objectives (SLOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Split Learning [10] is a distributed machine learning tech- nique that is characterized by a computational split of the neural network model into two partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Each partition could run on a separate computing node, hence splitting the computational resource demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To train a NN model with split learning, the NN must first be partitioned on different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, the forward propagation phase starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When the end of the NN partition at the first node is reached, the outputs of the last layer of activation functions are sent to the server node — step 1 (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then those outputs are used as inputs to the second NN partition and continue the forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After calculating the loss, the backward propagation in the server node is started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Once the backward propagation reaches the input layer of the server’s partition, the gradients of the inputs are sent to the last activation in the client to finish the backward propagation for client NN — step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Finally, the complete NN weights are updated with the gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, the server’s NN partition can also pair with other clients’ NN partition with the same client NN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' First, the trained weights on the last client are moved to the new client — step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then the new server and client pair trains as mentioned above — steps 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The SL algorithm preserves data privacy but suffers from a long convergence time with Non-Independent and Identi- cally Distributed (non-IID) data sources (Figure 2), and large intermediate data to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, training with more than one client is sequential, hence poorly scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Some solutions attempted to mitigate the transmission delay of such intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For example, in BottleNet [20] and BBNet [19], the authors aim at compressing the intermediate data with a particular NN design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Early-Exit [21] instead the idea is to add classifiers at the early layers to avoid computing the complete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Previous work [22] also used knowledge distillation to reduce the complexity of client models and the data to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this paper, we consider hybrid split-federated learning systems, to combine their benefits while minimizing their drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For example, we show that hybrid Federated Split Learning and Parallel Split Learning can converge with non-IID sources faster than Split Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Combining Split and Federated Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Other re- searchers have proposed combining the advantages of split and federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Parallel Split Learning (PSL) [13] (Figure 1c), they train the client NN partition on multiple AVG = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='" ∑#$% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='𝑤# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='WEIGHTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='INTERMEDIATE RESULTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='GRADIENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='(a) Federated Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='WEIGHTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='INTERMEDIATE RESULTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='GRADIENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='(b) Split Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='CLIENT B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='WEIGHTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='INTERMEDIATE RESULTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='GRADIENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='(c) Parallel Split Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='(d) Federated Split Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='Figure 1: Distributed NN Training Architectures: (a) Federated Learning: The NN is in the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The parameter server calculates the average weights among clients and overrides the local weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (b) Split Learning: The server partition sequentially trains with each of the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Client weights are shared with the next training client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (c) Parallel Split Learning: The server trains clients’ output in batches in parallel, but the client’s weights are kept private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (d) Federated Split Learning: Multiple Edge Server and Client pairs train simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The Edge Servers’ weights are averaged by a Parameter Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The clients weights are kept private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' edge nodes in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' During the Forward Propagation phase, the activation function results from different clients are sent to a single remote server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Such server then backward propagates the gradient from the loss function to the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Model specific approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FedSL [11] is one example of Recurrent Neural Networks (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The idea is to unroll the RNN’s feedback loop and split the recurrent NN partition to different nodes with the sequence data segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After each epoch, devices average and overwrite their weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This approach combines the benefit of SL and FL in the RNN training setting without introducing much overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FL Extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Some works extend the FL model instead of designing a hybrid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The authors of Federated Construction (FRC) [24], prioritize user data privacy trading off the efficiency of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Their model is partitioned into global and local shards and both deployed in each edge device, and the two partitions are trained alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' A parameter server then sends the global shard and retrieves the corresponding updated weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This de- sign makes the training stateless and, consequently, highly scalable for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Source Data Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' One of the advantages of these edge training systems is the high perceived level of source data privacy: user’s data doesn’t leave the edge device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, the adversaries can also learn the source data from the NN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' All systems mentioned except for FRC and PSL cannot maintain the privacy for NN weights [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' They maintain this privacy since they do not share the complete NN weights or the activation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Another threat discussed in NoPeek [29] targets the results of the activation function, which can be used to reconstruct the source data with an autoencoder NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Certain edge intelligence systems that partition Neural Networks and transmit intermediate data in between partitions are Figure 2: Accuracy degradation when training with Non-IID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' vulnerable to this threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Possible mitigation uses the Dis- tance Correlation (DC) loss [29] added to the original loss function to measure the difference between the source and the intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This approach maximizes DC loss and accuracy while updating the NN’s weights by solving a Multi-Objective Optimization Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Client-Based Privacy Approach (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' NoPeek’s loss function improves the resilience in reconstructing private source data from the intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, sharing the loss and constructing the distributed gradient graphs are either unsafe or need an extra management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To overcome this limitation, we propose an extension from NoPeek, namely Client-based Privacy Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The idea is to add noise with user specified methods to the results of activation functions in clients, so there would be less features to be used by the autoencoder NN (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We will mainly compare DC and Differential Privacy (DP-SGD) under CPA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' DP- SGD is a widely used lightweight algorithmic approach for data privacy [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The idea is to add a Gaussian noise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 ParallelSplit FederatedSplit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 SplitLearning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 X(percentage of unbalance)PARAMETERSERVER ↓Zl=o Wi SerAVG : 3 EDGE SERVER AEDGE SERVER B CLIENT A CLIENTB data data INTERMEDIATERESULTS GRADIENTS WEIGHTSto the gradients during a training phase of a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus the client’s output, generated by the updated weights, will confuse the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Comparing to Homomorphic Encryp- tion (HE) or Secure Multi-Party Computation (SMPC), we consider it fits better in our client and edge server setup, while the privacy level of encryption has been well discussed [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' It is not cost efficient at edge to consume a lot of battery for encryption and decryption steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' There are also methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', model compression( [19], [22], [32]), can potentially enhance privacy guarantee and consume power lower than HE but higher than DP-SGD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' But comparing existing privacy methods is not the focus of this project and we consider it as a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We mainly want to show the proposed CPA is a general approach and it provides high attack resilience with different methods including DC and DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, in this paper, we evaluate the traditional DP-SGD applied to the complete model and compare it with DP-SGD only applied to the clients using the CPA framework (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy-Oblivious FSL In this section, we detail our proposed Federated Split Learning (FSL) architecture, that we originally proposed in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL is a hybrid approach that combines the ad- vantages of SL and FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' It avoids sending users’ source data or sharing the complete NN parameters through the network while being scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our FSL architecture shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 1d has three types of entities: (i) edge servers, (ii) clients, and (iii) param- eter servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To train a NN with FSL, we first setup an authentication protocol [33], [34] among the entities to pair each client with one edge server, and edge servers with a parameter server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After pairs are found, the communications are sent without encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, in each client and server pair, we partition the complete NN into the client’s partition and (edge) server’s partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL has three training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In step 1, the client forward propagates with source data and transmits the intermediate data to the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The server then finishes the propa- gation and calculates the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In step 2, the server backward propagates to client source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In step 3, after epochs, the parameter server averages the weights in the edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consequently, FSL will have multiple advantages com- pared to the other approaches discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL clients will have a lower resource demand compared to FL since it will have fewer NN layers to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus FSL is a more practical scheme for edge intelligence application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Also, while FSL only averages the weights in the edge servers, FedAVG averages the complete weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Therefore FSL avoids po- tential vulnerability to model inversion attacks [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, FSL provides better scalability than SL, since client and server pairs can train independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Compared with the FRC [24] architecture, we note that the latter is inefficient in training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' It updates the parameters of one partition on each forward and backward propagation execution and runs multiple times to update the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Compared with Parallel Split Learning [13], we found four potential suboptimalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' First, since the edge devices have to synchronize with the central server, clients may have to wait until the server has finished with processing all the results of the activation functions in its queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In the worst case, assuming that all activation function results from n clients arrive at the same time, the lower bound of waiting time for each client to continue on with backward propagation is O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We hence conclude that PSL is not as scalable as our FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Second, we also observe that the PSL server would temporarily store multiple batches of the results of activation functions, so it needs a sophisticated logging and compaction storage system to recover from failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consequently, PSL is less robust than our FSL, since FSL has to maintain fewer states in each isolated pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Third, the PSL design may suffer from resiliency problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Meanwhile, in FSL, failure in one pair won’t prevent other pairs from training or inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Fourth, in PSL, since all intermediate data will be transmitted and processed by the single server, bandwidth and computation resource at the server node may get congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our findings are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy-Aware FSL We have discussed the efficiency and fault-tolerance properties of FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this section, we consider instead the privacy-preserving properties of different architectures and propose our privacy-aware FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, we discuss how to complement general split learning based architec- tures to mitigate the problem of sharing the output values of NN activation functions or weights over an honest but curious network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We first give a formal definition of our privacy attacker model, and then we discuss how a Client- based Privacy Approach and certain ways of partitioning Neural Network would help avoid such attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy Attacker Model and Assumptions We assume an attacker can capture the Intermediate Data (the results of last layer activation functions trans- mitted from client to edge server) in plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, we assume that attacker knows the client NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consequently, the adversary can implement an AutoEn- coder [35] NN to reproduce the source data fed into the client model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, to train the autoencoder NN, we assume that some datasets with features similar to the client source data are accessible to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Attack Resilience Given the attacker model, in order to compare the level of privacy guarantee among different privacy approaches, we define an Attack Resilience metric (τ) as: τ = 1 − ∥correct∥ ∥reconstructed∥ (1) It measures the misclassification rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' ∥correct∥ counts the number of images, reproduced by the attacker, which can be correctly classified by a trained classifier (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' And ∥reconstructed∥ is the total number of reproduced images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Client-Based Privacy Approach in Distributed Setting via Distance Correlation (CPA-DC) Motivated by the NoPeek approach [29], where weights are updated based on the sum of two loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', Cross-Entropy, and Distance Correlation (DC), we adopted the idea of using two loss functions into an alternately scheduling mechanism with two rounds: the regular round, minimizes the cross-entropy loss function in the (edge) server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The Distance Correlation round, maximizes the DC loss function in the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The idea is that we add noise that make source and intermediate data different to client’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We define this alternating behavior with loss func- tions in Equation 2: L = � Loss(g(f(x)), label) if e mod F == 0 m · DC(x, f(x)) otherwise, (2) where L is the measured loss value, e represents the epoch index, F represents the DC Frequency, Loss(·) is the loss function used to measure mis-classifications, DC is the distance correlation loss, m is the loss multiplier, f is the client NN, g is the server NN, x is the source data, and label is the labels in the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Notice that F controls the alternation frequency and m adds a weight to the DC loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The alternating loss function is a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our Client- Based Privacy Approach (CPA) can also work with other loss functions or methods adding random noise to the regular round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In the evaluation, we explored several methods to embed the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, we explored the trade-off between training time and the highest attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' How many layers do we assign to each neural network partition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this section, we discuss the problem of selecting how many layers need to be assigned for each NN partition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', client NN depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This tradeoff will tune training time (processing and transmission delay), privacy, and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Capturing the tradeoff between all these metrics is challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To illustrate, consider the tradeoff between processing delay and transmission delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The size difference among output layers in different partitions can be large, so a few partitioning policies may lead to significant transmission overhead, increasing training time and hence diminishing the gain of the hybrid FSL compared to the original Federated Learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In VGG-16 [7], the output size of the first convolutional layer is two times the size of the second convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, a system with a model cut after the second convolutional layer can tradeoff the extra pro- cessing delay at low-capacity clients while yielding a lower transmission delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We evaluate this effect in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Analytically, this effect is captured by solving Prob- lem 3, where α and β are developer-specified parameters that represent positive weights for the transmission delay of intermediate data (I) and computation delay (C), the pa- rameter d represents the depth of the client neural network, and finally, b represents the bandwidth, which is measured periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To efficiently solve this problem, we follow the approach in [36], where we build two regression models to predict the delays I and C, given the available bandwidth by profiling the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', computing the output size and processing time for each layer in the client NN, instead of training the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' min d (αI(d | b) + βC(d)) (3) The solution of Problem 3 is optimal with respect to de- lays, however, it can be sub-optimal with respect to privacy and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 shows, the client processing and transmission delay of FSL reach the minimum when the client NN depth is between 7 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 7a, instead, the client NN needs more than 16 layers to be above 90% attack resiliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, we conclude that an optimal NN partitioning decision should balance different objectives and constraints, including transmission delay, pro- cessing time, privacy, and accuracy, as shown in our Problem formulation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In such a problem, W represents the model weight vector, (I′, C′, A, R) is the tuple representing the observations for transmission delay, computation delay, ac- curacy, and resilience, (γ, κ) are new user-specified positive weights, and d is the client NN depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' max W,d (−αI′(W, d | b) − βC′(W, d) + γA(W, d) + κR(W, d)) (4) To solve Problem 4, we have to train W for each d until convergence and then find the best d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This brute force method is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' A more efficient approach would rely on predicting the delays, accuracy, and privacy without the full training of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Extending the approach in [36] to go beyond profiling delays, is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This is because the accuracy and attack resilience for each client and edge server pair is harder to profile and predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Specifically, their profiling depends on the weights trained on other pairs, the distribution of source data among clients, number of clients, number of layers to average in SerAVG, and training epochs (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Another work can predict the model accuracy [37], but it is based on the already trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Therefore, for our FSL architecture with SerAVG, a predic- tion method for partitioning remains an open question for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this paper, we experimentally demonstrate the best model partitioning that balances requirements on training time, accuracy, and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Results In this section, we describe the evaluation results re- lated to our Privacy-Oblivious FSL (POFSL) and Privacy- Aware FSL (PAFSL) architectures with our privacy-aware approaches (CPA in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3 and Neural Network Par- titioning in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our evaluation demonstrates the advantages of FSL over PSL and FRC in terms of training time, memory usage, and convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, we also show that our privacy-aware approaches can prevent the reconstruction of source images from intermediate data in the Split Learning-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We first discuss our experimental setup, then present our evaluation results of POFSL and PAFSL in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Experimental Setup This experiment set studies the convergence for POFSL and the privacy guarantee of PAFSL across different hard- ware and applications with different NNs and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For the hardware, we used two types of nodes on Chameleon Cloud [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' One has an RTX6000 GPU, two Intel Xeon Gold 6126 CPUs and 187 GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The other one has four NVIDIA V100 GPUs, two Intel Xeon Gold 6230 CPUs and 128 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We emulated the computer network among our distributed learning entities on the localhost interface on a physical machine, and each experiment was set to use a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' So that we can ignore the network bandwidth bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For the applications, we considered three classification tasks and implemented with PyTorch [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then the dis- tributed communication among entities of the systems was handled by PySyft [40] and PyGrid [41] and no encryption is applied on the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The first application runs a general image classification task with a VGG-16 [7] Convo- lutional Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The model was pre-trained using Imagenet [8] and then trained with the CIFAR-10 dataset [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We run this task on 5 clients running a NVIDIA V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The second task uses a LENET [42] CNN to recognize handwritten numbers in the MNIST dataset [43] on 20 clients running a NVIDIA RTX6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The third task classifies traffic, not images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, we decom- posed a one-dimensional-CNN, trained with the ISCX VPN- nonVPN (ISCX) traffic dataset [44], using 5 clients running on a RTX6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We partitioned the dataset and assigned among different clients with Independent and Identically Distributed (IID) probabilities and all our plots show 95% confidence intervals, unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our goal is to verify that FSL can always converge, with different tasks, different NNs, different devices, and different data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We verified the advantages in delay or privacy of FSL over existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Results for Privacy-Oblivious FSL This section illustrates the methodology and draws ob- servations of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Overall, our evaluations show that POFSL has less overhead and similar accuracy compar- ing to existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, we evaluate training time (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2), memory consumption (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3), and learner accuracy (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Experiment Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Given the size of the different datasets and number of clients, to reach at least 90% ac- curacy, the neural networks used for image classification needed 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' While for the traffic classification model, 80 epochs were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Training Time Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To evaluate training time, let us consider the experiment whose results are re- ported in Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis indicates the Cut Index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', the index of the last layer running in the client/local part of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When tested over the MNIST scenario, we can observe from Figure 3a and 3b that FSL has the shortest “Client Forward and Backward Propagation” (Client F&B) time among all other distributed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The Client F&B time includes transmission time for gradients and computing both activations and gradients in Client NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' And Server F&B time includes transmission time for hidden variable from client to server and computation in Server NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Notice that the weight update time is separately counted by “Client Update Time“ and “Server Update Time“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, we note that PSL is more vulnerable than FSL to limited bandwidth across splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' PSL is consistently the slowest, due to its inefficient server design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' the server has to synchronize the intermediate data, and it must process all batches of intermediate data in each training epoch sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Observing FRC and FL, we see the F&B times do not change along with the Cut Index (Figures 3a and 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note also that FRC is not training time efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' It updates its complete model with two almost full forward and backward steps [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This can be noted in the same figures: the FRC total F&B time for local and shared weights is almost doubled compared to the FL training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We were able to obtain similar results comparing the F&B times on another predicting scenario: the 1D CNN implemented by [44] (Figures 3d and 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Due to the limited size of this neural network (with only two convolutional layers), we evaluated the architectures with merely two Cut Indexes: at layer 3 and layer 6 of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Even in this experiment, we observe how our FSL still has the shortest Client F&B time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' PSL is the worst performant at each cut, and FL keeps performing better than FRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL consistently uses less time in each training epoch than the other analyzed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We found that PSL perform worse than FSL because of the single-server architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' PSL has similar results when comparing its client F&B time with FL and FRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FRC is not training time efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Its total F&B time almost doubles compared to FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When evaluating the training time on the CIFAR-10 scenario, we found a different trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4a): PSL had the longest training time, except for cut index of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, FSL did not always perform the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When most of the layers run within the client, FL has a shorter training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This is because the size of intermediate data changes as the cut moves, and with smaller data to send, the overall training time can be shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4b and 4c show the extra F&B time during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' And existing works for SL have discussed the similar behavior [19], [20], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, we show that FSL outperforms PSL as PSL F&B time is more vulnerable to the intermediate data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Figure 3, Client F&B time of FSL keeps decreasing with smaller Cut Index, while that of PSL still increases at Cut Index 6, 5 and 3, 2, although the intermediate data in this experiment is much smaller than 1 2 3 4 5 6 7 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 Client/Local F&B Time (s) PSL FSL FL FRC (a) LENET+MNIST Client Measurements 1 2 3 4 5 6 7 Cut Index 0 2 4 6 8 10 Server/Shared F&B Time (s) PSL FSL FL FRC (b) LENET+MNIST Server Measurements 1 2 3 4 5 6 7 Cut Index 0 50 100 150 200 250 300 Intermediate Data Size (KB) PSL FSL (c) LENET+MNIST Intermediate Data 3 6 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 Client F&B Time (s) PSL FSL FL FRC (d) VPN Workload Client Measurements 3 6 Cut Index 0 1 2 3 4 5 Server/Shared F&B Time (s) PSL FSL FL FRC (e) VPN Workload Server Measurements Figure 3: LeNet+MNIST: (a) Client Time, (b) Server Time, (c) Intermediate data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Observations: 1) Intermediate data size is correlated with the times taken by the PSL architecture while having little correlation under FSL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 2) FRC has almost twice the overall training time as FL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 3) Plots are obtained by averaging 20 clients’ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Intermediate data size is under batch size of 16 and each image was resized to (1,32,32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' VPN Workload: (d) Client Time (e) Server Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Similar considerations are valid for the VPN dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Tested with 5 clients with a one dimentional NN with input size of (1,784).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Still, FSL has the shortest Client F&B time compared to the other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4 7 16 30 Cut Index 0 50 100 150 200 Training Time (s) PSL FSL FL FRC (a) Pair Time 4 7 16 30 Cut Index 0 10 20 30 40 Client Time (s) PSL Client Update FSL Client Update PSL Client F&B FSL Client F&B (b) Client Time 4 7 16 30 Cut Index 0 20 40 60 80 100 Server Time (s) SerAVG PSL Server Update FSL Server Update PSL Server B&F FSL Server B&F (c) Server Time 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 7 16 30 Cut Index 0 1 2 3 4 5 6 7 8 Intermediate Data Size (MB) PSL FSL (d) Intermediate Data Size Figure 4: VGG + CiFar10: Plots show the effect of Cut Index over (a) averaged overall training time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (b) time spent in clients (5 clients average),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (c) time spent in server,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' (d) intermediate data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The transmission delay caused by intermediate data size can dominate the training time (occurring during F&B propagation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' using VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This behavior is caused by the single server bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, FSL is more scalable in terms of training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Such observation also explains why both client and server F&B times of PSL are consistently larger than FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The intermediate data and gradients can cause signifi- cant network overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Such overhead, however is better mitigated by our FSL than PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Existing work [19] [21] for Split Learning, as well as our partitioning strategies (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4) can further mitigate the communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, the additional delay in FSL is not considered a severe bottleneck compared to those systems training at the edge, like FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Memory Consumption Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Memory usage of each entity in the edge training and inference systems limits the scope of devices that can join the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To compare which system is more flexible to deploy in terms of memory capacity on devices, we show that each entity has calculated memory usage in the FSL, PSL, FL, and FRC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Real-world memory utilization can be highly variable as it depends on several implementation factors, such as libraries used and the Remote Procedure Calls (RPCs) im- plemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, the size of a model and its activation at each layer are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The following results show that FSL’s clients consume less memory compared to FL and FRC, and any of its servers occupy less memory than PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The two plots in Figure 5 show the memory demands computed at the client and the server for each architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis shows the Cut Index and the y-axis represents the corresponding expected memory usages in MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note that FL and FRC do not split the NN, so their memory demands are only shown in the left plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As shown in Figure 5a, the sizes of each client NN’s weight and the results at each layer are the same in FSL and PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Since FRC and FL compute the full NN in the client during training, they require more than five times the memory, for Cut Index 4 to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Figure 5b instead shows that the server memory demand decreases with the cut index, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, notice that PSL server need more memory to hold intermediate data from different clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Memory usage of FSL compared to other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To conclude, we found that FSL’s servers are lightweight com- pared to the PSL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consequently, state management would be easier in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Also, FSL’s clients are lightweight compared to FRC and FL, during training, so they are more suitable at edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Learner Accuracy Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this subsection, we focus on evaluating the convergence of SerAVG, pro- 4 7 16 30 Cut Index 0 100 200 300 400 500 600 Client Memory Usage [MB] PSL FSL FL FRC (a) 4 7 16 30 Cut Index 0 100 200 300 400 500 600 Server Memory Usage [MB] PSL FSL FL FRC (b) Figure 5: FSL’s server has lower memory demand compared to PSL, and FSL’s client has lower memory demand com- pared to FRC/FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' VGG + CiFar10: (a) Client and (b) Server memory demand (sum of model weights size and outputs at each layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Batch size is 32 and each image was resized to (3,32,32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' posed and detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Unlike FedAVG in Feder- ated Learning, SerAVG averages the server NN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We begin by discussing the correctness of SerAVG, comparing the accuracy of SerAVG PSL, and FedAVG (FL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, we compare the convergence rates based on different source data sizes and Cut Indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' SerAVG Evaluation: In this experiment set, we evaluate if SerAVG can enhance the accuracy of every model joining the training process, given that the source data at clients are non-IID distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our results are reported in Fig- ures 6a, 6b and 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To train the MNIST model, we split the training set in two parts, part 0 and part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The two parts represent two collections of skewed data sources at client ge- olocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We then let each part include data corresponding to half of the labels in the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We then split the MNIST test set into two parts in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note that the aforementioned way of splitting training set and test set is an extreme Non-IID case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For example, a model trained on part 0 of the training set has no knowledge of the labels in part 1 of training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To assess how well the systems may learn and predict on Non-IID data on more realistic data distributions, we further add 10% samples uniformly and randomly, selected from the complete MNIST dataset to all the four parts of datasets so that the model trained on either part of the training set may be able to classify labels in the other partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consider Figures 6a, 6b and 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis represents the data partitions used in training and validation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', 0 & 1 means data part 0 was used in training and part 1 was used during the validation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The y-axis shows the validation set accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' From left to right, the accuracy decreases when the server/shared NN for FSL, PSL, and FRC is shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When the Cut Index is 3, SerAVG, FedAvG, and PSL perform equally well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When the Cut Index is greater than 3, SerAVG is worse than FedAVG and lower but close to PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We conclude that SerAVG can enhance the accuracy in the Non-IID source data setting, while worse than FL and PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note that FedAVG averages all weights, while FSL never shares the client’s weight in the SerAVG setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, FSL clients cannot benefit from the gradients calculated at other clients and that may lead to lower accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Comparing PSL and FSL, the PSL server optimizes for minimal loss using all clients’ batch output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' On the other hand, SerAVG averages the trained weights on each server heuristically based on FedAVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, SerAVG will have lower accuracy compared with PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' And each client NN trained with a Non-IID dataset can extract little features from the other Non-IID dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note also a similar but smoother drop in accuracy based on Cut Indexes with Independent and Identically Distributed (IID) partitioned CIFAR10 dataset and VGG16 NN in Fig- ure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The two experiments with MNIST and CIFAR10 datasets suggest that SerAVG can preserve similar accuracy patterns even when applied to different NN models and Cut Indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Tradeoff between resource demand and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this experiment we evaluated the accuracy of FSL with small resource demand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', when the size of the input data is limited, and the client model runs on limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We found that with LENET and MNIST, the validation dataset can reach an accuracy range of 87% to 93%, as long as each client has enough data to train the machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' By varying the number of clients, we quantified the expected drop in accuracy for both architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our results are shown in Figure 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis indicates the index of epochs, and the y-axis shows the corresponding test set accuracy after a certain number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The dataset is IID among 20, 100, and 500 clients to study how the data size affects convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Figures 6a, 6b, 6c and 7a, we also noted that FSL (implementing the SerAVG mech- anism) keeps high accuracy when the Cut Index is small, allowing deployments over resource limited device given high performance requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We expect a higher accuracy for both architectures with more effort in tuning the hyper- parameters, given prior results in similar contexts [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' While our accuracy results show 95% confidence intervals, parameter tuning is out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Results for Privacy-Aware FSL In this section we present the evaluation results of our privacy-aware FSL (PAFSL) architecture and show that it can provide certain privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FSL clients do not share the source data and model weights, so adversaries cannot directly access the source data or reconstruct them with the model weights using model inversion attacks [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, when compared with Federated Reconstruc- tion (FRC) [24] which trains a complete model at the client, FSL still sends the intermediate data through a network to complete the forward and backward propagation between clients and (edge) servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' An adversary could use such data to reproduce the source data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', through an Autoen- coder [35] NN, trained with certain dataset, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To assess how our approach mitigates such vulnerabili- ties, we first introduce the evaluation setup, the design and usage of the attacker Autoencoder NN, and our experiment 0&0 1&0 Train Part & Val Part 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Accuracy FedAVG SerAVG NonAVG FRC PSL (a) Accuracy (Cut Index = 3) 0&0 1&0 Train Part & Val Part 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Accuracy FedAVG SerAVG NonAVG FRC PSL (b) Accuracy (Cut Index = 5) 0&0 1&0 Train Part & Val Part 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Accuracy FedAVG SerAVG NonAVG FRC PSL (c) Accuracy (Cut Index = 7) (d) Accuracy (different number of clients) Figure 6: In plots (a), (b) and (c), the accuracy of SerAVG is better than NonAVG but lower while mostly close to FedAVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' SerAVG: Average the server NN’s weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' NonAVG: Each pair trains on its own;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' FedAVG: Average the complete NN’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In plot (d), the FSL and PSL accuracy is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 4 7 16 30 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='88 Accuracy PSL Accu FSL Accu FRC Accu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Attack Resilience PSL Resi FSL Resi (a) Privacy Oblivious (VGG16+Cifar-10) 4 7 16 30 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Accuracy PSL Accu FSL Accu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 Attack Resilience PSL Resi FSL Resi (b) DC 1 & Muiltiplier=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 (VGG16+Cifar-10) 1 2 3 4 5 6 7 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Accuracy PSL Accu FSL Accu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Attack Resilience PSL Resi FSL Resi (c) Privacy Oblivious (Lenet+MNIST) 1 2 3 4 5 6 7 Cut Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Accuracy PSL Accu FSL Accu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Attack Resilience PSL Resi FSL Resi (d) DC 1 & Multiplier=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3 (Lenet+MNIST) Figure 7: Accuracy and Attack Resilience for Privacy Oblivious and Privacy Aware Architectures based on Cut Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Note: DC 1 in captions refer to DC Frequency = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' These figures show that to reach high accuracy and attack resilience, the Cut Index cannot be too big or too small and loss multiplier is another way to enhance attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Furthermore, loss multiplier can be more practical than DC Frequency, since it doesn’t introduce overhead to the training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, we discuss the results of different pri- vacy approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', NoPeek [29] and the Client Based Privacy Approach (CPA), and privacy level of different ways of paritioning the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' NoPeek solves a multi-objective opti- mization problem of two loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', one maximizes accuracy, and the other maximizes the differences between source images and intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For CPA, we evaluate CPA-DC and CPA-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' CPA-DC (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3) optimize the two loss function in NoPeek alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' CPA-DP applies a DP-SGD [30] algorithm in the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Finally we evaluate the privacy guarantee of different partitioning of client NN and server NN motivated by Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We conclude this section by presenting results that demonstrate the high resilience to privacy attacks of our proposed FSL and the advantages in training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our Privacy-Aware FSL and PSL extend our Privacy-Oblivious version by adding the CPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Partitioning the NN was made easy by considering only sequential NNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', LeNET and VGG16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We tested both systems with the same image classification workloads (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', MNIST and CIFAR10) on the same hardware (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', NVIDIA RTX6000 and NVIDIA V100, respectively) as the Privacy-Oblivious setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Setup the Attacker’s Auto-Encoder Neural Net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this subsection we explain how we define our pri- vacy attack, given the assumptions mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To understand how the privacy attack works, it is useful to recall how the Autoencoder NN that we used work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In particular, the Autoencoder is composed of two parts: an encoder and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We need a dataset similar to the source images to train the Autoencoder NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' While the encoder uses convolutional layers to extract latent variables from its input dataset, the decoder uses the last layer’s activation function outputs from the encoder and transposes the convolutional layers to reproduce the input dataset of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Consequently, the encoder NN structure is the same as the client NN structure, and the attacker is the decoder NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The ith layer of encoder would be the ith last layer of decoder, with transposed convo- lutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We assume the attacker’s decoder structure strictly mirrors the client NN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, we expect better attack resilience for SL-based systems in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy Evaluation: Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this subset of our evaluation, we want to show both CPA and carefully designed ways of partitioning NN provide high attack re- silience while preserving high accuracy, based on different datasets, NNs, for split learning based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Each experiment includes trials initializing the Autoen- coder’s weights and data-loaders with different random 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 FSL 20 FSL100 FSL500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 PSL_20 PSL 100 PSL 500 0 3 6 9 12 15 18 Epochsseeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In each trial, the attacker used a dataset containing similar features to the learner’s source dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To recon- struct MNIST, we selected EMNIST [47], and for CIFAR10 we selected the CIFAR100 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The EMNIST dataset con- tains hand-written characters instead of numbers in MNIST, so features like lines and curves are the same and attacker can decode those activation function outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Similarly, the CIFAR100 dataset contains 100 classes of RGB images in- stead of the 10 classes in CIFAR10, so the common features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' classifying cat or dog, can be used to reconstruct with the activation function outputs from the CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For each trial of the experiment, we first let the attacker learn to reproduce her datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Based on the MSE loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' a loss function that measures how different the original and reproduced images are, the attacker updates her weights in each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After 20 epochs, we used the decoder to reproduce the learner’s dataset from the intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When an autoencoder NN is trained, we train a new classifier with the same NN structure and source data as the learner to classify the reproduced images for 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The mean and standard deviation of this classifier’s attack resilience τ in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We show the results comparing the four systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', POFSL, POPSL, PAFSL, and PAPSL, with different CPAs and Cut Indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Then, we evaluate the trade-off between accuracy and attack resilience for FSL and PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy Evaluation using NoPeek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As we illustrated in Section 2, NoPeek solves a multi-objective optimization problem that takes in the source data and intermediate data to maximize the difference with a Distance Correlation (DC) loss function, as well as the prediction and labels to maximize the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To solve such optimization prob- lem, NoPeek has to share the value of the loss over an network which may cause vulnerability or added complexity of maintaining the gradient graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As shown in Table 1, this approach has both high attack resilience (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', 97% for PSL and 98% for FSL) and high learner’s accuracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', 97% for PSL and 96% for FSL) when trained for the same number of epochs and clients as the Privacy Oblivious experiment with the MNIST dataset and LENET NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' cases PSL FSL attack resilience(τ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9837 learner accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9614 TABLE 1: NoPeek stats with 20 clients training 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Result using Client-Based Privacy Ap- proach via Distance Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' To mitigate the draw- backs of the loss value sharing, we consider a new approach that prevents transmitting data outside clients, improving upon NoPeek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We optimize for the similar two objectives in NoPeek alternatively in the Client-Based Privacy Ap- proach (CPA) via DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As shown in Equation 2, there are DC Frequency (F) and Loss Multiplier (m) to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' DC Frequency (F) defines how many times the DC loss function is optimized after the loss function for accuracy is optimized once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Loss Multiplier (m) is applied to the loss function result but has the equivalent effect of multiplying the learning rate by a factor m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' These two parameters control how different the intermediate data and source data will be, by changing the frequency of optimizing the DC loss and by changing the learning rate of gradients applied during that optimization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We note multiple tradeoffs in CPA-DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' First, CPA-DC is not as training time-efficient as NoPeek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' NoPeek can optimize its two objectives simultaneously while CPA-DC has to solve them sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, we consider that NoPeek transfers more information than necessary over a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Second, there are tradeoffs for DC Frequency (F) and Loss Multiplier (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Increasing F adds more epochs to optimize for DC loss, so the attack resilience would be higher at the expense of a longer training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, we introduced m, so that we can keep a small F and only increase m, which reduces training time while maintaining attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We multiply larger m by the DC loss, similar to increasing the gradient descent step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, we need less DC epochs to maintain the attack resilience, given a larger m, while the DC loss becomes less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Based on the discussion, intuitively we expect that a large m combined with F of 1 can balance between training time efficiency and DC loss gradients’ accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' These two parameters should be carefully designed in a production environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We first experimented MNIST classification with differ- ent DC Frequencies with a constant loss multiplier of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1 (equivalent to reducing the learning rate by 10%) and stud- ied the tradeoff between accuracy and attack resilience, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis represents the DC Frequency, the left y-axis shows the learner accuracy and the right y- axis shows the corresponding attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' From the top plot of Figure 81, as DC Frequency is in- creasing, for both Privacy Aware FSL (PAFSL) and Privacy Aware PSL (PAPSL) systems, the attack resiliency increases and the learner accuracy decreases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Notice that PAFSL achieves better accuracy and good resilience for most DC Frequency values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' For DC Frequency from 10 to 20, given that the attack resilience of PAFSL and PAPSL are close within 10% difference, PAFSL achieves more than 90% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' From 25 to 35, PAPSL does not learn any features while PAFSL still has about 80% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' When DC Frequency is five, the PAPSL has an advantage over PAFSL, with close accuracy, and PAPSL has around 20% more attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The result shows that PAFSL with CPA-DC is easier to tune for high accuracy and attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Within wider domain of DC Frequency, PAFSL has higher accuracy and good attack resilience compared to PAPSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This is because of the learning rate (step size) in SerAVG and PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The server weight update rule of PSL is shown in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' W t+1 g = W t g − η NC � i=1 ∂g(f(xi)) ∂Wg (5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' A DC Frequency of zero corresponds to POFSL and POPSL without the privacy-aware approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The server weight update rule of FSL is shown in Equa- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' W t+1 g = �NC i=1(W t g − η ∂g(f(xi)) ∂Wg ) NC = W t g − η NC NC � i=1 ∂g(f(xi)) ∂Wg , (6) where W t g indicates the weights in the server at iteration t, g is the server NN, f is the client NN, xi represent the i-th batch of data, NC is the number of clients, and η is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Intuitively, since PSL has a larger step size, its server NN can be confused quicker than FSL servers by the intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, the confused server NN can further confuse the client NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' It justifies our observation that PSL’s accuracy and attack resilience become unstable quickly when increasing the DC Frequency (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Therefore, we conclude that FSL is easier to tune compared to PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Figures 7b and 7d, with a fixed DC Frequency (F), we show the accuracy (left y-axis) and attack resilience (right y-axis) based on different Loss Multiplier (m) for different models and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Furthermore, we compared the privacy oblivious cases (Figure 7a and Figure 7c), and the privacy aware cases at different Cut Indexes (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As expected, increasing the Loss Multiplier (m) enhances attack resilience but reduces accuracy, especially when the client NN is deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Overall our evaluation of CPA-DC shows good attack resilience and accuracy with a combination of small DC Frequency (F) and big Loss Multiplier (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' And our FSL has better accuracy and similar attack resilience to PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We hence conclude that our CPA-DC can defend against our attacker model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Privacy Evaluation with Differential Privacy Ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The previous section has discussed the CPA-DC, but instead of DC there are other lightweight methods that can enhance the privacy guarantee which adds noise to the client NN while prevent depleting the client battery quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this section, we compare CPA-DP (using the popular DP- SGD [30] algorithm inside clients) and CPA-DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Also, we show that na¨ıvely using DP-SGD in an FSL system would lead to low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The implementation extends POFSL with a DP-SGD optimizer, provided by the Opacus [48] library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This method would add normally distributed ran- dom noise to the gradients during backward propagation based on noise multiplier ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This parameter controls the magnitude of the noise added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Notice that DC generates the gradients in a specific direction to reduce correlation between intermediate data and source data in each Distance Correlation round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' So we expect CPA-DP to have a worse level of privacy, given the same level of learner accuracy, compared to CPA-DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, the focus of this section is to show that CPA can be applied with other privacy methods like DP, despite DP’s worse privacy compared to DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We summarize the results of CPA-DP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' This plot shows the result when Cut Index equals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The x-axis is the noise multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The attack resiliency of FSL and (a) CPA-DC (Multiplier=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='1) (b) CPA-DP (c) DP-SGD in Complete NN Figure 8: LeNet+MNIST: Learner accuracy and attack re- silience (τ) with 20 clients and Cut Index of 3 for Client- Based Privacy Approaches (via DC (a) and via DP (b)) and DP-SGD on the global learner model (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Our FSL with both client-based policies guarantee high-level of privacy and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' PSL with noise multiplier > 0 is consistently better by nearly 5% than FSL and PSL with noise multiplier = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' At the same time, the accuracy decreases by less than 1% in either FSL or PSL from noise multiplier = 0 to noise multiplier = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' CPA is a general approach and can be customized with different methods to enhance attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The evaluation shows that CPA-DP can also improve attack resilience, while the learner accuracy does not change much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Comparing with CPA-DC, both methods provide similar learner accuracy, but CPA-DC’s attack resilience is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We now compare CPA-DP against applying DP-SGD in both client NN and server NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Figure 8c shows high attack resilience, but the learner accuracy of FSL drops below 60% when noise multiplier ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Meanwhile, PSL shows a similar behavior as using CPA-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' So, CPA- DP is considered a better method for FSL to enhance its attack resilience than with DP-SGD applied in both clients and servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' The reason for FSL’s lower accuracy under DP- SGD and high noise can be attributed to its SerAVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After applying SerAVG, the distribution of the random noise in the server NN can be arbitrary, as shown in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', 7, while the noise in the client NN stays intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, the resulting complete NN in FSL may not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' PSLAccuracy PSL Resilience FsLAccuracy Client-Based Approach: DC Loss in Client FSLResilience 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='9 e lience e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content="8' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='6 Re: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4 u Atta a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2 0 5 10 15 20 25 30 35 40 DC FrequencyPSL Accuracy PSLResilience FSLAccuracy DifferentialPrivacy (Client + Server) FSLResilience 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 NoiseMultiplierPSL Accuracy PSL Resilience FSL Accuracy Client-Based Approach: DP-SGD in Client FSL Resilience ence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='75 esilier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='95 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Noise MultiplierPSL Accuracy PSL Resilience FSL Accuracy Client-Based Approach: DP-SGD in Client FSL Resilience ence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='75 esilier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='95 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='0 Noise MultiplierW t+1 g = W t g − η NC NC � i=1 (∂g(f(xi)) ∂Wg + et gi) (7) The variable et gi indicates the gaussian noise added for the i-th server NN at iteration t according to et gi ∼ N(0, (noise multiplier × max gradient norm)2) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Evaluation Using Different ways of Partitioning NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In Figure 7a and Figure 7c, the right y-axis shows the attack resilience, and the x-axis indicates the Cut Indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Overall, if we have a deeper client NN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', moving from smaller to bigger Cut Indexes), the attack resilience in- creases and accuracy decreases (consistent with the SerAVG Evaluation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' After adding more layers, the intermediate data would have less features from the source data, but only keeps those that can improve the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Thus, less features are preserved and the attacker’s ability to reconstruct the source’s data is hindered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We also want to emphasize that with CIFAR10 work- load, when there are 7 layers in the clients, the attack resilience reaches about 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We reach the same result with MNIST workload at the Cut Index of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' No extra privacy- aware method was used, and as we discussed earlier, the transmission delay can also be reduced with a deeper NN in the client due to potentially smaller intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Cut Index is an important hyper-parameter for train- ing delay, accuracy and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Different Cut Indexes bring the following tradeoff: the deeper client NN adds more resource demand at the edge, but reduces the transmission time and enhances attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' On the other hand, a shallower server NN may lead to lower accuracy with SerAVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Furthermore, system architects can combine the ap- proaches mentioned above, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=', having a moderately deep NN in clients and using the CPA-DC, to find a balance between resource demand and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As in Fig- ure 7d, with cut index = 4, DC Frequency = 1 and loss multiplier = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content='3, we still get about 80% attack resilience and more than 90% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' On the other hand, when comparing FSL and PSL, we note that FSL has more hyper-parameters to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' But, in all experiments reported in Figure 7, when the Cut Index is large, FSL has a better accuracy than PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' So we conclude that a carefully specified way of partitioning the NN can benefit the most when applied in hybrid federated-split learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Conclusion Systems like Federated Learning (FL), Split Learning (SL), and later works aim to fit specific scenarios such as distributed model training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' However, they are not flexible enough to fit some use cases with the recent development in edge and constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' In this work, we propose and extensively evaluate Feder- ated Split Learning (FSL), a system for efficient training and inference with high-level privacy for clients’ source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We present a Client-based Privacy Approach (CPA) for split learning-based systems to provide high attack resilience by adding noise to the intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' We also study the training time, accuracy and privacy level of different ways to partitioning a NN in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' As further works, we aim to research prediction based NN partitioning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Moreover, comparisons between FSL and existing NN training and inference systems at the edge are carried out, along with explanations of their pros and cons against FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Among the main analyzed systems, the Parallel Split Learning (PSL)’s principal limitation is a slow and stateful server, and the Federated Reconstruction (FRC) system is inefficient in training time and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' Acknowledgements This work has been supported by National Science Foun- dation Awards CNS-1908574 and CNS-1908677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQf2_7N/content/2301.01824v1.pdf'} +page_content=' H.' metadata={'source': 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Mathematical Sciences and Applications, Tsinghua University, Beijing, +China +eLyndon Baines Johnson School of Public Affairs, University of Texas at Austin, Texas, +United States +fDepartment of Physics, Southern Methodist University, Texas, United States +Abstract +This paper introduces new methods for studying the prevalence of terrorism +around the world and over time. +Our analysis treats spatial prevalence of +terrorism, the changing profile of groups carrying out the acts of terrorism, +and trends in how many attacks take place over time. First, we use a time- +evolving cluster analysis to show that the geographic distribution of regions of +high terrorist activity remain relatively consistent over time. Secondly, we use +new metrics, inspired from geometry and probability, to track changes in the +distributions of which groups are performing the terrorism. We identify times at +which this distribution changes significantly and countries where the time-varying +breakdown is most and least homogeneous. We observe startling geographic +patterns, with the greatest heterogeneity from Africa. Finally, we use a new +implementation of distances between distributions to group countries according to +their incidence profiles over time. This analysis can aid in highlighting structural +similarities in outbreaks of extreme behavior and the most and least significant +public policies in minimizing a country’s terrorism. +Keywords: +Terrorism, Cluster analysis, Time series analysis, Metric geometry +1. Introduction +Terrorism and other violent coordinated behavior have a highly deleterious +effect on society. Terrorism not only may cause loss of life and injury, it may +also thoroughly destabilize communities and governments [1], cause enormous +economic costs [2], affect financial markets [3], and damage key industries such +as tourism [4]. +Email address: max.menzies@alumni.harvard.edu (Max Menzies) +1 +arXiv:2301.00408v1 [physics.soc-ph] 1 Jan 2023 + +Understanding trends in terrorism over time assists governments in antic- +ipating and responding to its acute impact. The observation of geographic +spread may inspire new approaches to geopolitical engagement [5]; a changing +composition of active terrorist organizations may require changes in negotiation +or military response [6]; an increasing number of terrorist activities is cause for +concern and could warrant a variety of measured responses. +Out data comes from the Global Terrorism Database (GTD) [7], a com- +prehensive catalog of more than 200 000 incidents from 1970 through 2018. +Incident data include date, location, perpetrator (including group responsible), +and several other features. We remark that the commonly used term “terrorism” +is a complex concept with emotional, political, and legal dimensions. The organi- +zation maintaining the GTD defines terrorism as “the threatened or actual use of +illegal force and violence by a non-state actor to attain a political, economic, reli- +gious, or social goal through fear, coercion, or intimidation.” The data collecting +organization “uses a series of inclusion criteria to systematically identify events +for inclusion in the database.” We do not opine on the definition of terrorism, +only including events from this database without additional selection criteria. +This paper builds on a long literature of multivariate time series analysis, +developing a new mathematical method and a more extensive analysis of terrorism +dynamics than previously performed. Existing methods of time series analysis +include parametric models [8] such as exponential [9] or power-law models, +[10] and nonparametric methods such as distance analysis [11, 12], distance +correlation [13–15], Bayesian approaches [16] and network models [17]. We draw +upon numerous approaches that have been successfully applied in numerous +disparate fields such as epidemiology [18–24], finance [25–32], cryptocurrency [33– +37], crime [38, 39] and other fields [40–45]. We hope our work can complement +other mathematical analyses of terrorism, which are relatively small in number +compared to other fields but cross dynamical systems [46], network analysis +[47, 48], power models [49, 50], Poisson models [51], agent-based models [52] and +other parametric models [53, 54]. +Cluster analysis is another common statistical method with successful appli- +cations to numerous fields. Designed to group data points according to similarity, +clustering algorithms are highly varied - common examples are K-means [55] and +spectral clustering [56], which partition elements into discrete sets, and hierar- +chical clustering [57, 58], which does not specify a precise number of clusters. In +this paper, we will use hierarchical clustering and K-means; the latter requires +an initial choice of the number of clusters k. We draw upon several methods +to address the subtle question of how to select this k. The primary application +of cluster analysis in this paper is to reveal geographic persistence of terrorist +activity over time, and to investigate relationships between different years re- +garding the composition of which perpetrator organizations are responsible for +the totality of attacks. +This paper is structured as follows. In Section 2, we use cluster analysis +to reveal geographic persistence, namely consistent patterns where terrorism +is occurring over time. In Section 3, we analyze changes in the composition +of perpetrator organizations, that is who is committing terrorism, both within +2 + +specified countries and in observed clusters of Section 2. We identify countries +with the greatest diversity of terrorist actors, as well as key times in which +particular countries’ perpetrator profile changed. In Section 4, we analyze the +changing number of attacks over time, and analyze similarity between different +countries’ trajectories of attacks over time. +Together, these analyses provide insight about geographic persistence of +terrorism over time (where it is occurring), the changing composition of terrorist +organizations committing the attacks (who is committing the terrorism) and the +change in number of attacks over time (how much terrorism is occurring), all +analyzed over time. This analysis may assist geopolitical and social researchers +and policymakers working to identify the most and least successful countries +in combating any increasing terrorist activity, and could provide blueprints for +increased international collaboration and better relations between countries. +2. Geographic persistence via clustering +In this section, we investigate where terrorism occurs, specifically, the geo- +graphic persistence of terrorism over time, grouping terrorist attacks by year. Our +data spans 2000-2018 inclusive, a period of T = 19 years. We aim to demonstrate +a considerable persistence in the geographic distribution of terrorist activities +across the world. For this purpose, we use cluster analysis to group events, as +recorded in the GTD [7], by their geographic proximity on a year-by-year basis. +By calculating various estimates of the total number of clusters, we show a +considerable persistence in the geographic patterns of terrorist attacks each year. +Specifically, we index our years t = 1, ..., T. For each year, we consider the +full collection of Nt attacks observed in that year. Using the haversine formula +[59], we calculate all pairwise geodesic distances between the locations of events +for that year, generating a Nt × Nt matrix. We make several adjustments to the +data set, such as excluding Australia, New Zealand and Japan from the data, +each of which reports very few attacks, and separate the Western Hemisphere +(the Americas) from the Eastern Hemisphere (Europe, Asia, Africa and Oceania). +Finally, we draw upon m = 16 different methods [60] for selecting the number of +clusters each year, to produce estimates k1(t), ..., km(t) for each t. We record the +estimated cluster numbers for each year of data and utilized method in Table +1 for the Western Hemisphere and Table 2 for the Eastern Hemisphere. We +provide an overview of clustering and some of the methods used in Appendix A. +Almost every chosen method exhibits a considerable persistence in the de- +tected number of clusters, suggesting a persistence in the geographic localization +of terrorist attacks over time. By averaging over all chosen methods across all +years, we estimate a persistent total of K = 4 and 11 regions for the Western +and Eastern hemispheres, respectively. +Having observed the persistence in these cluster numbers on a yearly basis, we +now wish to aggregate the clusters from each year so we can study the changing +composition of these persistent regions over time. For this purpose, we apply the +K-medoids algorithm over the entire set of attacks over 2000-2018, and cluster +them into K = 4 groups for the Americas (the Western Hemisphere) and K = 11 +3 + +2000 +’01 +’02 +’03 +’04 +’05 +’06 +’07 +’08 +’09 +’10 +’11 +’12 +’13 +’14 +’15 +’16 +’17 +’18 +ptbiserial +4 +5 +5 +4 +3 +5 +5 +4 +4 +3 +5 +5 +5 +4 +4 +5 +5 +4 +3 +silhouette +4 +4 +4 +5 +2 +4 +5 +5 +3 +3 +5 +4 +3 +3 +3 +3 +3 +3 +5 +kl +4 +2 +4 +2 +4 +4 +5 +3 +2 +3 +3 +5 +2 +4 +3 +3 +5 +5 +4 +cindex +2 +2 +5 +5 +3 +5 +5 +3 +3 +5 +4 +3 +5 +2 +4 +4 +5 +5 +4 +mcclain +4 +3 +2 +3 +2 +5 +2 +2 +4 +2 +2 +2 +5 +2 +2 +2 +2 +2 +2 +dunn +3 +5 +2 +3 +5 +5 +4 +5 +3 +3 +5 +3 +5 +5 +5 +3 +5 +2 +5 +ch +5 +5 +3 +4 +4 +4 +5 +3 +5 +4 +5 +5 +3 +4 +4 +4 +5 +5 +5 +hartigan +3 +3 +3 +4 +4 +4 +3 +3 +4 +3 +3 +3 +5 +3 +3 +4 +3 +3 +4 +ccc +2 +2 +3 +2 +2 +5 +2 +5 +2 +2 +5 +5 +2 +5 +5 +5 +5 +5 +5 +trcovw +4 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +tracew +4 +4 +3 +4 +3 +4 +4 +3 +4 +3 +3 +3 +3 +3 +3 +3 +3 +3 +4 +friedman +4 +5 +5 +4 +5 +4 +4 +4 +4 +4 +5 +5 +3 +4 +5 +4 +5 +4 +4 +rubin +4 +4 +3 +4 +4 +4 +4 +3 +4 +4 +4 +3 +3 +4 +3 +4 +3 +3 +4 +db +5 +4 +4 +5 +3 +5 +2 +4 +3 +5 +5 +4 +5 +4 +4 +3 +4 +5 +3 +duda +2 +5 +3 +2 +2 +3 +2 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +0 +gap +2 +2 +3 +2 +2 +2 +2 +3 +2 +2 +5 +5 +2 +3 +3 +3 +3 +3 +2 +Table 1: Estimates of the number of clusters of terrorist attacks on a year-by-year basis across +the Americas (Western Hemisphere). 16 different methods are compared, with considerable +persistence over time in the number of clusters. +2000 +’01 +’02 +’03 +’04 +’05 +’06 +’07 +’08 +’09 +’10 +’11 +’12 +’13 +’14 +’15 +’16 +’17 +’18 +ptbiserial +4 +4 +6 +4 +4 +5 +7 +5 +9 +6 +5 +5 +4 +5 +9 +5 +6 +6 +5 +silhouette +25 +25 +25 +25 +15 +14 +17 +13 +17 +20 +13 +16 +10 +14 +24 +23 +14 +14 +23 +kl +17 +11 +7 +7 +15 +7 +20 +8 +10 +24 +13 +16 +7 +14 +23 +18 +9 +9 +13 +cindex +25 +25 +25 +25 +25 +25 +25 +25 +25 +25 +25 +25 +25 +21 +25 +25 +25 +25 +25 +mcclain +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +dunn +8 +25 +12 +22 +25 +21 +22 +24 +21 +11 +21 +15 +20 +13 +25 +16 +8 +11 +2 +ch +25 +25 +24 +25 +17 +17 +18 +14 +17 +20 +13 +17 +10 +14 +13 +17 +14 +14 +13 +hartigan +8 +7 +6 +4 +6 +7 +9 +8 +3 +5 +6 +7 +6 +5 +8 +5 +6 +8 +7 +ccc +25 +25 +24 +25 +17 +17 +18 +14 +17 +20 +13 +17 +10 +14 +13 +17 +14 +14 +13 +trcovw +6 +3 +4 +4 +3 +3 +4 +3 +4 +5 +3 +7 +4 +6 +8 +6 +7 +9 +3 +tracew +4 +7 +4 +4 +3 +3 +4 +8 +4 +5 +6 +7 +4 +6 +8 +6 +7 +4 +4 +friedman +17 +24 +23 +17 +15 +14 +18 +13 +16 +20 +13 +16 +10 +14 +13 +12 +14 +14 +13 +rubin +17 +24 +24 +7 +15 +7 +16 +8 +16 +16 +13 +16 +10 +14 +23 +17 +12 +14 +13 +db +25 +24 +25 +9 +24 +25 +19 +15 +2 +22 +9 +13 +11 +5 +2 +2 +20 +2 +11 +duda +2 +2 +2 +2 +4 +3 +2 +3 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +gap +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +Table 2: Estimates of the number of clusters of terrorist attacks on a year-by-year basis across +the Eastern Hemisphere, comprising Europe, Africa and Asia, but excluding Australia, New +Zealand and Japan. 16 different methods are compared, with considerable persistence over +time in the number of clusters. +4 + +for the Eastern Hemisphere. Two resulting clusters (one from each hemisphere) +have fewer than 10 events from across the entire period, and are removed as +outlier clusters. This yields 13 geographic clusters or regions of the world’s +terrorist attacks over time. +In the Western Hemisphere, the regions observed are North America, Colom- +bia and Ecuador, and southern South America. In the Eastern Hemisphere, +the regions observed are southern Africa, East Africa, West Africa, Europe, the +Middle East, South Asia, East China, central Russia and Siberia, Southeast +Asia, and the Philippines (and nearby Oceanic islands). We make use of these +determined regions in subsequent sections, where we study changes in the com- +position of terrorist attacks, both on a country-by-country basis and within our +identified regions. +3. Analysis of the composition of perpetrators +In this section, we turn to an analysis of which actors and groups are +committing the acts of terrorism within each region. Specifically, among the top +40 countries with the greatest total counts of terrorist events, we investigate the +changes of the composition of terror perpetrators on a yearly basis. For each +determined country, we construct a T × T distance matrix between different +distributions of perpetrators by year. In addition, we perform the same analysis +for the determined regions from Section 2. +3.1. Distance between compositional distributions +Given a year t and a given country or region (where at least one attack +occurred), let At be the sets of terror perpetrators of attacks in that year, as +recorded in the GTD database. For any x ∈ At, let p(t) +x +be the proportion +of attacks attributable to the group x. Many attacks are unknown in their +attribution, thus, “unknown” is a permissible group. As every attack carries a +unique primary attribution, � +x∈At p(t) +x += 1. So we may understand p(t) as a +probability vector of length |At|. +Analogously, let p(s) +y +be the proportion of attacks attributable to the group +y ∈ As. Define a distance between the two distributions of terrorism by +d(s, t) = 1 +2 +� +z∈As∪At +|p(s) +z +− p(t) +z |. +(1) +This has the property that d(s, t) = 0 if and only if years s, t have an identical +proportion of terrorist attacks among perpetrator groups. In addition, d(s, t) ≤ 1, +with equality if and only if As and At are disjoint, with no intersection in +perpetrators between the two years. If As is the empty set, we exclude the year s +from consideration. Thus, a single country or region R produces a S × S matrix +DR, where S is the number of years where at least one attack was recorded; for +most regions and countries, S = T. +5 + +There are several enlightening interpretations of the distance defined in (1). +As mentioned, the vector of proportions for any given year is a probability vector +p(t) of dimension |At|. As the sum of the elements is 1, p(t) naturally lies on +a simplex ∆ = {(x1, ..., xN) : � xi = 1, xi ≥ 0} of dimension N = |At|. When +we compute the distance (1), we are implicitly embedding both p(t) and p(s) +in a larger simplex of dimension N = |At ∪ As|. In fact, when we compute all +distances d(s, t) one may consider the vectors p(t) as having been embedded in a +larger simplex of dimension N = |A| where A = ∪T +t=1At is the set of all recorded +groups in the region across all time. +Now simplices (including the higher-dimensional embedded simplex) are +convex sets, that is, any line segment between two points in a simplex lies entirely +within the simplex. Thus, we were initially motivated to use an alternative +distance to (1) that bears explaining. As each probability vector p(t) lies within +a larger simplex and the line segment between p(s), p(t) lies entirely within the +simplex, an initial choice of distance could be the geometric straight line distance +along the line segment, that is, +� +� +z∈As∪At +|p(s) +z +− p(t) +z |2 +� 1 +2 +. +(2) +This distance may have other uses in future works, as it can be interpreted as +the straight line distance along a path of deformation (in the geometric sense) +between p(s) and p(t). But we think it is necessary to justify why we use (1) +over (2). We present both a geometric and probabilistic explanation. +First, the simplex may be interpreted as (the positive quadrant of) a unit +sphere in the vector space RN under the L1 norm ∥x∥ = |x1|+...+|xN| and then +imbuing RN with this norm naturally motivates using it to measure distance +between p(s) and p(t), as we have done in (1) up to a constant factor. And +secondly, the quantity in (1) can be reinterpreted precisely (including the factor +of 1 +2) probabilistically as the discrete Wasserstein distance between p(s) and p(t). +Essentially, let the set A of all perpetrator groups in a region be endowed with +a metric such that d(x, y) = 1 if x ̸= y and 0 otherwise. Then the Wasserstein +metric between two distributions on A, which measures the “work” (in the sense +of physics) to change one distribution into the other, coincides with the formula +(1). This is discussed in greater detail in [61] and Appendix B. +Finally, we define ν(R), a measure of the total heterogeneity of the composi- +tion of attacks across our period of analysis. Given an S × S matrix A, we define +its Frobenius norm by ∥A∥ = +��S +i,j=1 A2 +ij +� 1 +2 . Let ν(R) = ∥DR∥, intuitively, +regions with a low ν(R) experience attacks rather consistently from the same +set of organizations year after year; those with a high ν(R) experience attacks +from organizations that change their composition substantially with time. +We remark that we could repeat the above analysis by removing all attacks +with “unknown” perpetrator prior to any computations of px and ν(R). However, +an “unknown” perpetrator should not merely be considered as randomly missing +6 + +data. Terrorist groups regularly take credit for their attacks for a complex +variety of reasons and “credit-taking has the potential to tell observers a great +deal about the nature of the threats groups pose” [62]. +Thus, significantly +different proportions of known vs unknown attacks between countries or regions +should be taken into account in any analysis of compositional differences between +perpetrators. +In addition, we remark that the value of ν(R) may reduce considerably +with smaller values of S, as a smaller S results in far fewer terms in the sum +∥A∥ = +��S +i,j=1 A2 +ij +� 1 +2 . We consider this appropriate: countries or regions that +have entire years with no terrorist activity should be considered to display less +heterogeneity in composition of attacks. Simply put, with many years of no +activity, less terrorism is occurring in absolute terms, so we should consider less +compositional changes as occurring. +3.2. Results +In Table 3, we rank the 40 countries with the greatest total counts of terrorist +events according to their value of ν(R), while Table 4 ranks the 13 determined +clusters of attacks from Section 2. In Figure 1, we present bar plots of changing +perpetrator composition over time for various countries R, while Figure 2 plots +hierarchical clustering on DR for the same countries. +The countries with the lowest norm are South Sudan, Ukraine, Pakistan, +Syria and Afghanistan. While the first four countries are characterized a large +number of unknown attacks, Afghanistan, however, is dominated by attacks +from the Taliban after 2003 (Figure 1a). The difference in composition between +2000-2002 and subsequent years can be seen in Figure 2a. The persistence of +such a concentration in the perpetration of terrorism indicates how prolific they +were as a terror organization since the turn of the century [63]. The countries +with the highest norm are the United States, Somalia, Nepal, Spain, and the +Democratic Republic of the Congo. The United States (Figures 1b and 2b) has +attacks spread over many groups rather than any consistency between actors, +while other countries, particularly Somalia, exhibit a regime change in which +different groups claim the majority of attacks in different years. Specifically, +from 2009-2018 inclusive, al-Shabaab [64] is consistently the top perpetrator of +terrorism in Somalia, accounting for over 80% of attacks from 2013 onwards +(Figures 1c and 2c). From 2000-2008, most attacks are characterized as unknown, +with al-Shabaab responsible for at most 15% of attacks in any single year. +There are several other countries that exhibit noteworthy trends in the +composition of perpetrator organizations. In Sri Lanka, ranked 31st in ν(R), the +Liberation Tigers of Tamil Eelam, commonly known as the Tamil Tigers [65], +constituted at least 69% of attack attributions from 2000-2009 inclusive, but that +organization ceased to exist following the end of the Sri Lankan civil war [66]. This +considerable shift in composition can be seen in Figures 1d and 2d. In Colombia, +the Revolutionary Armed Forces of Colombia or FARC dominated terrorism +between 2000-2015 inclusive, constituting the top perpetrator of terrorism for +all but one of these years. After an historic peace deal [67], FARC combatants +7 + +Rank +Country +Norm +1 +South Sudan +9.17 +2 +Ukraine +21.71 +3 +Pakistan +28.4 +4 +Syria +30.59 +5 +Afghanistan +31.66 +6 +Libya +44.54 +7 +Thailand +48.55 +8 +Lebanon +50.46 +9 +Iraq +51.05 +10 +The Philippines +52.68 +11 +Egypt +57.14 +12 +Bangladesh +57.25 +13 +Russia +63.93 +14 +Colombia +66.91 +15 +United Kingdom +68.02 +16 +Mali +72.76 +17 +Turkey +73.36 +18 +Cameroon +73.83 +19 +India +91.39 +20 +Indonesia +94.43 +21 +Greece +94.73 +22 +Saudi Arabia +96.02 +23 +West Bank and Gaza Strip +105.18 +24 +Yemen +108.38 +25 +Sudan +112.23 +26 +Central African Republic +113.80 +27 +Myanmar +116.98 +28 +Israel +117.52 +29 +Nigeria +125.06 +30 +Algeria +126.61 +31 +Sri Lanka +127.01 +32 +Burundi +129.00 +33 +France +131.04 +34 +Germany +131.84 +35 +Kenya +150.94 +36 +United States +157.05 +37 +Somalia +160.44 +38 +Nepal +169.58 +39 +Spain +175.93 +40 +Democratic Republic of the Congo +188.99 +Table 3: Ranking of the 40 countries with the greatest total counts of terrorist events according +to their value of ν(R), defined in Section 3.1. Countries with smaller values display more +homogeneity in the composition of terrorist attacks from year to year. +8 + +Rank +Cluster +Norm +1 +East China +22.16 +2 +Central Russia and Siberia +32.88 +3 +South Asia +45.96 +4 +The Philippines and proximity +48.10 +5 +The Middle East +60.88 +6 +Colombia and Ecuador +64.97 +7 +Southeast Asia +75.11 +8 +Europe +89.61 +9 +Southern Africa +113.21 +10 +East Africa +114.43 +11 +North America +114.92 +12 +Southern South America +117.95 +13 +West Africa +129.90 +Table 4: Ranking of the K = 13 determined clusters/regions with the greatest total counts of +terrorist events according to their value of ν(R), defined in Section 3.1. Regions with smaller +values display more homogeneity in the composition of terrorist attacks from year to year. +were reintegrated into society and attacks ceased. Beyond 2016, the National +Liberation Army of Colombia (ELN) [68] was responsible for the majority of +terrorist attacks (Figures 1e and 2e). Finally, in the Philippines, the New People’s +Army [69] accounts for a majority of terrorism of known groups from 2000-2018. +However, the recorded composition of perpetrators changes significantly after +2005, as the modal number of attacks are “unknown” every year from 2006 +(Figures 1f and 2f). +Finally, regarding the determined regions from Section 2, we see startling +geographic trends in the ordering of composition heterogeneity in Table 4. The +five clusters/regions with the least ν(R) (indicating greater heterogeneity in +perpetrator attribution) are all in Asia, while the three African regions all fall +within the five clusters with the greatest value of ν(R). +4. Time series analysis of attack counts +In this final section, we analyze the prevalence of terrorism over time by +country, specifically, how many attacks are occurring throughout the analysis +window. +4.1. Distances between time series +Consider a fixed country or region R. To examine the trajectories of attack +counts on a finer basis, we divide our 19-year period of analysis into individual +months t = 1, 2, ..., P, where P = 12T = 228. Let zR(t) be the number of attacks, +regardless of attribution, observed in month t. To the time series z(t), t = 1, ..., P, +9 + +we associate the following probability distribution: +fR = +1 +�P +s=1 zR(s) +P +� +t=1 +zR(t)δt, +(3) +where δt is a Dirac delta distribution at t. That is, fR is a distribution that +apportions to month t the weight of the attacks observed in that month as a +proportion of the total attacks across the whole period. We can then compare +different trajectories as distributions by using the L1-Wasserstein metric [70]: +D(R1, R2) = W1(fR1, fR2). We remark that this is the more traditional Wasser- +stein distance, between distributions on the real number line with its standard +metric, than the discrete Wasserstein distance discussed in Section 3. +This distance has several advantageous properties over more commonly used +discrepancy measures between normalised trajectories. For example, previous +work [71] has used the L1 norm and metric between normalised trajectories, +defined as follows: +∥zi∥1 = +T +� +t=1 +zi(t) +(4) +vi = +zi +∥zi∥1 +(5) +dij = ∥vi − vj∥1 +(6) +This treats each time series zi(t) as a vector in RP , normalises by its L1 norm, +and compares these normalised vectors with the L1 metric [72]. This distance is +suitable in most instances but has some undesirable properties when measuring +discrepancy between noisy time series. Specifically, this L1 distance dij has +maximal possible value equal to 2 when zi(t) and zj(t) have disjoint support. +Practically, this would mean that two countries’ trajectories would receive a large +L1 discrepancy measure if the attacks fell around the same time but not in exactly +the same months. For example, if country i and j had broadly similar trends +in events, but in country i more attacks were reported in January and March +while country j reported more on February and April, then the L1 distance +measure would be larger than their similarity. Further smoothing and averaging +can resolve some of these issues, but the Wasserstein metric ameliorates this +issue even more, as it is robust to small translations of distributions. That is, if +f is a distribution and fδ(x) = f(x + δ), then W1(f, fδ) = |δ|, as shown in [73]. +This means the Wasserstein metric assigns a low value in the case that countries +i and j have similar trajectories where attacks just fall in nearby but distinct +months. +4.2. Results +We apply hierarchical clustering to this metric across the 40 countries with the +greatest total counts of terrorist events (Figure 3a) and the 13 regions identified +in Section 2 (Figure 3b). Various insights can be gleaned from the cluster +10 + +structure and pairings. First, the prototypical trajectory of terrorist events +among our collection of countries is captured in a large cluster of similarity that +extends from Nigeria down to India in Figure 3a. Containing 26 countries, the +primary feature of this cluster is a general increase in terrorist activity over time. +A limited amount of heterogeneity exists within this majority cluster, visible +in the slight differences between Nigeria (Figure 4a) and India (4b). However, +essentially these 26 countries display time series that, considered as distributions, +have most of the terrorist attacks occurring toward the end of the period of +analysis. The increase in terrorist activity in Africa (with many of the time series +resembling that of Nigeria) is particularly noticeable, and has been of growing +concern in recent years [74, 75]. +Other than the majority behavior, some outlier countries are revealed. Sri +Lanka, Algeria and Spain all share some unusual and striking commonalities in +trends in terrorist attacks over time - a striking (relative) dearth of attacks after +2010. This is illustrated in Figures 4c and 4d, where the time series (considered as +distributions) have the majority of their “weight” before 2010. Next, the United +States and the West Bank and Gaza Strip are observed to be quite different +from the majority collection. Indeed, as seen in Figures 4e and 4f respectively, +these time series yield distributions that are rather flat over the entire period +of analysis, a rather rare feature among the countries studied. Other outlier +countries include Indonesia and to a lesser extent Russia and Greece - all three +of these exhibit rather idiosyncratic trajectories of terrorism over time, with +bursts over differing periods and little relationship to the trends observed in +other countries. +These geographic trends are also made apparent in Figure 3b. There, the +regions where terrorist attacks mostly occur toward the end of the period group +together in the bottom cluster, including regions in Africa, south Asia, and the +Middle East. Europe and North America cluster together as being characterized +more by terrorism remaining more flat across the period of analysis. +5. Conclusion +This paper has applied numerous mathematical approaches to analyze world- +wide terrorism, drawing from metric geometry, unsupervised learning, and +analysis of distributions and time series. We both reveal persistence and sub- +stantial changes in the global profile of terrorism. In Section 2, we showed +substantial persistence in the cluster structure of terrorism on a geographic basis +over time, and determined a convenient and interpretable list of 13 regions into +which to divide all the world’s terrorist events, other than a negligible proportion +of outliers. We must remark that our use of the word “cluster” is informed by +the machine learning/statistical literature, rather than the standard security +or foreign policy literature. In the latter, a “terrorist cluster” commonly refers +to a collection of attacks or perpetrators on a very small geographic level, or a +network of connected actors. The former would yield far too many clusters on +the scale of the entire world to be informative, while the latter is not the focus +of this paper (and is not summarized in the Global Terrorism Database). These +11 + +(a) +(b) +(c) +(d) +(e) +(f) +Figure 1: Changing composition of terrorist event perpetrators over time for (a) Afghanistan +(b) the United States (c) Somalia (d) Sri Lanka (e) Colombia (f) the Philippines. +12 + +2000 +2001 +2002 +8'0 +2003 +2004 +2005 +2006 +0.6 +2007 +2008 +2009 +2010 +0.4 +2011 +2012 +2013 +2014 +2015 +0.2 +2016 +2017 +2018 + Gholam Yahya Akbar . +0.0 +Unknown +The Northern Aliance (or United Islamic Front for Salvation of Afghanistan - UIFSA) +Taliban . +Oppasition Group +epleo-n +Hizb-l-Islami +Anti-United States extremists +Saif-ul-Muslimeen +Islamist extremists + Jaish al-Muslimin (Army of the Muslims) , + Tela Mohammed . + Haqqani Network +Muslim extremists +Mahaz Fedai Tahrik Islami Afghanistan + Al-Fatah +Khorasan jihadi group +Militants + Insurgents +Lashkar-e-Jhangvi +Islamic Movement of Uzbekistan (IMU) , +Mullah Dadullah Front +Lashkar-e-Taiba (LeT) +Khorasan Chapter of the Islamic State . +Jaish-e-Mohammad (JeM) . + Jundallah (Pakistan) + Haji Fateh2000 +0.8 +2001 +2002 +0.7 +2003 +2004 . +0.6 +2005 +2006 +2007 +0.5 +2008 +2009 +0.4 +2010 +2011 +E'O +2012 +2013 +2014 +0.2 +2015 +2016 +0.1 +2017 +2018 +0.0 +Earth Liberation Front (ELF) +Animal Liberation Front (ALF) +Anti-Abortion extremists +Anti-white extremists +Unknown +Coalition to Save the Preserves (csP) +White supremacists/nationalists +Revenge of the Trees +Environmentalists +epieo-n +Anti-Immigrant extremists +extemists +Anti-Govemment extremists + Revolutionary Cells-Animal Liberation Brigade +Anti-Semitic extremists + Neo-Nazi extremists +Ku Klux Klan +Anti-Liberal extremists +Minutemen American Defense +Incel extremists +Al-Qaida in the Arabian Peninsula (AQAP) +Tehrik-i-Taliban Pakistan (TTP) +The Justice Department +Anti-Muslim extremists +Sovereign Citizen +Pro-LGBT Rights extremists +extremists +Anarchists +Anti-Gun Control extremists + Students For Insurrection +Veterans United for Non-Religious Memorials +United Aryan Empire +Right-wing extremists , +Anti-sikh extremists +Citizens for Constitutional Freedom +Court Reform extremists +Anti-Trump extremists +Black Hebrew Israelites +Conspiracy theory extremists +Anti-LGBT extremists +Anti-Arab extremists +Anti-Republican extremists +White Rabbit Three Percent Ilinois Patriot Freedom Fighters Militia + Male supremacists +jihadi-inspired +Anti-Israeli +Anti-Kim Jong-il +Muslim +bejl +Anti-Police 2000 + 1.0 +2001 +2002 +2003 +2004 +8'0- +2005 +2006 +2007 +0.6 +2008. +2009 +2010 +2011 + 0.4 +2012 +2013 +2014 +2015 +0.2 +2016 +2017 +2018 +Ahlu-sunah Wal-jamea (Somalia) . +0.0 +Unknown +U/l Somali Militiamen +Rahanwein Resistance Army (RRA) +Marhan Clan + qiqeh Jewn pewweynw jo sayoddns + Supporters of Muse Sudi Yalahow + Musa Sudi Yalahow Militia + Habr Gedir Clan +Al-Ittihaad al-Islami (AIAl) +Islamic Courts Union (ICU) +Islamic Tendency +ep!eo-n +Mujahideen Youth Movement (MyM) +Muslim extremists + Al-Shabaab + Hizbul al Islam (Somalia) + Somali Islamic Front + 23 May Democratic Alliance (Algeria) + Sunni Muslim extremists +Islamic Party (Somalia) + Shabelle Valley militia +Raskamboni Movement +Awdal Regional Administration Army (ARAA) + Islamic State of Iraq and the Levant (ISIL) + Jabha East Africa2000 +- 1.0 +2001 +2002 +2003. +B'0 +2004 +2005 +2006 +2007 +0.6 +2008 +2009 +2010 +0.4 +2012 +2013 +2014 +2015 +0.2 +2016 +2017 +2018 + Calonel Karuna Faction . +Tamil Makkal Viduthalai Pulikal (TMVP) . +0.0 +Unknown +Liberation Tigers of Tamil Eelam (LTTE) +National Front Against Tigers (NFAT) +Govemment Supporters +Christian extremists + Muslim Militants +Buddhist Monks +Bodu Bala Sena +Anti-Muslim extremists + Sinhale Jathika Balamuluwa 2000 +2001 +0.8 +2002 +2003 +2004 . + 0.7 +2005 +2006 +0.6 +2007 +2008 + 0.5 +2009 +2010 + 0.4 +2011 +2012 + 0.3 +2013 +2014 + 0.2 +2015 +2016 + 0.1 +2017 +2018 +0.0 +Unknown +Revolutionary Armed Forces of Calombia (FARC) +National Liberation Army of Colombia (ELN) +Popular Liberation Army (EPL) +United Self Defense Units of Calombia (AUC) + Right-wing Death Squad + Paramilitaries +Gunmen +Right-wing Paramilitaries , + Civil Association for Peace in Colombia, Asocipaz +Left-wing extremists + Death Squad +People's Revolutionary Army (ERP). + Rebels +Left-wing Guerillas . + Black Eagles + (dda) Kwiy s,aidoad uekenbeled +Los Rastrojos (Colombia) +People's Revolutionary Movement (MRP) +Revolutionary Armed Forces of Colombia (FARC) dissidents + Patria Grande Ejercito del Pueblo +Gaitanista Self-Defense Forces of Colombia (AGC) 2000 +0.6 +2001 +2002 +2003 + 0.5 +2004 +2005 +2006 +0.4 +2007 +2008 +2009 +0.3 +2010 +2011 +2012 +2013 +0.2 +2014 : +2015 +2016 + 0.1 +2017 +2018 +0.0 +New People's Army (NPA) , +Moro Islamic Liberation Front (MILF) +(s) d jeres +Moro National Liberation Front (MNLF) +Unknown +Muslim extremists +Left-wing extremists +sw!isnw + Jemaah Islamiya (J) +Pentagon Kidnap Group + Muslim Rebels +Muslim Separatists +Communists +(sauiddiyd an jo Jaipios) seuidild bu lemey +Young Officer Union of the New Generation and Reformist Armed Forces of the Philippines (YOU-RAFP) + Al-Khobar +Alcubar group +Alex Boncayao Brigade (ABB) +Farmer's Movement of the Philippines (KMP) +Bangsamoro National Liberation Army +Aniban ng Ayaw sa Komunista (ANAK) + Rebels +Partido Marxista-Leninista ng Pilipinas (PMLP) +Bangsamoro Islamic Freedom Movement (BIFM) , +Kilafah Islamic Movement , +National Democratic Front-Bicol (NDF-Bicol) + Moro Ghuraba +Magahat Militia +Maute Group +Ansar Al-Khilafa (Philippines) + Jundul Khilafah (Philippines) +Islamic State of Iraq and the Levant (ISIL) +New Indigenous People's Army +East Asia Division of the Islamic State(a) +(b) +(c) +(d) +(e) +(f) +Figure 2: Hierarchical clustering on the perpetrator composition matrix DR for R being one +of six countries: (a) Afghanistan (b) the United States (c) Somalia (d) Sri Lanka (e) Colombia +(f) the Philippines. +13 + +Sri Lanka +1.0 +2012 +2012 +2010 +2010 +2015 +2015 +2016 +2016 +0.8 +2018 +2018 +2013 +2013 +2014 +2014 +2007 + 0.6 +2007 +2005 +2005 +2001 +2001 +2000 +2000 + 0.4 +2008 +2008 +2004 +2004 +2009 +2009 +2006 +2006 + 0.2 +2003 +2003 +2002 +2002 +2017 +2017 +0.2 +9°0 +201220102015201620182013201420072005200120002008200420092006200320022017 +0.0 +0.0 +0.4Colombia +2018 +2018 +2017 +2017 +0.8 +2016 +2016 +2012 +2012 +0. 7 +2009 +2009 +2006 +2006 +0.6 +2007 +2007 +2013 +2013 +2005 +2005 + 0.5 +2002 +2002 +2004 +2004 +0.4 +2008 +2008 +2014 +2014 +0.3 +2011 +2011 +2010 +2010 + 0.2 +2003 +2003 +2015 +2015 +0.1 +2000 +2000 +2001 +2001 +0.5 +0. 0 +0'0 +0.1 +0.2 +E'O +0.4Philippines +2005 +2005 +0. 7 +2002 +2002 +2001 +2001 +0. 6 +2016 +2016 +2013 +2013 +2012 +2012 + 0.5 +2011 +2011 +2006 +2006 +2010 +2010 + 0.4 +2009 +2009 +2008 +2008 +0.3 +2015 +2015 +2014 +2014 +2018 +2018 + 0.2 +2017 +2017 +2007 +2007 +2004 +2004 + 0.1 +2003 +2003 +2000 +2000 +0.0 +0.1 +0.2 +E0 +0.4Afghanistan +2013 +2013 +2009 +2009 +2012 +2012 +2014 +2014 +2006 +2006 +0.6 +2010 +2010 +2007 +2007 + 0.5 +2008 +2008 +2004 +2004 +2016 +2016 +0.4 +2015 +2015 +2011 +2011 + 0.3 +2017 +2017 +2018 +2018 +2005 +2005 +02 +2003 +2003 +2001 +2001 + 0.1 +2002 +2002 +2000 +2000 +0.0 +0.0 +0.1 +0.2 +EOUnitedStates +1.0 +2014 +2014 +2010 +2010 +2009 +2009 +2018 +2018 +0.8 +2016 +2016 +2015 +2015 +2004 +2004 +2003 +2003 + 0.6 +2007 +2007 +2005 +2005 +2001 +2001 +2008 +2008 + 0.4 +2000 +2000 +2011 +2011 +2006 +2006 + 0.2 +2017 +2017 +2013 +2013 +2012 +2012 +2002 +2002 +0.0 +0.0 +0.1 +0.2 +0.4 +9050Somalia +2003 +2003 +2000 +2000 +2007 +2007 +2008 +2008 +0.8 +2006 +2006 +2005 +2005 +2012 +2012 +2009 +2009 + 0.6 +2010 +2010 +2016 +2016 +2013 +2013 +2017 +2017 + 0.4 +2015 +2015 +2018 +2018 +2014 +2014 + 0.2 +2011 +2011 +2002 +2002 +2004 +2001 +2001 +0.0 +0.0 +0.2 +0.4 +9:0 +0.8 +1.0(a) +(b) +Figure 3: Hierarchical clustering applied to the Wasserstein metric D between time series +distributions fR as R ranges over our collection of (a) 40 countries and (b) 13 regions. +14 + +Sri Lanka +Ageria +Spain + United States +West Bank and Gaza Strip +140 +Burundi +Nigeria +Somalia +Kenya +Yemen +120 +Afghanistan +Sudan +Pilippines +Egypt +Libya +100 +eus +Central African Republic +Turkey +Ukraine +Bangladesh + 80 +Iraq +South Sudan +Mali +Germany +Pakistan + 60 +Democratic Republic of the Congc +Saudi Arabia +Cameroon +Lebanon +40 +United Kingdom +hdia +brael +Nepal +France +20 +Colombia +Thailand +Greece +Russia +hdonesia +5 +10 +15 +20 +25Central Russia +Colombiaand Ecuador +60 +Europe +North America +50 +Southeast Asia +East China + 40 +Southern Africa +West Africa +30 +East Africa +The Philippines +20 +Southern South America +10 +The Middle East +South Asia +10 +15(a) +(b) +(c) +(d) +(e) +(f) +Figure 4: Time series (and smoothing curve) of monthly terrorist events for (a) Nigeria (b) +India (c) Sri Lanka (d) Spain (e) the United States (f) the West Bank and Gaza Strip. +15 + +80 +60 +40 +20 +0140 +120 +100 +80 +60 +40 +2040 +30 +20 +1030 +25 +20 +15 +1025 +20 +15 +10 +5 70 +60 +50 +40 +30 +20 +10determined regions have then been used in other analysis in the paper, but could +be used in concert with countless other methods in the applied mathematics and +statistics literature, some of which have been applied to terrorism, but many of +which have not. +In Section 3, we reveal that despite geographic persistence, there are ample +changes in the composition of terrorist actors over time. And yet this is far +from uniform among different countries and regions. +We discuss geometric +interpretations of distributions over perpetrator groups and settle on an ap- +propriate discrepancy measurement that has both geometric and probabilistic +interpretations. Our results show substantial differences in the heterogeneity of +changing perpetrator composition over time. In particular, regions and countries +of Africa show the greatest heterogeneity, with African regions and countries +represented among the highest ranks of heterogeneity in both Table 3 and 4. We +also discuss numerous high-profile examples of changing composition of terrorist +activity, including the Tamil Tigers in Sri Lanka and the FARC in Colombia. +Such insights are revealed to even authors with little specialized knowledge in +terrorist activity, so we are confident our methodology may be used by experts +in other fields to discern distributional changes both of a more subtle nature and +in various other contexts. +Finally, Section 4 judiciously selects an appropriate metric to understand the +discrepancy between the noisy time series of monthly terrorist attack counts. +Applying this to both countries and our determined regions, hierarchical clus- +tering is able to quickly reveal noteworthy similarities and differences between +distributions over time. The majority of countries, particularly from Africa, are +revealed to have terrorist activity dominated by the last decade, but that is +not uniform, with a more consistent pattern from the United States and West +Bank/Gaza, and the opposite pattern from countries such as Algeria, Sri Lanka +and Spain. Applying the same analysis to the determined regions reveals these +geographic differences even more clearly. +The primary limitation of this paper is its global scope. Future work could +use these methods, including with more reliable and focused data sets, on a +much more granular level, investigating for example terrorist activity within a +single country over time. With greater precision in location reporting, geographic +clusters of activity within a particular country could conceivably be far less +consistent over time. One could repeat our analysis of compositional changes +by removing “unknown” as the affiliation, or could aim to use investigative or +supervised learning techniques to identify or guess the affiliation as much as +possible. Further, one could incorporate relationships between different groups +to modify the distance between distributions over perpetrators, such that related +groups are considered closer than unrelated groups. And the number of attacks +by different groups could be incorporated into the time series analysis of Section +4, including the use of parametric methods such as ARIMA models or frequency +analysis. All our methods could be used in conjunction with a careful comparison +of policy responses to identify outlier countries where the composition of terror +perpetrators or the number of events is notably greater or lesser. This could +reveal which policies were the most and least successful at a variety of goals, +16 + +either the overall reduction of terrorism or the disruption and management of +certain groups. +Overall, we believe this paper has added several mathematical approaches +to the as-of-yet-limited use of mathematics, statistics and machine learning to +analyze persistence and trends in terrorism, revealed elegant insights therein, +and could be incorporated with the work of both other mathematicians and +security experts to glean further insights into worldwide terrorism. Given the +vast expenditures spent every year on combating terrorism, we would welcome +further approaches to its research. +Conflict of interest +The authors declare that they have no conflict of interest. +Data availability statement +All data analyzed in this study are available at the Global Terrorism Database +[7]. +Appendix A. Existing cluster theory +In this section, we provide an overview of the three clustering algorithms +required in the above implementations: hierarchical clustering, K-means and +K-medoids, and methods to select the optimal number of clusters. In our most +general setup, x1, . . . , xn are elements of some normed or metric space X. +Hierarchical clustering [57] is an iterative clustering technique that does not +specify discrete partitions of elements. Rather, the methodology seeks to build a +hierarchy of similarity between elements. In this paper we use agglomerative +hierarchical clustering with the average linkage method [76], where each element +xi begins in its own cluster and branches between them are successively built. +The results of hierarchical clustering are commonly displayed in dendrograms, as +we use in this paper. Unlike K-means, hierarchical clustering does not require +the choice of a number of clusters k. Further, hierarchical clustering can be +implemented on any distance, not necessarily Euclidean space. +K-means and K-medoids clustering seek to minimize appropriate sums of +square distances. With k chosen a priori, we investigate all possible partitions +(disjoint unions) C1 ∪ C2 ∪ · · · ∪ Ck of {x1, . . . , xn}. Let zj be the centroid +(average) of the subset Cj. For K-means, one seeks to minimize the sum of +square distances within each cluster to its centroid: +k +� +j=1 +� +x∈Cj +||x − zj||2 +For a normed space with dimension at least 2, it is NP-hard to find the global +optimum of this problem. The K-means algorithm [55] is an iterative algorithm +17 + +that converges quickly and suitably to a locally optimal solution. K-means can +typically only be implemented on Euclidean space. K-medoids is a substitute +where zj is restricted to be one of the data points, and selected such that its +distance from other elements of the cluster is minimal. Thus, K-medoids can be +implemented more generally on any metric space, not necessarily on a normed +space with the Euclidean metric. +How to best choose the number of clusters k for the K-means or K-medoids +algorithm is a difficult problem. Different methods for estimating k may produce +considerably differing results. +In this paper, we draw upon 16 methods to +determine the appropriate number of clusters before implementing K-means +or K-medoids. +These methods include among others, Ptbiserial index [77], +silhouette score [78], KL index [79], C index [80], McClain-Rao index [81] and +Dunn index [82]. The methodology described above is flexible however, and may +use any combination of existing methods. +Appendix B. Probability distribution distance +Let (X, d) be any metric space, µ, ν two probability measures on X, and +q ≥ 1. The Wasserstein metric between µ, ν is defined as +W q(µ, ν) = inf +γ +� � +X×X +d(x, y)qdγ +� 1 +q +, +(B.1) +where the infimum is taken over all probability measures γ on X × X with +marginal distributions µ and ν. Henceforth, let q = 1. By the Kantorovich- +Rubinstein formula [83], there is an alternative formulation when X is compact +(for example, finite): +W 1(µ, ν) = sup +F +���� +� +X +Fdµ − +� +X +Fdν +���� , +(B.2) +where the supremum is taken over all 1-Lipschitz functions F : X → R. +Proposition Appendix B.1. Let (X, d) be a finite discrete metric space, with +d(x, y) = 1 for all x ̸= y and 0 otherwise. Let W 1(µ, ν) be the L1-Wasserstein +metric between two probability measures µ, ν on X, with corresponding distribu- +tion functions f, g, expressed as in (B.2). That is, +W 1(µ, ν) = sup +F +���� +� +X +Fdµ − +� +X +Fdν +���� . +(B.3) +Then, this supremum is optimized by the following choice of F: +F(x) = +� +1, +f(x) ≥ g(x), +0, +f(x) < g(x). +(B.4) +18 + +Thus, W 1(f, g) reduces to the same form of (1), namely +W 1(f, g) = 1 +2||f − g||1. +(B.5) +Proof. Let F be an arbitrary 1-Lipschitz function on X. That is, F : X → R +and |F(x) − F(y)| ≤ d(x, y) ≤ 1 for all x, y. +Let M = supx∈X F(x) and +m = infy∈X F(y). By taking the supremum over x and the infimum over y, the +Lipschitz condition implies that M − m ≤ 1. So +� +X +Fdµ − +� +X +Fdν = +� +x∈X +F(x)(f(x) − g(x)) +(B.6) +≤ +� +x:f(x)≥g(x) +M(f(x) − g(x)) + +� +x:f(x) 0, i = 1, . . . , n, +where Ω ⊂ Rd (d ≥ 1) is a bounded domain and u = (u1, . . . , un). This model was derived +rigorously from a moderately interacting stochastic particle system in a mean-field-type +limit [11]. The parameter γ > 0 is related to the stochastic diffusion of the particle system, +and aij ∈ R describes the strength of the repulsive or attractive interaction between the +ith and the jth species. We impose initial and no-flux boundary conditions, +(2) +ui(0) = u0 +i +in Ω, +∇ui · ν = 0 +on ∂Ω, t > 0, i = 1, . . . , n, +where ν is the exterior unit normal vector to ∂Ω. In the absence of the diffusion parameter +γ, (1) can be interpreted as a mass conservation equation with the partial velocity ∇pi(u), +which is determined according to Darcy’s law by the partial pressure pi(u). System (1) in +one space dimension for two species, γ = 0, and det(aij) = 0 was first studied in [4], proving +the global existence of segregated solutions (i.e., the supports of u1 and u2 do not intersect +for all times if this holds true initially). This result was generalized to arbitrary space +dimensions in [5], still for two species. For an arbitrary number of species, the existence +of global weak solutions to (1)–(2) was shown in [25, Appendix B] if det(aij) > 0 and the +existence of local strong solutions was proved in [18] if det(aij) = 0. +The matrix A = (aij) ∈ Rn×n does not need to be symmetric nor positive definite so that +the diffusion matrix associated to (1) is generally neither symmetric nor positive definite +too. A minimal requirement for local solvability at the linear level is the parabolicity in +the sense of Petrovskii, which is satisfied if all eigenvalues of A have a positive real part +[1]. Global solvability is guaranteed under the detailed-balance condition, i.e., there exist +π1, . . . , πn > 0 such that πiaij = πjaji for all i ̸= j [25, Theorem 17]. This condition also +appears in the theory of time-continuous Markov chains generated by A, and (π1, . . . , πn) +is the associated invariant measure. We assume this condition throughout this paper. It +implies that �ui := πiui solves the system +∂t�ui = div +� +�ui +n +� +j=1 +aij +πj +∇�uj +� +, +with a symmetric positive definite matrix (aij/πj). Consequently, we may assume, without +loss of generality, that the matrix A in (1) is already symmetric and positive definite. +Due to the nonlinear cross-diffusion structure, the analysis of (1) is highly nontrivial. +The key idea of the analysis is to exploit the entropy structure of (1). This means that +there exist Lyapunov functionals, called entropies, that are nonincreasing in time along +solutions to (1)–(2) and that provide gradient estimates. In the present situation, these +functionals are given by the Boltzmann (or Shannon) entropy HB and the Rao entropy + +BDF2 FINITE-VOLUME SCHEME +3 +HR, +HB(u) = +n +� +i=1 +� +Ω +ui(log ui − 1)dx, +HR(u) = 1 +2 +� +Ω +uTAudx, +giving formally the entropy equalities +dHB +dt ++ +� +Ω +� +4γ +n +� +i=1 +|∇√ui|2 + +n +� +i,j=1 +aij∇ui · ∇uj +� +dx = 0, +(3) +dHR +dt ++ +� +Ω +� +γ +n +� +i,j=1 +aij∇ui · ∇uj + +n +� +i=1 +ui|∇pi(u)|2 +� +dx = 0, +(4) +and thus providing gradient bounds for ui. +The Boltzmann entropy is related to the +thermodynamic entropy of the system, while the Rao entropy measures the functional +diversity of the species [30]. +Since the Boltzmann entropy HR is convex, the implicit Euler scheme preserves the +entropy inequality (3) (see, e.g., [27] for a related system). The logarithmic structure of +HR seems to prevent entropy stability in higher-order schemes like BDF or Crank–Nicolson +approximations [22]. However, thanks to the quadratic structure of the Rao entropy HR, +we are able to prove stability of HR for the BFD2 approximation. To explain the idea, let +T be a triangulation of Ω into control volumes K ⊂ Ω with measure m(K) and let ∆t be +the time step size. Furthermore, let uk +i,K be an approximation of ui(xK, tk), where xK ∈ K +and tk = k∆t. We write the BDF2 discretization of (1) as +(5) +m(K) +∆t +�3 +2uk +i,K − 2uk +i,K + 1 +2uk−2 +i,K +� ++ +� +σ∈EK +Fk +i,K,σ = 0, +where EK is the set of the edges (or faces) of K and Fk +i,K,σ is the numerical flux, defined in +(18) below. The usual idea to derive a priori bounds is to choose the test function uk +i,K in +(5) and to use the inequality +(6) +�3 +2uk +i,K − 2uk +i,K + 1 +2uk−2 +i,K +� +uk +i,K ≥ h0(uk +i,K, uk−1 +i,K ) − h0(uk−1 +i,K , uk−2 +i,K ), +where +h0(a, b) = 1 +4 +� +5a2 − 4ab + b2� += 1 +4 +� +a +b +�T � +5 +−2 +−2 +1 +� � +a +b +� +, +a, b ∈ R, +is a positive definite quadratic form. Assuming that Fk +i,K,σuk +i,K can be bounded from below, +this gives a priori bounds for (uk +i,K)2. Inequality (6) can be explained in the framework of +Dahlquist’s G-stability theory [23]. +In our case, we need the test function pi(uk +K) to derive the discrete analog of (4). Then +the question is whether there exists a functional h(u, v) such that +(7) +n +� +i=1 +�3 +2uk +i,K − 2uk +i,K + 1 +2uk−2 +i,K +� +pi(uk +K) ≥ h(uk +K, uk−1 +K ) − h(uk−1 +K , uk−2 +K ). + +4 +A. J¨UNGEL AND M. VETTER +Note that we need to sum over all species in this inequality. The main novelty of this +paper is the observation that the scalar inequality (6) can be extended to inequality (7) +for vectors u, v ∈ Rn. Indeed, we show in Lemma 7 that (7) holds for +(8) +h(u, v) = 1 +4(5uTAu − 4uTAv + vTAv) = 1 +4 +� +u +v +�T � +5A +−2A +−2A +A +� � +u +v +� +with u, v ∈ Rn. Introducing the discrete Rao entropy by H(u, v) = � +K∈T m(K)h(u, v) +for piecewise constant functions u and v, this yields the BDF2 analog of the Rao entropy +inequality +H(uk, uk−1) + c∆t|uk|2 +1,2,T ≤ H(uk−1, uk−2) +for k ≥ 2, +where |·|1,2,T is the discrete H1(Ω) norm, defined in Section 2.3, and c > 0 depends on the +smallest eigenvalue of A and on γ. This inequality is the key for proving our main results: +• Existence and uniqueness of discrete solutions: There exists a solution uk +i to the +BDF2 finite-volume scheme (5), which conserves the mass � +K∈T m(K)uk +i,K of the +ith species and dissipates the discrete Rao entropy. Moreover, the solution is unique +if ∆t/(∆x)d+2 is sufficiently small, where ∆x is the size of the mesh (Theorem 3). +This unusual quotient comes from an inverse inequality needed to bound higher- +order norms. +• Large-time behavior: The discrete solution uk +i converges for large times k → ∞ +to the constant steady state ¯ui = m(Ω)−1 � +Ω u0 +i dx with a quasi-explicit exponential +rate (Theorem 4). The proof uses the well-established relative entropy (or energy) +method, but the two-step scheme requires an iteration of this argument. +• Convergence of the discrete scheme: The fully discrete solution converges to a solu- +tion to the semidiscrete problem if ∆x → 0, and the semidiscrete solution converges +to a weak (nonnegative) solution to (1)–(2) as ∆t → 0 (up to subsequences; see +Theorem 5). +• Convergence rate: If the solution to (1)–(2) is sufficiently smooth, the semidiscrete +solution converges with order two, as expected for the BDF2 scheme (Theorem 6). +The paper is organized as follows. +The numerical scheme and our main results are +detailed in Section 2. In Section 3, we prove the existence and uniqueness of a discrete +solution, while its large-time behavior is analyzed in Section 4. Section 5 is devoted to +the convergence of the full scheme, and the second-order convergence in time is verified in +Section 6. Finally, we present in Section 7 some numerical examples in one and two space +dimensions. +2. Numerical scheme and main results +We need some simple auxiliary results and some notation before formulating the numer- +ical scheme and the main results. +2.1. Some linear algebra. We denote by | · | the Euclidean norm on Rn. Given a sym- +metric positive matrix A ∈ Rn×n, we introduce the weighted norm |u|2 +A := uTAu and the + +BDF2 FINITE-VOLUME SCHEME +5 +weighted inner product (u, v)A := uTAv for u, v ∈ Rn. With this notation, the discrete +Rao entropy density can be written as +(9) +h(u, v) = 1 +4(5|u|2 +A − 4(u, v)A + |v|2 +A) +for u, v ∈ Rn. +Denoting by λm > 0 the smallest and by λM > 0 the largest eigenvalue of A, it holds that +(10) +λm|u|2 ≤ |u|2 +A ≤ λM|u|2 +for u ∈ Rn. +Let λ1, . . . , λn > 0 be the eigenvalues of A. Then the eigenvalues of the matrix in (8) equal +(3 ± +√ +8)λi for i = 1, . . . , n. This shows that for u, v ∈ Rn, +(11) +1 +4(3 − +√ +8)(|u|2 +A + |v|2 +A) ≤ h(u, v) ≤ 1 +4(3 + +√ +8)(|u|2 +A + |v|2 +A), +1 +4(3 − +√ +8)λm(|u|2 + |v|2) ≤ h(u, v) ≤ 1 +4(3 + +√ +8)λM(|u|2 + |v|2). +2.2. Spatial domain and mesh. Let d ≥ 1 and let Ω ⊂ Rd be a bounded polygonal (if +d = 2) or polyhedral (if d ≥ 3) domain. We associate to this domain an admissible mesh, +given by (i) a family T of open polygonal or polyhedral control volumes, which are also +called cells, (ii) a family E of edges (or faces if d ≥ 3), and (iii) a family of points (xK)K∈T +associated to the control volumes and satisfying [21, Definition 9.1]. This definition implies +that the straight line xKxL between two centers of neighboring cells is orthogonal to the +edge (or face) σ = K|L between two cells. For instance, triangular meshes with acute +angles, Delaunay meshes, rectangular meshes, and Vorono¨ı meshes satisfy this condition +[21, Example 9.2]. The size of the mesh is given by ∆x = maxK∈T diam(K). The family +of edges E is assumed to consist of interior edges Eint satisfying σ ⊂ Ω and boundary edges +σ ∈ Eext satisfying σ ⊂ ∂Ω. For a given K ∈ T , EK denotes the set of edges of K with +EK = Eint,K ∪ Eext,K. For any σ ∈ E, there exists at least one cell K ∈ T such that σ ∈ EK. +For given σ ∈ E, we define the distance +dσ = +� +d(xK, xL) +if σ = K|L ∈ Eint,K, +d(xK, σ) +if σ ∈ Eext,K, +where d is the Euclidean distance in Rd, and the transmissibility coefficient +(12) +τσ = m(σ) +dσ +, +where m(σ) denotes the (d − 1)-dimensional Lebesgue measure of σ. +We suppose the +following mesh regularity condition: There exists 0 < ζ ≤ 1/2 such that for all K ∈ T and +σ ∈ EK, +(13) +d(xK, σ) ≥ ζdσ. +This is equivalent to +η ≤ d(xK, σ) +d(xL, σ) ≤ 1 +η +for all σ = K|L, + +6 +A. J¨UNGEL AND M. VETTER +where η = ζ/(1−ζ) ∈ (0, 1]. The statement follows by observing that d(xK, σ)+d(xL, σ) = +d(xK, xL) holds, which is a consequence of the orthogonality of σ = K|L and xKxL. Hence, +the mesh regularity (13) means that the mesh is locally quasi-uniform. A consequence of +the mesh regularity is the following estimate +(14) +� +σ∈Eint,K +m(σ)dσ ≤ 1 +ζ +� +σ∈Eint,K +m(σ)d(xK, σ) = d +ζ m(K) +for K ∈ T , +where we used in the last step the formula for the volume of a (hyper-)pyramid. +2.3. Function spaces. Given a triangulation T , let T > 0, NT ∈ N and introduce the +time step size ∆t = T/NT and the time steps tk = k∆t for k = 0, . . . , NT. +We set +ΩT = Ω × (0, T). The space of piecewise constant functions is defined by +VT = +� +v : Ω → R : ∃(vK)K∈T ⊂ R, v(x) = +� +K∈T +vK1K(x) +� +, +where 1K is the indicator function on K. To define a norm on this space, we define for +K ∈ T , σ ∈ EK, +vK,σ = +� +vL +if σ = K|L ∈ Eint,K, +vK +if σ ∈ Eext,K, +DK,σv := vK,σ − vK, +Dσv := |DK,σv|. +Let 1 ≤ q < ∞ and v ∈ VT . The discrete W 1,q(Ω) norm on VT is given by +∥v∥1,q,T = +� +∥v∥q +0,q,T + |v|q +1,q,T +�1/q, +where +∥v∥q +0,q,T = +� +K∈T +m(K)|vK|q, +|v|q +1,q,T = +� +σ∈Eint +m(σ)dσ +���� +Dσv +dσ +���� +q +for v ∈ VT . +When q = ∞, we define |v|1,∞,T = maxσ∈Eint |Dσv|/dσ. +If v = (v1, . . . , vn) ∈ V n +T is a +vector-valued function, we write for notational convenience +∥v∥0,q,T = +n +� +i=1 +∥vi∥0,q,T , +∥∇v∥0,q,T = +n +� +i=1 +∥∇vi∥0,q,T . +We associate to the discrete W 1,q norm a dual norm with respect to the L2 inner product: +∥v∥−1,q,T = sup +� � +Ω +vwdx : w ∈ VT , ∥w∥1,q,T = 1 +� +. +Finally, we introduce the space VT ,∆t of piecewise constant functions with values in VT , +VT ,∆t = +� +v : ΩT → R : ∃(vk)k=1,...,NT ⊂ VT , v(x, t) = +NT +� +k=1 +vk(x)1[tk−1,tk)(t) +� +, + +BDF2 FINITE-VOLUME SCHEME +7 +equipped with the L2(0, T; H1(Ω)) norm +∥v∥L2(0,T;H1(Ω)) = +� NT +� +k=1 +∆t∥vk∥2 +1,2,T +�1/2 +for all v ∈ VT ,∆t. +2.4. Discrete gradient. The discrete gradient is defined on a dual mesh. For this, we +define the cell TK,σ of the dual mesh for K ∈ T and σ ∈ EK: +• “Diamond”: Let σ = K|L ∈ Eint,K. Then TK,σ is that cell whose vertices are given +by xK, xL, and the end points of the edge σ. In higher dimensions, they might be +(double) (hyper-)pyramids. +• “Triangle”: Let σ ∈ Eext,K. Then TK,σ is that cell whose vertices are given by xK +and the end points of the edge σ. +The union of all “diamonds” and “triangles” TK,σ equals the domain Ω (up to a set of +measure zero). The property that the straight line xKxL is orthogonal to the edge σ = K|L +implies that +m(σ)d(xK, xL) = d m(TK,σ) +for all σ = K|L ∈ Eint. +The approximate gradient of v ∈ VT ,∆t is then defined by +∇T v(x, t) = +m(σ) +m(TK,σ)DK,σ(vk)νK,σ +for x ∈ TK,σ, t ∈ (tk−1, tk], +where νK,σ is the unit vector that is normal to σ and points outwards of K. +2.5. Numerical scheme. The initial functions are approximated by their L2(Ω)-orthogo- +nal projection on VT : +(15) +u0 +i,K = +1 +m(K) +� +K +u0 +i (x)dx +for all K ∈ T , i = 0, . . . , n. +Let uk−1 +K += (uk−1 +1,K , . . . , uk−1 +n,K) for K ∈ T be given. Since the BDF2 scheme is a two-step +method, we need a first time step which is computed from the implicit Euler method. The +following time steps are determined from the BDF2 method. The finite-volume scheme +reads as +m(K) +∆t (u1 +i,K − u0 +i,K) + +� +σ∈EK +F1 +i,K,σ = 0, +(16) +m(K) +∆t +�3 +2uk +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� ++ +� +σ∈EK +Fk +i,K,σ = 0, +k ≥ 2, +(17) +for i = 1, . . . , n, K ∈ T , and the numerical fluxes are given by +(18) +Fk +i,K,σ = −τσ +� +γDK,σuk +i + (uk +i,σ)+DK,σpi(uk) +� +, +where τσ is defined in (12) and z+ = max{0, z} denotes the positive part of z ∈ R. Finally, +the so-called mobility is given by +(19) +uk +i,σ = M(uk +i,K, uk +i,L) +for σ = K|L, +uk +i,σ = 0 +else, + +8 +A. J¨UNGEL AND M. VETTER +where M is a general mean function satisfying +(i) M : [0, ∞)2 → [0, ∞) is Lipschitz continuous, satisfies M(u, u) = u (consistency), +and has linear growth in the sense M(u, v) ≤ |u| + |v| for u, v ≥ 0. +(ii) There exists C > 0 such that |M(u, v) − u| ≤ C|u − v| for all u, v ≥ 0. +Examples for M are M(u, v) = (u + v)/2 or M(u, v) = max{u, v}. Note that we do not +need logarithmic mean functions like in [27], since we do not use the chain rule in the +cross-diffusion part, so that we can use simpler expressions. +Remark 1 (Nonnegativity). We truncate the mobility by (uk +i,σ)+ in the numerical flux +(18) to ensure the discrete Rao entropy inequality (see (21) below). Indeed, when testing +(17) with pi(uk), we need that the sum � +σ∈Eint τσ(uk +i,σ)+|DK,σpi(uk)|2 is nonnegative. Un- +fortunately, the quadratic Rao entropy does not allow us to prove the nonnegativity of the +discrete solution, and standard maximum principle arguments do not apply here, so that +the truncation cannot be removed. A positivity-preserving BDF2 finite-difference scheme +was proposed in [13], but the proof relies on discrete L∞(Ω) bounds for uk−1, which are not +available in our case. Also the Shannon entropy does not help (as in [27]), since it is not +compatible with the BDF2 discretization. Indeed, when we wish to derive a discrete analog +of (3), we need a finite continuous functional h(u, v) satisfying h(u, u) = �n +i=1 ui(log ui−1) +(consistency condition) such that +n +� +i=1 +�3 +2uk +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� +log uk +i,K ≥ h(uk +K, uk−1 +K ) − h(uk−1 +K , uk−2 +K ). +If uk +i,K = uk−1 +i,K → 0 and uk−2 +i,K > 0 for all i ∈ {1, . . . , n}, the previous inequality converges +to −∞ ≥ −h(0, uk−2 +K ), which is absurd. At least, we obtain nonnegative solutions in the +limit (∆x, ∆t) → 0; see Theorem 5 below. +Remark 2 (Discrete integration by parts). The fluxes Fk +i,K,σ are consistent approximations +of the exact fluxes through the edges if we impose the conservation Fi,K,σ + Fi,L,σ = 0 for +all edges σ = K|L, requiring that they vanish on the Neumann boundary edges, i.e., +Fi,K,σ = 0 for all σ ∈ Eext,K. In particular, for v = (vK) ∈ VT , the following discrete +integration-by-parts formulas hold: +(20) +� +K∈T +� +σ∈EK +Fi,K,σvK = − +� +σ∈Eint +σ=K|L +Fi,K,σDK,σv, +� +K∈T +� +σ∈EK +τσ(DK,σv)vK = −|v|2 +1,2,T . +2.6. Main results. We impose the following hypotheses. +(H1) Data: Ω ⊂ Rd with d ≥ 1 is a bounded polygonal (d = 2) or polyhedral (d ≥ 3) +domain, T > 0, and u0 ∈ L2(Ω; Rn). We set ΩT = Ω × (0, T). +(H2) Discretization: T is an admissible discretization of Ω satisfying (13) and tk = k∆t +for k = 1, . . . , NT. +(H3) Coefficients: Let γ > 0, and A = (aij) ∈ Rn×n is symmetric and positive definite. +Let λm > 0 and λM > 0 be the smallest and largest eigenvalue of A, respectively. + +BDF2 FINITE-VOLUME SCHEME +9 +The positivity of γ is not needed for the existence analysis but for the convergence +result, where we need higher-order integrability that is deduced via the discrete Gagliardo– +Nirenberg inequality from the gradient bound. +As mentioned in the introduction, the +symmetry and positive definiteness of A can be replaced by the positivity of the real parts +of the eigenvalues of A and the detailed-balance condition. +Recall the discrete BDF2 Rao entropy (see (9)) +H(u, v) = +� +K∈T +m(K)h(uK, vK) = 1 +4 +� +K∈T +m(K) +� +5|uK|2 +A − 4(uK, vK)A + |vK|2 +A +� +for u, v ∈ VT . If u = v, this expression reduces to the usual discrete Rao entropy, used for +the implicit Euler scheme, H(u) := H(u, u) = 1 +2 +� +K∈T m(K)|uK|2 +A. Our first result is the +existence of a discrete solution. +Theorem 3 (Existence and uniqueness of discrete solutions). Let Hypotheses (H1)–(H3) +hold, let k ∈ N, and let uk−1 ∈ V n +T be given. Then there exists a solution uk = (uk +1, . . . , uk +n) ∈ +V n +T to scheme (15)–(19) satisfying the discrete entropy inequality +H(uk, uk−1) + γ∆t|A1/2uk|2 +1,2,T ≤ H(uk−1, uk−2) +for k ≥ 2, +(21) +H(u1) + γ∆t|A1/2u1|2 +1,2,T ≤ H(u0), +(22) +and the scheme preserves the mass, � +K∈T m(K)uk +i,K = +� +Ω u0 +i (x)dx for i = 1, . . . , n, k ≥ 1. +These results also hold if γ = 0. Furthermore, the solution is unique if γ > 0, minσ∈Eint dσ ≥ +ξ∆x for some ξ > 0, and +∆t +(∆x)d+2 < C(d, ξ, ζ)γλ2 +m +λ2 +ML2H(u0) , +where ζ is defined in (13) and L is the Lipschitz constant of the mean function M, defined +in (19). +The existence of a discrete solution is proved by a fixed-point argument using the Brouwer +degree theorem. Uniform estimates are obtained from the discrete Rao entropy inequality +(21), where the BDF2 time approximation is estimated according to (7). This inequality, +which is the key of our analysis, is proved in Lemma 7. +The uniqueness of solutions is proved by using the relative entropy method, which is +equivalent to the energy method in the present case, since the Rao entropy is quadratic. In +other words, we use the test function pi(uk) − pi(vk) in the difference of the equations (17) +satisfied by two discrete solutions uk and vk. The cross-diffusion part contains cubic ex- +pressions, which turn into quadratic ones if |A1/2vk|1,∞,T is bounded (similar as in [12]). By +an inverse inequality, this norm is bounded, up to some factor, by (∆x)−d/2−1∥A1/2vk∥0,2,T , +and ∥A1/2vk∥0,2,T is bounded because of (21)–(22). The remaining quadratic expression +is estimated by using the gradient bounds (which requires γ > 0) and the discrete L2(Ω) +bound coming from the time discretization (and introducing the factor ∆t). The condition +minσ∈Eint dσ ≤ ξ∆x is discussed in Remark 8. +For the next result, we set ¯ui = m(Ω)−1 � +Ω u0 +i dx and recall the discrete Poincar´e– +Wirtinger inequality ∥v − ¯v∥0,2,T ≤ CPζ−1/2|v|1,2,T for v ∈ VT [7, Theorem 3.6]. Then, + +10 +A. J¨UNGEL AND M. VETTER +in view of (10), +(23) +∥A1/2(v − ¯v)∥0,2,T ≤ CP +� λM +λmζ +�1/2 +|A1/2v|1,2,T +for v ∈ VT . +Theorem 4 (Large-time behavior). Let uk be a solution to scheme (15)–(19). Then, for +k ≥ 2, +∥A1/2(uk − ¯u)∥0,2,T ≤ +√ +2∥A1/2(u0 − ¯u)∥0,2,T (1 + κ∆t)−(k−2)/4, +where κ = 4γλmζ/((3 + +√ +8)C2 +PλM) and CP > 0 is the constant of the Poincar´e–Wirtinger +inequality (23). +The theorem states that uk converges exponentially fast to the constant steady state ¯u. +Indeed, setting λ∆t := log(1 + κ∆t)/(∆t) ↗ κ as ∆t → 0, we have +∥A1/2(uk − ¯u)∥0,2,T ≤ +√ +2∥A1/2(u0 − ¯u)∥0,2,T exp(−λ∆ttk), +k ≥ 2. +The proof of Theorem 4 is based on the discrete entropy inequality for the discrete relative +Rao entropy H(uk−¯u, uk−1−¯u), similar to (21). Indeed, by the discrete Poincar´e–Wirtinger +inequality, the discrete gradient term is bounded from below by the discrete L2(Ω) norm +of uk − ¯u. As H(uk − ¯u, uk−1 − ¯u) can be estimated in terms of the discrete L2(Ω) norms of +uk − ¯u and uk−1 − ¯u, we need to iterate the entropy inequality a second time. Then, using +(11), we arrive at the inequality +H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−1H(uk−2 − ¯u, uk−3 − ¯u), +and solving this recursion shows the result. +The numerial convergence of the scheme is proved in two steps. First, we show that the +fully discrete solution uk +m ∈ V n +T , indexed with the space grid size ∆xm → 0 as m → ∞, +converges, up to a subsequence, to a solution uk ∈ H1(Ω) to the semidiscrete system +1 +∆t(u1 +i − u0 +i ) = div(γi∇u1 +i + u1 +i ∇pi(u1)), +1 +∆t +�3 +2uk +i − 2uk−1 +i ++ 1 +2uk−2 +i +� += div(γi∇uk +i + (uk +i )+∇pi(uk)) +in Ω, +with no-flux boundary conditions ∇uk +i · ν = 0 on ∂Ω, i = 1, . . . , n. Second, we prove that +a subsequence of the sequence of semidiscrete solutions converges to a weak solution to +(1)–(2) as ∆t → 0. Both steps may be summarized as follows (the precise convergence +statements can be found in Propositions 9 and 11). +Theorem 5 (Convergence of the scheme). Let Hypotheses (H1)–(H3) hold and let (Tm)m∈N +be a sequence of admissible discretizations of Ω satisfying (13) uniformly in m and ∆xm → +0, ∆tm → 0 as m → ∞. Then the solution (um) to (15)–(19), constructed in Theorem 3, +converges, up to a subsequence, as m → ∞ to a function u = (u1, . . . , un) satisfying ui ≥ 0 +in ΩT for i = 1, . . . , n, ui ∈ L2(0, T; H1(Ω)), ∂tui ∈ L2d+4(0, T; W 1,2d+4(Ω)′), and u is a +weak solution to (1)–(2). + +BDF2 FINITE-VOLUME SCHEME +11 +The proof is based on suitable estimates uniform with respect to ∆xm and ∆tm, derived +from the entropy inequality (21). For the limit ∆xm → 0, we follow the strategy of [10]. +The compactness argument is different, since we still keep the time discretization. The limit +∆tm → 0 is based on a higher-order integrability property derived from the Gagliardo– +Nirenberg inequality and on the Aubin–Lions compactness lemma in the version of [16]. +We need the condition γ > 0 since the application of the discrete Gagliardo–Nirenberg +inequality requires discrete gradient bounds. However, the term involving pi(u) only pro- +vides a bound for the discrete kinetic energy � +σ∈Eint τσ(uk +i,σ)+|DK,σpi(uk)|2, from which we +are unable to conclude gradient bounds. For the Euler scheme, this issue can be overcome +by using the Boltzmann entropy inequality, which provides bounds in L2(0, T; H1(Ω)) and +L∞(0, T; L1(Ω)) (see (3)), and consequently in L2+2/d(ΩT), which is the required higher- +order integrability bound. As mentioned in the introduction, this entropy is not compatible +with the BDF2 discretization. Therefore, the restriction γ > 0 seems to be unavoidable +with our approach. +Finally, we verify that the convergence of the semidiscrete system is of second order. +Theorem 6 (Second-order convergence). Let uk be a solution to (31) and assume that the +solution to (1)–(2) satisfies u ∈ C3([0, T]; L2(Ω)) ∩ L∞(0, T; W 1,∞(Ω)). Furthermore, let +ε > 0 be arbitrary and assume that +∆t < +4(3 − +√ +8)γλm +λ3 +M∥∇u∥2 +L∞(ΩT ) + 4γλmε. +Then there exists C(ε) > 0, which is of order ε−1/2 as ε → 0 but independent of ∆t, such +that +max +k=1,...,NT ∥A1/2(uk +i − ui(tk))∥L2(Ω) ≤ C(ε)(∆t)2 +for i = 1, . . . , n. +We allow for the parameter ε > 0 to minimize the time step size constraint; however, +optimizing this constraint gives large constants C(ε). The theorem is proved by analyzing +the relative entropy H(u(tk) − uk, u(tk−1) − uk−1), using a Taylor expansion for ui up to +order (∆t)3 (which requires a bound for ∂3 +t ui), and iterating the entropy inequality once +more. The resulting recursive inequality for the relative entropy can be solved, leading to +the desired second-order bound. +3. Proof of Theorem 3 +First, we make precise inequality (7). Recall definition (8) of h(u, v) and let H(u, v) = +� +K∈T m(K)h(u, v) be the discrete Rao entropy. +Lemma 7 (BDF2 inequality). It holds for u, v, w ∈ Rn that +�3 +2u − 2v + 1 +2w +�T +Au = h(u, v) − h(v, w) + 1 +4|u − 2v + w|2 +A. + +12 +A. J¨UNGEL AND M. VETTER +In particular, for uk, uk−1, uk−2 ∈ V n +T , +n +� +i,j=1 +� +K∈T +m(K) +�3 +2uk +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� +aijuk +j,K ≥ H(uk, uk−1) − H(uk−1, uk−2). +Proof. The proof follows by a direct computation. +□ +3.1. Definition and continuity of the fixed-point operator. We assume that k ≥ 2, +since the existence of a solution u1 ∈ V n +T to the Euler scheme (17) satisfying (22) follows +from [26, Theorem 1]. Let uk−1 ∈ V n +T be given and let R > 0, δ > 0. We set +ZR = +� +w = (w1, . . . , wn) ∈ V n +T : ∥wi∥1,2,T < R for i = 1, . . . , n +� +, +and let w ∈ ZR. We consider the linear regularized problem +(24) ε +� � +σ∈EK +τσDK,σwε +i − m(K)wε +i,K +� += m(K) +∆t +�3 +2wi,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� ++ +� +σ∈EK +F+ +i,K,σ(w) +for i = 1, . . . , n, K ∈ T , where +F+ +i,K,σ(w) = −τσ +� +γDK,σwi + w+ +i,σDK,σpi(w) +� +. +The ε-regularization guarantees the coercivity of the associated bilinear form, while the +truncation w+ +i,σ is needed to obtain the nonnegativity of the entropy dissipation (see the +estimate of I6 below). +We claim that (24) has a unique solution wε ∈ V n +T . Indeed, since the mapping g(wε) = +ε(� +σ∈EK τσDK,σwε +i − m(K)wε +i,K) is linear and acting on finite-dimensional spaces, we only +need to verify its injectivity. Let wε be in the kernel of this mapping. Multiplying g(wε) = 0 +by wε +i,K, summing over K ∈ T , and using the discrete integration-by-parts formula (20) +gives +0 = +� +K∈T +� +σ∈EK +τσ(DK,σwε +i )wε +i,K − +� +K∈T +m(K)(wε +i,K)2 = −∥wε +i ∥2 +1,2,T . +This yields wε = 0 and proves the claim. +Next, we show that the fixed-point mapping F : ZR → V n +T , F(w) = wε, is continuous. +For this, we multiply (24) by −wε +i,K, sum over K ∈ T , and use discrete integration by +parts and the Cauchy–Schwarz inequality: +ε∥wε +i ∥2 +1,2,T = − 1 +∆t +� +K∈T +m(K) +�3 +2wi,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� +wε +i,K + +� +σ∈Eint +σ=K|L +F+ +i,K,σ(w)DK,σwε +i +≤ 1 +∆t +���� +3 +2wi − 2uk−1 +i ++ 1 +2uk−2 +i +���� +0,2,T +∥wε +i ∥0,2,T + γ|wi|1,2,T |wε +i |1,2,T +(25) +− +� +σ∈Eint +σ=K|L +τσ(wi,σ)+DK,σpi(w)DK,σwε +i . + +BDF2 FINITE-VOLUME SCHEME +13 +For the last term, we use the Cauchy–Schwarz inequality and the fact that any norm on +V n +T is equivalent: +− +� +σ∈Eint +σ=K|L +τσ(wi,σ)+DK,σpi(w)DK,σwε +i = − +n +� +j=1 +� +σ∈Eint +σ=K|L +τσaijw+ +i,σDK,σwiDK,σwε +i +≤ +n +� +j=1 +� � +σ∈Eint +σ=K|L +τσ|Dσwε +i |2 +�1/2� � +σ∈Eint +σ=K|L +τσa2 +ij(w+ +i,σ)2|Dσwj|2 +�1/2 +≤ C(A)∥w∥0,∞,T +n +� +j=1 +|wε +i |1,2,T |wj|1,2,T ≤ C(A, R)∥wε +i ∥1,2,T , +where we took into account the linear growth of w+ +i,σ with respect to wi,K and wi,L (see (19)) +and the definition of ZR. Inserting these estimates into (25) and dividing by ∥wε +i ∥1,2,T , it +follows that ε∥wε +i ∥1,2,T ≤ C(A, R). +This bound allows us to verify the continuity of F. Indeed, let wℓ → w as ℓ → ∞ +and set wε,ℓ = F(wℓ). Then (wε,ℓ)ℓ∈N is uniformly bounded in the discrete H1(Ω) norm. +Therefore, there exists a subsequence, which is not relabeled, such that wε,ℓ → wε as +ℓ → ∞. Passing to the limit ℓ → ∞ in scheme (24), we see that wε is a solution of +the scheme and consequently wε = F(w). Since the solution to the linear scheme (24) is +unique, the entire sequence (wε,ℓ)ℓ∈N converges to wε, which shows the continuity of F. +3.2. Existence of a fixed point. According to the Brouwer degree fixed-point theorem, +it is sufficient to show that for all (wε, ρ) ∈ ZR × [0, 1] such that wε = ρF(wε), it holds +that wε ̸∈ ∂ZR or, equivalently, ∥wε∥1,2,T < R. We claim that this is true for sufficiently +large R > 0. Indeed, let wε be such a fixed point. It satisfies +ε +� � +σ∈EK +τσDK,σwε +i − m(K)wε +i,K +� += ρ +∆t m(K) +�3 +2wε +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� ++ ρ +� +σ∈EK +F+ +i,K,σ(wε). +We multiply this equation by −(∆t)pi(wε) and sum over i = 1, . . . , n, K ∈ T . Then +0 = I1 + I2 + I3, where +I1 = −ε∆t +n +� +i,j=1 +� +K∈T +� � +σ∈EK +τσDK,σwε +i − m(K)wε +i,K +� +aijwε +j,K, +I2 = ρ +n +� +i,j=1 +� +K∈T +aij +�3 +2wε +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� +wε +j,K, + +14 +A. J¨UNGEL AND M. VETTER +I3 = −ρ∆t +n +� +i=1 +� +σ∈Eint +σ=K|L +F+ +i,K,σ(wε)DK,σpi(wε). +By discrete integration by parts, +I1 = ε∆t +n +� +i,j=1 +� � +σ∈Eint +σ=K|L +τσaijDK,σwε +i DK,σwε +j + +� +K∈T +m(K)aijwε +i,Kwε +j,K +� +≥ ελm∆t(|wε|2 +1,2,T + ∥wε∥2 +0,2,T ) = ελm∆t∥wε∥2 +1,2,T , +and by Lemma 7, +I2 ≥ H(wε, uk−1) − H(uk−1, uk−2). +For the third term, we obtain +I3 = ρ∆t +n +� +i,j=1 +� +σ∈Eint +σ=K|L +τσγaijDK,σwε +i DK,σwε +j + ρ∆t +n +� +i=1 +� +σ∈Eint +σ=K|L +τσ(wε +i,σ)+ +� +n +� +j=1 +aijDK,σwε +j +�2 +≥ γρ∆t +� +σ∈Eint +σ=K|L +τσ|A1/2DK,σwε|2 = γρ∆t|A1/2wε|2 +1,2,T . +Collecting these estimates gives +(26) +ε∆t∥wε∥2 +1,2,T + H(wε, uk−1) + γ∆tρ|A1/2wε|2 +1,2,T ≤ H(uk−1, uk−2). +Setting R = (ε∆t)−1/2H(uk−1, uk−2)1/2 + 1, we infer that ∥wε∥2 +1,2,T ≤ (R − 1)2 < R2 and +thus wε ̸∈ ∂ZR, which shows the claim. Hence, there exists a fixed point wε to F, which is +a solution to +(27) ε +� � +σ∈EK +τσDK,σwε +i −m(K)wε +i,K +� += m(K) +∆t +�3 +2wε +i,K −2ui,K + 1 +2uk−2 +i,K +� ++ +� +σ∈EK +Fi,K,σ(wε). +3.3. Limit ε → 0. The solution wε to (27) satisfies the regularized entropy inequality +(26) with ρ = 1, and the right-hand side is independent of ε and M. It follows from the +Bolzano–Weierstraß theorem that there exists a subsequence of wε, which is not relabeled, +such that wε → w as ε → 0. +In particular, ε1/2wε → 0. +Since the problem is finite +dimensional, we can pass to the limit ε → 0 in (27). Consequently, uk := w is a solution +to (17)–(19). The same limit in (26) with ρ = 1 leads to the discrete entropy inequality of +Theorem 3, which finishes the proof. +3.4. Uniqueness of solutions. Let uk, vk ∈ V n +T be two solutions to (15)–(19) with the +same initial data u0 = v0. We take the difference of the equations satisfied by uk and +vk, multiply the resulting equation by pi(uk +K) − pi(vk +K) = �n +j=1 aij(uk +j,K − vk +j,K), sum over + +BDF2 FINITE-VOLUME SCHEME +15 +i = 1, . . . , n, K ∈ T , and use discrete integration by parts. This leads to 0 = I4 + I5 + I6, +where +I4 = +3 +2∆t +n +� +i,j=1 +� +K∈T +m(K)aij(uk +i,K − vk +i,K)(uk +j,K − vk +j,K) +I5 = +n +� +i,j=1 +� +σ∈Eint +σ=K|L +τσγaijDK,σ(uk +i − vk +i )DK,σ(uk +j − vk +j ) +I6 = +n +� +i,j,ℓ=1 +� +σ∈Eint +σ=K|L +τσaij +� +(uk +i,σ)+DK,σuk +j − (vk +i,σ)+DK,σvk +j +� +aiℓDK,σ(uk +ℓ − vk +ℓ ). +By the definition of the weighted norm, I4 = (3/(2∆t))∥A1/2(uk − vk)∥2 +0,2,T . Furthermore, +I5 ≥ γ +� +σ∈Eint +σ=K|L +τσ|(A1/2DK,σ(uk − vk))|2 = γ|A1/2(uk − vk)|2 +1,2,T . +We add and subtract the term (uk +i,σ)+DK,σvk +j in I6 and apply the Cauchy–Schwarz inequal- +ity: +I6 = +n +� +i,j,ℓ=1 +� +σ∈Eint +σ=K|L +τσaijaiℓ +� +(uk +i,σ)+DK,σ(uk +j − vk +j ) ++ +� +(uk +i,σ)+ − (vk +i,σ)+� +DK,σvk +j +� +DK,σ(uk +ℓ − vk +ℓ ) += +n +� +i=1 +� +σ∈Eint +σ=K|L +τσ(uk +i,σ)+ +� +n +� +j=1 +aijDK,σ(uk +j − vk +j ) +�� +n +� +ℓ=1 +aiℓDK,σ(uk +ℓ − vk +ℓ ) +� +− +� +σ∈Eint +σ=K|L +τσ(ADK,σvk)T� +diag +� +(uk +i,σ)+ − (vk +i,σ)+� +A1/2� +(A1/2DK,σ(uk − vk)) +≥ − +� � +σ∈Eint +σ=K|L +τσ|ADK,σvk|2�� diag +� +(uk +i,σ)+ − (vk +i,σ)+� +A1/2��2 +�1/2 +× +� � +σ∈Eint +σ=K|L +τσ|A1/2DK,σ(uk − vk)|2 +�1/2 +, +where diag((uk +i,σ)+ − (vk +i,σ)+) denotes the diagonal matrix with the entries (uk +i,σ)+ − (vk +i,σ)+ +for i = 1, . . . , n. Together with +|ADK,σvk| ≤ |A1/2||A1/2DK,σvk| ≤ λ1/2 +M |DK,σvk|A +and + +16 +A. J¨UNGEL AND M. VETTER +�� diag +� +(uk +i,σ)+ − (vk +i,σ)+� +A1/2�� ≤ +�� diag +� +(uk +i,σ)+ − (vk +i,σ)+���|A1/2| +≤ λ1/2 +M +max +i=1,...,n |(uk +i,σ)+ − (vk +i,σ)+| ≤ λ1/2 +M +max +i=1,...,n |uk +i,σ − vk +i,σ|, +we find that +I6 ≥ −λM|A1/2vk|1,∞,T max +i=1,...,n +� � +σ∈Eint +σ=K|L +m(σ)dσ|uk +i,σ − vk +i,σ|2 +�1/2 +|A1/2(uk − vk)|1,2,T . +It remains to estimate the term involving the difference |uk +i,σ − vk +i,σ|. By the Lipschitz +continuity of the mean function M(uk +i,K, uk +i,L) = uk +i,σ with Lipschitz constant L > 0 and the +mesh regularity (14), +� +σ∈Eint +σ=K|L +m(σ)dσ|uk +i,σ − vk +i,σ|2 = 1 +2 +� +K∈T +� +σ∈Eint,K +m(σ)dσ|uk +i,σ − vk +i,σ|2 +≤ L2 +2 +� +K∈T +� +σ∈Eint +σ=K|L +m(σ)dσ +� +|uk +i,K − vk +i,K| + |uk +i,L − vk +i,L| +�2 +≤ 2L2 � +K∈T +� +σ∈Eint +σ=K|L +m(σ)dσ|uk +i,K − vk +i,K|2 ≤ 2dL2 +ζ +� +K∈T +m(K)|uk +i,K − vk +i,K|2 += 2dL2 +ζ +∥uk − vk∥2 +0,2,T ≤ 2dL2 +λmζ ∥A1/2(uk − vk)∥2 +0,2,T , +This shows that +I6 ≥ −λML +λ1/2 +m +�2d +ζ +�1/2 +|A1/2vk|1,∞,T ∥A1/2(uk − vk)∥0,2,T |A1/2(uk − vk)|1,2,T . +Collecting the estimates for I4, I5, and I6 and using Young’s inequality gives +3 +2∆t∥A1/2(uk − vk)∥2 +0,2,T + γ|A1/2(uk − vk)|2 +1,2,T +(28) +≤ λML +λ1/2 +m +�2d +ζ +�1/2 +|A1/2vk|1,∞,T ∥A1/2(uk − vk)∥0,2,T |A1/2(uk − vk)|1,2,T +≤ +3 +2∆t∥A1/2(uk − vk)∥2 +0,2,T + ∆t +3 +dλ2 +ML2 +λmζ |A1/2vk|2 +1,∞,T |A1/2(uk − vk)|2 +1,2,T . +Now, the inverse inequality |A1/2vk|1,∞,T ≤ C′(d)(∆x)−d/2ζ−1/2|A1/2vk|1,2,T [14, Prop. 3.10] +and condition dσ ≥ ξ∆x imply that +|A1/2vk|2 +1,∞,T ≤ C′(d)2 +(∆x)dζ +� +σ∈Eint +σ=K|L +m(σ) +dσ +|DK,σ(A1/2vk)|2 + +BDF2 FINITE-VOLUME SCHEME +17 +≤ C′(d)2 +(∆x)dζ +� +σ∈Eint +σ=K|L +m(σ)dσ +ξ2(∆x)2|DK,σ(A1/2vk)|2. +It follows from (14) and |DK,σ(A1/2vk)|2 ≤ 2(|vk +K|2 +A + |vk +L|2 +A) that +|A1/2vk|2 +1,∞,T ≤ +C′(d)2 +(∆x)d+2ξ2ζ +� +K∈T +d +ζ m(K)|DK,σ(A1/2vk)|2 +≤ +2dC′(d)2 +(∆x)d+2(ξζ)2 +� +K∈T +m(K)|A1/2vk +K|2 = C(d, ξ, ζ) +(∆x)d+2 ∥A1/2vk∥2 +0,2,T . +Using this inequality as well as the bound +(3 − +√ +8)∥A1/2vk∥2 +0,2,T ≤ 4H(v1, v0) ≤ 2(3 + +√ +8)(H(v1) + H(v0)) ≤ 4(3 + +√ +8)H(u0), +obtained from (21)–(22), we deduce from (28), for another constant C(d, ξ, ζ) that +γ|A1/2(uk − vk)|2 +1,2,T ≤ C(d, ξ, ζ)λ2 +ML2 +λm +∆t +(∆x)d+2H(u0)|A1/2(uk − vk)|2 +1,2,T . +Then our smallness condition on ∆t/(∆x)d+2 implies that |A1/2(uk − vk)|1,2,T = 0 and +consequently, uk = vk, finishing the proof. +Remark 8. The quasi-uniform condition minσ∈Eint dσ ≥ ξ∆x > 0 implies condition (23) in +[20], since the mesh regularity (13) gives minK∈T minσ∈EK d(xK, σ)/ diam(K) ≥ ζdσ/∆x ≥ +ζξ > 0. It also implies the mesh regularity condition diam(K)/d(xK, σ) ≤ ξ0 in [20, (9)], +since, because of (13) again, diam(K)/d(xK, σ) ≤ ∆x/(ζdσ) ≤ 1/(ξζ) =: ξ0. It can be +seen by considering quadratic cells that the quasi-uniform condition minσ∈Eint dσ ≥ ξ∆x > +0 generally does not imply the mesh regularity condition (13) and vice versa, so both +conditions are independent from each other. +4. Proof of Theorem 4 +We infer from mass conservation, � +K∈T m(K)uk +i,K = � +K∈T m(K)u0 +i,K = m(Ω)¯ui, that +H(uk − ¯u, uk−1 − ¯u) = H(uk, uk−1) + 1 +2 +� +K∈T +m(K) +� +|¯u|2 +A − 3(uk +K, ¯u)A + (uk−1 +K , ¯u)A +� += H(uk, uk−1) − 1 +2 m(Ω)|¯u|2 +A. +Then the entropy inequality (21) shows that +(29) +H(uk − ¯u, uk−1 − ¯u) + γ∆t|A1/2uk|2 +1,2,T ≤ H(uk−1 − ¯u, uk−2 − ¯u) +for k ≥ 2. Another iteration gives, for k ≥ 3, +H(uk − ¯u, uk−1 − ¯u) + γ∆t +� +|A1/2uk|2 +1,2,T + |A1/2uk−1|2 +1,2,T +� +≤ H(uk−2 − ¯u, uk−3 − ¯u). + +18 +A. J¨UNGEL AND M. VETTER +Hence, taking into account the discrete Poincar´e–Wirtinger inequality (23), +H(uk − ¯u, uk−1 − ¯u) + γλmζ +C2 +PλM +∆t +� +∥A1/2(uk − ¯u)∥2 +0,2,T + ∥A1/2(uk−1 − ¯u)∥2 +0,2,T +� +≤ H(uk−2 − ¯u, uk−3 − ¯u), +and the norm equivalence (10), +H(uk − ¯u, uk−1 − ¯u) + +4γλmζ∆t +(3 + +√ +8)C2 +PλM +H(uk − ¯u, uk−1 − ¯u) ≤ H(uk−2 − ¯u, uk−3 − ¯u). +This can be written as +H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−1H(uk−2 − ¯u, uk−3 − ¯u), +where κ = 4γλmζ/((3 + +√ +8)C2 +PλM). Depending on whether k is odd or even, we resolve +this iteration as follows: +H(u2ℓ+1 − ¯u, u2ℓ − ¯u) ≤ (1 + κ∆t)−ℓH(u1 − ¯u, u0 − ¯u), +H(u2ℓ+2 − ¯u, u2ℓ+1 − ¯u) ≤ (1 + κ∆t)−ℓH(u2 − ¯u, u1 − ¯u) +≤ (1 + κ∆t)−ℓH(u1 − ¯u, u0 − ¯u), +where we used (29) in the last step. As in both cases ℓ ≥ (k − 2)/2, we conclude that +(30) +H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−(k−2)/2H(u1 − ¯u, u0 − ¯u). +We want to express this inequality in terms of the ∥A1/2(·)∥0,2,T norm. We observe that, +by Young’s inequality, ∥A1/2(uk − ¯u)∥2 +0,2,T ≤ 4H(uk − ¯u, uk−1 − ¯u) and, in view of (22), +H(u1 − ¯u, u0 − ¯u) = H(u1) − H(¯u) ≤ H(u0) − H(¯u) += H(u0 − ¯u) = 1 +2∥A1/2(u0 − ¯u)∥2 +0,2,T . +Then we deduce from (30) that +∥A1/2(uk − ¯u)∥2 +0,2,T ≤ 4H(uk − ¯u, uk−1 − ¯u) ≤ 4(1 + κ∆t)−(k−2)/2H(u1 − ¯u, u0 − ¯u) +≤ 2(1 + κ∆t)−(k−2)/2∥A1/2(u0 − ¯u)∥2 +0,2,T , +which concludes the proof. +5. Proof of Theorem 5 +We split the proof into two parts. We first prove the convergence in the space variable +and then the convergence in the time variable. An alternative is to show the convergence +in both variables simultaneously; see, e.g., [27]. + +BDF2 FINITE-VOLUME SCHEME +19 +5.1. Convergence in space. We show the following result for ∆x → 0. +Proposition 9 (Convergence in space). Let the assumptions of Theorem 5 hold and let +(uk +m) be the sequence of solutions to (15)–(19) constructed in Theorem 3 associated to an +admissible mesh Tm with mesh size ∆xm for m ∈ N satisfying ∆xm → 0 as m → ∞. Then +there exists a subsequence which is not relabeled such that uk +i,m → uk +i strongly in L2(Ω) as +m → ∞ and uk +i solves for all φi ∈ W 1,max{2,d}(Ω), i = 1, . . . , n, +(31) +1 +∆t +� +Ω +�3 +2uk +i − 2uk−1 +i ++ 1 +2uk−2 +i +� +φidx + +� +Ω +� +γ∇uk +i + (uk +i )+∇pi(uk) +� +· ∇φidx = 0. +Proof. For fixed ∆t, the discrete entropy inequality in Theorem 3 provides a uniform +bound for ∥uk +m∥1,2,Tm. Then, by the discrete Rellich–Kondrachov compactness theorem [21, +Lemma 5.6], there exists a subsequence of (uk +m) = (uk +1,m, . . . , uk +n,m), which is not relabeled, +such that uk +m → uk strongly in L2(Ω) as m → ∞. Moreover, the sequence of discrete +gradients (∇muk +m) converges weakly in L2(Ω) to some function which can be identified by +∇uk; see [10, Lemma 4.4]. Let φi ∈ C2(Ω) and set φi,K := φi(xK) for K ∈ T . Then the +limit ∆xm → 0 in the BDF2 approximation becomes +1 +∆t +� +K∈T +m(K) +�3 +2uk +i,K − 2uk−1 +i,K + 1 +2uk−2 +i,K +� +φi,K → 1 +∆t +� +Ω +�3 +2uk +i − 2uk−1 +i ++ 1 +2uk +i +� +φidx. +Next, we set F m = F m +1 + F m +2 + F m +3 , where +F m +1 = −γ +� +K∈T +� +σ∈EK +τσDK,σuk +i,mφi,K, +F m +2 = − +� +K∈T +� +σ∈EK +τσ(uk +i,m,K)+DK,σpi(uk +m)φi,K, +F m +3 = − +� +K∈T +� +σ∈EK +τσ +� +(uk +i,m,σ)+ − (uk +i,m,K)+� +DK,σpi(uk +m)φi,K. +We introduce the intermediate integral F m +0 = F m +01 + F m +02, where +F m +01 = γ +� +Ω +∇muk +m,i · ∇φidx, +F m +02 = +� +Ω +(uk +i,m)+∇mpi(uk +m) · ∇φidx. +It follows from the weak convergence of the discrete gradients and the strong convergence +in L2(Ω) that F m +0 → F as m → ∞, where +F = γ +� +Ω +∇uk +i · ∇φidx + +� +Ω +(uk +i )+∇pi(uk) · ∇φidx. +Thus, if we can show that F m +0 − F m → 0, then |F m − F| ≤ |F m − F m +0 | + |F m +0 − F| → 0, +proving the claim. +By discrete integration by parts and the definition of the discrete gradient, +F m +1 = γ +� +σ∈Eint +σ=K|L +τσDK,σuk +i,mDK,σφi, + +20 +A. J¨UNGEL AND M. VETTER +F m +01 = γ +� +σ∈Eint +σ=K|L +m(σ) +m(TK,σ)DK,σuk +i,m +� +TK,σ +∇φi · νK,σdx. +Using the Taylor expansion (here we need φi ∈ C2(Ω)) +DK,σφi +dσ += φi,L − φi,K +d(xK, xL) = ∇φi · νK,σ + O(∆xm) +for σ = K|L, +where we have taken into account the property xK − xL = d(xK, xL)νK,σ, we obtain +|F m +01 − F m +1 | ≤ γ +� +σ∈Eint +σ=K|L +m(σ)|DK,σuk +i,m| +���� +1 +m(TK,σ) +� +TK,σ +∇φi · νK,σdx − DK,σφi +dσ +���� +≤ Cγ∆xm +� +σ∈Eint +m(σ)|Dσuk +i,m|, +where C > 0 depends on the L∞ norm of D2φi. We apply the Cauchy–Schwarz inequality +and use the mesh property (14) to find that +|F m +01 − F m +1 | ≤ Cγ∆xm +� � +σ∈Eint +m(σ) +dσ +|Dσuk +i,m|2 +�1/2� � +σ∈Eint +m(σ)dσ +�1/2 +≤ Cγ∆xm|uk +i,m|1,2,Tm +�d +ζ m(Ω) +�1/2 +→ 0 +as m → ∞. +Similar arguments lead to +|F m +02 − F m +2 | ≤ C∆xm +� +K∈Tm +� +σ∈Eint,K +m(σ)(uk +i,m,K)+|DK,σpi(uk +m)| +≤ C∆xm +� � +K∈Tm +|(uk +i,m,K)+|2 +� +σ∈Eint,K +m(σ)dσ +�1/2 +|pi(uk +m)|1,2,Tm +≤ C∆xm +�d +ζ +� +K∈Tm +m(K)|(uk +i,m,K)+|2 +�1/2 +|pi(uk +m)|1,2,Tm +≤ C(ζ)∆xm∥uk +i,m∥0,2,Tm|pi(uk +m)|1,2,Tm. +The right-hand side converges to zero since +|pi(uk +m)|2 +1,2,Tm = +� +σ∈Eint +σ=K|L +τσ +� +n +� +j=1 +aijDK,σuk +j,m +�2 +≤ C(A)|uk +m|2 +1,2,Tm ≤ C. +Finally, using |DK,σφi| ≤ C(φi)∆xm and property (ii) of the mean function, +|F m +3 | ≤ +� +σ∈Eint +σ=K|L +τσ|uk +i,m,σ − uk +i,m,K||DK,σpi(uk +m)||DK,σφi| + +BDF2 FINITE-VOLUME SCHEME +21 +≤ C(φi)∆xm +� +σ∈Eint +σ=K|L +τσ|Dσuk +i,m||DK,σpi(uk +m)| +≤ C(φi, A)∆xm +� � +σ∈E +τσ|Dσuk +i,m|2 +�1/2� +n +� +j=1 +� +σ∈E +τσ|Dσuk +j,m|2 +�1/2 +→ 0. +This shows that F m +0 − F → 0 as m → ∞, concluding the proof. +□ +5.2. Convergence in time. We wish to perform the limit ∆t → 0 in (31). For this, we +need an estimate in a better space than L2(ΩT), provided by the following lemma. +Lemma 10 (Higher-order integrability). Let (u(τ)) be a family of solutions to (31) associ- +ated to the time step size τ := ∆t, constructed in Proposition 9. Then there exists C > 0 +independent of τ such that +∥u(τ)∥Lp(ΩT ) ≤ C +for p = 2 + 4/d. +Proof. The lemma follows from the discrete entropy inequalities (21)–(22) and the Gagliar- +do–Nirenberg inequality. Indeed, we infer from the entropy inequalities after summation +over k = 2, . . . , NT that +∥u(τ)∥L∞(0,T;L2(Ω)) + ∥u(τ)∥L2(0,T;H1(Ω)) ≤ C. +Then it follows from the Gagliardo–Nirenberg inequality with θ = d/2 − d/p that +∥u(τ)∥p +Lp(0,T;Lp(Ω)) ≤ C +� T +0 +∥u(τ)∥pθ +H1(Ω)∥u(τ)∥p(1−θ) +L2(Ω) dt +≤ C∥u(τ)∥p(1−θ) +L∞(0,T;L2(Ω)) +� T +0 +∥u(τ)∥2 +H1(Ω)dt ≤ C, +since pθ = 2. This finishes the proof. +□ +Proposition 11 (Convergence in time). Let (u(τ)) be a family of solutions to (31) with +τ = ∆t. Then u(τ) converges to a weak solution u to (1)–(2) satisfying +ui ∈ L2(0, T; H1(Ω)) ∩ L∞(0, T; L2(Ω)), +∂tui ∈ Lr(0, T; W 1,2d+4(Ω)′), +where r = (2d + 4)/(2d + 3) > 1. +Proof. We estimate the discrete time derivative Dτu(τ) +i (t) := +3 +2uk +i − 2uk−1 +i ++ 1 +2uk−2 +i +for +t ∈ [kτ, (k + 1)τ) for k ≥ 2. Let φi ∈ L2d+4(0, T; W 1,2d+4(Ω)). Then +1 +τ +� T +2τ +��⟨Dτu(τ) +i , φi⟩W 1,d+2(Ω)′ +��rdt +≤ γrC +� T +2τ +� +Ω +|∇u(τ) +i +· ∇φi|rdxdt + C +� T +2τ +� +Ω +|(u(τ) +i )+∇pi(u(τ)) · ∇φi|rdxdt +≤ γrC∥∇u(τ) +i ∥r +L2(ΩT )∥∇φi∥r +L2d+4(ΩT ) ++ C∥u(τ) +i ∥r +L(2d+4)/d(ΩT )∥∇pi(u(τ))∥r +L2(ΩT )∥∇φi∥r +L2d+4(ΩT ) + +22 +A. J¨UNGEL AND M. VETTER +≤ C∥φi∥r +L2d+4(0,T;W 1,2d+4(Ω)), +where we used the fact that pi(u(τ)) is a linear combination of all u(τ) +j +for j = 1, . . . , n. +This implies the bound τ −1∥Dτu(τ) +i ∥Lr(2τ,T;W 1,2d+4(Ω)′) ≤ C. +Let πτu(τ)(t) = u(τ)(t − τ) be a shift operator. We relate the implicit Euler scheme and +the BDF2 scheme by +uk +i − uk−1 +i += 2 +3 +�3 +2uk +i − 2uk−1 +i ++ 1 +2uk−2 +i +� ++ 1 +3(uk−1 +i +− uk−2 +i +). +Then +∥u(τ) − πτu(τ)∥Lr(2τ,T;W 1,2d+4(Ω)′) = +���2 +3Dτu(τ) + 1 +3πτ(u(τ) − πτu(τ)) +��� +Lr(2τ,T;W 1,2d+4(Ω)′) +≤ 2 +3∥Dτu(τ)∥Lr(2τ,T;W 1,2d+4(Ω)′) + 1 +3∥u(τ) − πτu(τ)∥Lr(τ,T−τ;W 1,2d+4(Ω)′). +Adding ∥u(τ) − πτu(τ)∥Lr(2τ,T;W 1,2d+4(Ω)′) ≤ C1 from the first Euler step (proved in a similar +way as above) to the left-hand side and absorbing the last term on the right-hand side by +the left-hand side, we find that +2 +3τ ∥u(τ) − πτu(τ)∥Lr(2τ,T;W 1,2d+4(Ω)′) ≤ 2 +3τ ∥Dτu(τ)∥Lr(2τ,T;W 1,2d+4(Ω)′) ≤ C. +Together with the uniform L2(0, T; H1(Ω)) bound for u(τ), we can apply the Aubin–Lions +compactness lemma in the version of [16] to conclude that, up to a subsequence, as τ → 0, +u(τ) → u +strongly in L2(ΩT). +In view of the higher-order estimate of Lemma 10, this convergence also holds in Lq(ΩT) +for all q < 2 + 4/d. Furthermore, again up to a subsequence, +Dτu(τ) ⇀ ∂tu +weakly in Lr(2τ, T; W 1,2d+4(Ω)′). +These convergences are sufficient to pass to the limit τ → 0 in (31) for test functions +φi ∈ L2d+4(2τ, T; W 1,2d+4(Ω)′). +□ +6. Second-order convergence +As in the previous section, we set D∆tuk +i = 3 +2uk +i − 2uk−1 +i ++ 1 +2uk−2 +i +and write (31) as +(32) +1 +∆t +� +Ω +D∆tuk +i φidx + +� +Ω +� +γ∇uk +i + (uk +i )+∇pi(uk) +� +· ∇φidx = 0. +A Taylor expansion shows that, for some ξk ∈ (0, T), +D∆tui(tk) := 3 +2ui(tk) − 2ui(tk−1) + 1 +2ui(tk−2) = (∆t)∂tui(tk) − (∆t)3 +3 +∂3 +t ui(ξk). +Then, using a test function φi ∈ H1(Ω) in (1), +(33) +1 +∆t +� +Ω +D∆tui(tk)φidx + +� +Ω +(γ∇ui + ui∇pi(u))(tk) · ∇φidx = (∆t)2 +3 +� +Ω +∂3 +t ui(ξk)φidx. + +BDF2 FINITE-VOLUME SCHEME +23 +We take the difference of (32) and (33), choose the test function φi = pi(u(tk)) − pi(uk) = +(A(u(tk) − uk))i, and sum over i = 1, . . . , n: +1 +∆t +� +Ω +D∆t(u(tk) − uk)TA(u(tk) − uk)dx = I7 + I8, +where, +(34) +I7 = − +n +� +i=1 +� +Ω +� +γ∇(ui(tk) − uk +i ) + ui(tk)∇pi(u(tk)) − (uk +i )+∇pi(uk) +� +×∇(A(u(tk) − uk))idx, +I8 = (∆t)2 +3 +n +� +i=1 +� +Ω +∂3 +t ui(ξk)(A(u(tk) − uk))idx. +Set vk := u(tk) − uk +i . It follows from the BDF2 inequality in Lemma 7, applied to the +left-hand side, that +1 +∆t +� +Ω +D∆t(u(tk) − uk)TA(u(tk) − uk)dx ≥ 1 +∆t +� +H(vk, vk−1) − H(vk−1, vk−2) +� +. +For the terms I7 and I8, we use the definition pi(uk) = (Auk)i, the Lipschitz continuity of +z �→ z+, the nonnegativity of ui, and Young’s inequality: +I7 = − +n +� +i=1 +� +Ω +γ∇(A1/2vk)i · ∇(A1/2vk)idx +− +n +� +i=1 +� +Ω +� +(ui(tk) − (uk +i )+)∇(Au(tk))i + (uk +i )+∇(A(u(tk) − uk))i +� +· ∇(Avk)idx +≤ −γ∥∇(A1/2vk)∥2 +L2(Ω) + λ−1/2 +m +∥A1/2vk∥L2(Ω)λ3/2 +M ∥∇u(tk)∥L∞(Ω)∥∇(A1/2vk)∥L2(Ω) +− +n +� +i=1 +� +Ω +(uk +i )+|∇(Avk)i|2dx ≤ +λ3 +M +4γλm +∥∇u∥2 +L∞(ΩT )∥A1/2vk∥2 +L2(Ω) +and +I8 ≤ (∆t)2 +3λ1/2 +m +∥∂3 +t u∥L∞(0,T;L2(Ω))∥A1/2vk∥L2(Ω). +Summarizing, we obtain from (34) +H(vk, vk−1) − H(vk−1, vk−2) ≤ C1∆t∥A1/2vk∥2 +L2(Ω) + C2(∆t)3∥A1/2vk∥L2(Ω), +(35) +where +C1 = +λ3 +M +4γλm +∥∇u∥2 +L∞(ΩT ), +C2 = +1 +3λ1/2 +m +∥∂3 +t u∥L∞(0,T;L2(Ω)). +We iterate this inequality once more and use the inequality a + b ≤ +� +2(a2 + b2) as well as +the norm equivalence (11): +H(vk, vk−1) − H(vk−2, vk−3) ≤ C1∆t +� +∥A1/2vk∥2 +L2(Ω) + ∥A1/2vk−1∥2 +L2(Ω) +� ++ C2(∆t)3� +∥A1/2vk∥L2(Ω) + ∥A1/2vk−1∥L2(Ω) +� +≤ C1∆t +� +∥A1/2vk∥2 +L2(Ω) + ∥A1/2vk−1∥2 +L2(Ω) +� + +24 +A. J¨UNGEL AND M. VETTER ++ +√ +2C2(∆t)3� +∥A1/2vk∥2 +L2(Ω) + ∥A1/2vk−1∥2 +L2(Ω) +�1/2 +≤ 4C1∆t +3 − +√ +8H(vk, vk−1) + 4 +√ +2C2(∆t)3 +3 − +√ +8 +H(vk, vk−1)1/2. +We apply Young’s inequality for ε > 0: +� +1 − 4(C1 + ε) +3 − +√ +8 ∆t +� +H(vk, vk−1) ≤ H(vk−2, vk−3) + 2C2 +2(∆t)5 +(3 − +√ +8)ε, +and assume that ∆t < (3 − +√ +8)/(4(C1 + ε)). This recursion is of the form ak ≤ bak−2 + +bc(∆t)5, where ak = H(vk, vk−1) and +b = +� +1 − 4(C1 + ε) +3 − +√ +8 ∆t +�−1 +, +c = 2C2 +2(∆t)5 +(3 − +√ +8)ε, +and it can be resolved explicitly depending on whether k is odd or even: +a2ℓ+1 ≤ bℓa1 + c(∆t)5 +ℓ−1 +� +j=0 +bj, +a2ℓ+2 ≤ bℓa2 + c(∆t)5 +ℓ−1 +� +j=0 +bj. +The sum can be estimated according to +ℓ−1 +� +j=0 +bj = bℓ − 1 +b − 1 ≤ +� +1 − +4∆t +3 − +√ +8(C1 + ε) +�−ℓ+1 +3 − +√ +8 +4∆t(C1 + ε). +Since ℓ = tℓ/∆t ≤ T/∆t, the bracket approximates the exponential function and can be +bounded by a constant depending only on C1 + ε and T. This shows that there exist +constants K1, K2 > 0 such that +H(v2ℓ+1, v2ℓ) ≤ K1(C1, ε, T)H(v1, v0) + K2(C1, C2, ε−1, T)(∆t)4, +H(v2ℓ+2, v2ℓ+1) ≤ K1(C1, ε, T)H(v2, v1) + K2(C1, C2, ε−1, T)(∆t)4. +Going back to inequality (35) for k = 2, we can argue in a similar way as before that +H(v2, v1) is bounded by K3H(v1, v0) + K4(∆t)5 for some constants K3, K4 > 0, which are +independent of ∆t. Furthermore, since v0 = 0, we have H(v1, v0) = (5/4)∥A1/2(u(t1) − +u1)∥L2(Ω) ≤ K5(∆t)4 for some K5 > 0 independent of ∆t. This shows that H(vk, vk−1) ≤ +K6(∆t)4, where K6 depends on C1, C2, ε−1, and T. Taking the square root and using (11) +shows the result. +7. Numerical examples +The finite-volume scheme (15)–(19) is implemented in Matlab, using the mobility M(u, v) += 1 +2(u + v). As the numerical scheme is implicit, we have solved the nonlinear system of +equations at each time step by using the Matlab routine fsolve, based on Newton’s method +with trust regions. The optimality tolerance was chosen as 10−14. + +BDF2 FINITE-VOLUME SCHEME +25 +7.1. First example: one-dimensional domain, three species. We choose the domain +Ω = (0, 1), the parameter γ = 1/2, as well as the positive definite matrix A and the initial +data u0 according to +A = +� +� +2 +1 +1/2 +1 +3 +3/2 +1/2 +3/2 +1 +� +� , +u0(x) = +� +� +cos(πx) + 2 +2 − cos(2πx) +2 +� +� . +The numerical parameters are ∆x = 1/12 800 and ∆t = 1/128. The numerical solution is +illustrated in Figure 1 at various times. All components converge to the constant steady +state ¯u = 2. Interestingly, although initially equal to the steady state, the density u3 +becomes nonconstant for positive times before it tends to the constant steady state for +large times. +Such a phenomenon is sometimes called uphill diffusion, which typically +appears in thermodynamic multicomponent systems due to cross diffusion [28]. +Figure 1. Densities u1(t) (darker blue line), u2(t) (lighter green line), u3(t) +(dashed black line) at times t = 0, 0.01, 0.1 (from left to right) versus space. +7.2. Second example: two-dimensional domain, two species. We take Ω = (0, 1)2, +∆x = +√ +2 · 2−5 ≈ 0.0044, ∆t = 1/256, γ = 1/2, and +A = +� +1 +1/2 +1/2 +1 +� +, +u0(x) = +� +1(0,1/2)2(x) +1(1/2,1)2(x) +� +. +Figure 2 shows the evolution of u = (u1, u2) at various times. Although being discontinuous +and segregated initially, the solution becomes smooth and mixes the densities for positive +times. This is not surprising, as full segregation (i.e., the supports of u1 and u2 do not +intersect) is expected only when γ = 0 and det A = 0. The numerical scheme preserved +the nonnegativity in all our experiments, even for the initial data of this example. The +numerical solutions are the same with or without the cutoff used in (18). +7.3. Third example: exponential time decay. We choose the one-dimensional domain +Ω = (0, 1), γ = 0.1, ∆x = 2−7, ∆t = (10 · 27)−1, and +A = +� +β +2 +2 +1 +� +, +u0(x) = +� +2 − cos(πx) +2 + cos(πx) +� +, + +3 +2.5 +2 +1.5 +0.2 +0.4 +0.6 +0.8 +0 +13 +2.5 +2 +1.5 +0.2 +0.4 +0.6 +0.8 +0 +13 +2.5 +2 +1.5 +0.2 +0.4 +0.6 +0.8 +0 +126 +A. J¨UNGEL AND M. VETTER +Figure 2. Density u1(t) (upper row) and u2(t) (lower row) at times t = +0, 0.02, 0.2 (from left to right) versus space. +where β > 4. The distance ∥A1/2(uk−¯u)∥L2(Ω) presented in Figure 3 for β = 5 and β = 4.01 +shows that the time decay behaves exponentially, as predicted by Theorem 4. The decay +rates (excluding the initial decay) are −4.37 for β = 5 and −1.03 for β = 4.01, and they +decrease for smaller values of det A. We have also observed an exponential decay when +γ = 0 with smaller decay rates. +Figure 3. Semilogarithmic plot for ∥A1/2(uk − ¯u)∥L2(Ω) versus time tk. +7.4. Fourth example: Convergence rate in time. We choose the values for A and u0 as +in the previous example as well as γ = 0, ∆x = 2−9, and ∆t = (10·2p)−1 with p = 1, . . . , 8. + +0 +-Logarithm of distance +-1 +- Reference line +-2 +-3 +-4 +-5 +-6 +0.2 +0.4 +0.6 +0.8 +1.2 +0 +1 +Time0 +- Logarithm of distance +-Reference line +-1 +-2 +-3 +-4 +-5 +-6 +0.2 +0.4 +0.6 +0.8 +1.2 +0 +1 +Time0.5 +0.5 +0.50.5 +0 +0.5 +0.5 +00.5 +0 +0.5 +1 +0.5 +00.5 +0 +0.5 +0.5 +00.5 +0 +0.5 +0.5 +O +00.5 +0 +0.5 +7 +0.5 +0BDF2 FINITE-VOLUME SCHEME +27 +The reference solution uref is computed with the time step size ∆t = (10 · 29)−1. +As +expected, the convergence rate at time T = 0.02, shown in Figure 4 for two different values +of β, is about two, even in the case det A = 0. +Figure 4. Discrete L2(Ω) error ∥A1/2(u(∆t)−uref)(T)∥L2(Ω) versus time step +size ∆t = (10 · 2p)−1 for p = 1, . . . , 8 for β = 5 (left) and β = 4 (right). +References +[1] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. +In: H. J. 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Biol. 21 (1982), +24–43. + +BDF2 FINITE-VOLUME SCHEME +29 +Institute of Analysis and Scientific Computing, Technische Universit¨at Wien, Wiedner +Hauptstraße 8–10, 1040 Wien, Austria +Email address: juengel@tuwien.ac.at +Institute of Analysis and Scientific Computing, Technische Universit¨at Wien, Wiedner +Hauptstraße 8–10, 1040 Wien, Austria +Email address: martin.vetter@tuwien.ac.at + diff --git a/dNE1T4oBgHgl3EQfdwTr/content/tmp_files/load_file.txt b/dNE1T4oBgHgl3EQfdwTr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1102a67b34ae4ae0afc7a62347293946be5dd6fa --- /dev/null +++ b/dNE1T4oBgHgl3EQfdwTr/content/tmp_files/load_file.txt @@ -0,0 +1,1048 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf,len=1047 +page_content='A CONVERGENT ENTROPY-DISSIPATING BDF2 FINITE-VOLUME SCHEME FOR A POPULATION CROSS-DIFFUSION SYSTEM ANSGAR J¨UNGEL AND MARTIN VETTER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A second-order backward differentiation formula (BDF2) finite-volume dis- cretization for a nonlinear cross-diffusion system arising in population dynamics is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical scheme preserves the Rao entropy structure and conserves the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suit- able a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The novelty is the extension of this inequality to the system case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Some numerical experiments in one and two space dimensions underline the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Introduction The design of structure-preserving finite-volume schemes for parabolic equations is fun- damental to describe accurately the behavior of the numerical solutions to these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In the literature, usually implicit Euler time discretization are used to derive such schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', [2, 3, 6, 9, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' However, implicit Euler schemes are only first order accurate in time, while finite-volume implementations often show second-order accuracy in space [9, 27] (also see [17] for an analytical result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In order to match the convergence rates in space and time, there is the need to design second-order time approximations, which lead to structure- preserving and convergent schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Some works suggest higher-order time discretizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' [8, 15, 19, 24, 29]), but they are only concerned with semidiscrete equations or dif- ferent numerical methods, or they do not contain any numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In this paper, we propose a second-order BDF two-point flux approximation finite-volume scheme, which conserves the mass and dissipates the Rao entropy, for a nonlinear cross-diffusion system arising in population dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The quadratic structure of the Rao entropy allows us to extend the G-stability theory of Dahlquist to the system case, leading to existence, uniqueness, and convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 65L06, 65M08, 65M12, 35Q92, 92D25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Cross-diffusion equations, Rao entropy, discrete entropy dissipation, linear multistep method, finite-volume method, population dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P33010 and F65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, ERC Advanced Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 101018153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='03200v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='NA] 9 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER The dynamics of the population density ui(x, t) of the ith species is modeled by the cross-diffusion equation (1) ∂tui = div(γ∇ui + ui∇pi(u)), pi(u) := n � j=1 aijuj in Ω, t > 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, where Ω ⊂ Rd (d ≥ 1) is a bounded domain and u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This model was derived rigorously from a moderately interacting stochastic particle system in a mean-field-type limit [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The parameter γ > 0 is related to the stochastic diffusion of the particle system, and aij ∈ R describes the strength of the repulsive or attractive interaction between the ith and the jth species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We impose initial and no-flux boundary conditions, (2) ui(0) = u0 i in Ω, ∇ui · ν = 0 on ∂Ω, t > 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, where ν is the exterior unit normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In the absence of the diffusion parameter γ, (1) can be interpreted as a mass conservation equation with the partial velocity ∇pi(u), which is determined according to Darcy’s law by the partial pressure pi(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' System (1) in one space dimension for two species, γ = 0, and det(aij) = 0 was first studied in [4], proving the global existence of segregated solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', the supports of u1 and u2 do not intersect for all times if this holds true initially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This result was generalized to arbitrary space dimensions in [5], still for two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For an arbitrary number of species, the existence of global weak solutions to (1)–(2) was shown in [25, Appendix B] if det(aij) > 0 and the existence of local strong solutions was proved in [18] if det(aij) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The matrix A = (aij) ∈ Rn×n does not need to be symmetric nor positive definite so that the diffusion matrix associated to (1) is generally neither symmetric nor positive definite too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A minimal requirement for local solvability at the linear level is the parabolicity in the sense of Petrovskii, which is satisfied if all eigenvalues of A have a positive real part [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Global solvability is guaranteed under the detailed-balance condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', there exist π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , πn > 0 such that πiaij = πjaji for all i ̸= j [25, Theorem 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This condition also appears in the theory of time-continuous Markov chains generated by A, and (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , πn) is the associated invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We assume this condition throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It implies that �ui := πiui solves the system ∂t�ui = div � �ui n � j=1 aij πj ∇�uj � , with a symmetric positive definite matrix (aij/πj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Consequently, we may assume, without loss of generality, that the matrix A in (1) is already symmetric and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Due to the nonlinear cross-diffusion structure, the analysis of (1) is highly nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The key idea of the analysis is to exploit the entropy structure of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This means that there exist Lyapunov functionals, called entropies, that are nonincreasing in time along solutions to (1)–(2) and that provide gradient estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In the present situation, these functionals are given by the Boltzmann (or Shannon) entropy HB and the Rao entropy BDF2 FINITE-VOLUME SCHEME 3 HR, HB(u) = n � i=1 � Ω ui(log ui − 1)dx, HR(u) = 1 2 � Ω uTAudx, giving formally the entropy equalities dHB dt + � Ω � 4γ n � i=1 |∇√ui|2 + n � i,j=1 aij∇ui · ∇uj � dx = 0, (3) dHR dt + � Ω � γ n � i,j=1 aij∇ui · ∇uj + n � i=1 ui|∇pi(u)|2 � dx = 0, (4) and thus providing gradient bounds for ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The Boltzmann entropy is related to the thermodynamic entropy of the system, while the Rao entropy measures the functional diversity of the species [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Since the Boltzmann entropy HR is convex, the implicit Euler scheme preserves the entropy inequality (3) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', [27] for a related system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The logarithmic structure of HR seems to prevent entropy stability in higher-order schemes like BDF or Crank–Nicolson approximations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' However, thanks to the quadratic structure of the Rao entropy HR, we are able to prove stability of HR for the BFD2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' To explain the idea, let T be a triangulation of Ω into control volumes K ⊂ Ω with measure m(K) and let ∆t be the time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, let uk i,K be an approximation of ui(xK, tk), where xK ∈ K and tk = k∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We write the BDF2 discretization of (1) as (5) m(K) ∆t �3 2uk i,K − 2uk i,K + 1 2uk−2 i,K � + � σ∈EK Fk i,K,σ = 0, where EK is the set of the edges (or faces) of K and Fk i,K,σ is the numerical flux, defined in (18) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The usual idea to derive a priori bounds is to choose the test function uk i,K in (5) and to use the inequality (6) �3 2uk i,K − 2uk i,K + 1 2uk−2 i,K � uk i,K ≥ h0(uk i,K, uk−1 i,K ) − h0(uk−1 i,K , uk−2 i,K ), where h0(a, b) = 1 4 � 5a2 − 4ab + b2� = 1 4 � a b �T � 5 −2 −2 1 � � a b � , a, b ∈ R, is a positive definite quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Assuming that Fk i,K,σuk i,K can be bounded from below, this gives a priori bounds for (uk i,K)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Inequality (6) can be explained in the framework of Dahlquist’s G-stability theory [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In our case, we need the test function pi(uk K) to derive the discrete analog of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then the question is whether there exists a functional h(u, v) such that (7) n � i=1 �3 2uk i,K − 2uk i,K + 1 2uk−2 i,K � pi(uk K) ≥ h(uk K, uk−1 K ) − h(uk−1 K , uk−2 K ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER Note that we need to sum over all species in this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The main novelty of this paper is the observation that the scalar inequality (6) can be extended to inequality (7) for vectors u, v ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, we show in Lemma 7 that (7) holds for (8) h(u, v) = 1 4(5uTAu − 4uTAv + vTAv) = 1 4 � u v �T � 5A −2A −2A A � � u v � with u, v ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Introducing the discrete Rao entropy by H(u, v) = � K∈T m(K)h(u, v) for piecewise constant functions u and v, this yields the BDF2 analog of the Rao entropy inequality H(uk, uk−1) + c∆t|uk|2 1,2,T ≤ H(uk−1, uk−2) for k ≥ 2, where |·|1,2,T is the discrete H1(Ω) norm, defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='3, and c > 0 depends on the smallest eigenvalue of A and on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This inequality is the key for proving our main results: Existence and uniqueness of discrete solutions: There exists a solution uk i to the BDF2 finite-volume scheme (5), which conserves the mass � K∈T m(K)uk i,K of the ith species and dissipates the discrete Rao entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Moreover, the solution is unique if ∆t/(∆x)d+2 is sufficiently small, where ∆x is the size of the mesh (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This unusual quotient comes from an inverse inequality needed to bound higher- order norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Large-time behavior: The discrete solution uk i converges for large times k → ∞ to the constant steady state ¯ui = m(Ω)−1 � Ω u0 i dx with a quasi-explicit exponential rate (Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The proof uses the well-established relative entropy (or energy) method, but the two-step scheme requires an iteration of this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Convergence of the discrete scheme: The fully discrete solution converges to a solu- tion to the semidiscrete problem if ∆x → 0, and the semidiscrete solution converges to a weak (nonnegative) solution to (1)–(2) as ∆t → 0 (up to subsequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' see Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Convergence rate: If the solution to (1)–(2) is sufficiently smooth, the semidiscrete solution converges with order two, as expected for the BDF2 scheme (Theorem 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical scheme and our main results are detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In Section 3, we prove the existence and uniqueness of a discrete solution, while its large-time behavior is analyzed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Section 5 is devoted to the convergence of the full scheme, and the second-order convergence in time is verified in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finally, we present in Section 7 some numerical examples in one and two space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Numerical scheme and main results We need some simple auxiliary results and some notation before formulating the numer- ical scheme and the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Some linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We denote by | · | the Euclidean norm on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Given a sym- metric positive matrix A ∈ Rn×n, we introduce the weighted norm |u|2 A := uTAu and the BDF2 FINITE-VOLUME SCHEME 5 weighted inner product (u, v)A := uTAv for u, v ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' With this notation, the discrete Rao entropy density can be written as (9) h(u, v) = 1 4(5|u|2 A − 4(u, v)A + |v|2 A) for u, v ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Denoting by λm > 0 the smallest and by λM > 0 the largest eigenvalue of A, it holds that (10) λm|u|2 ≤ |u|2 A ≤ λM|u|2 for u ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , λn > 0 be the eigenvalues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then the eigenvalues of the matrix in (8) equal (3 ± √ 8)λi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This shows that for u, v ∈ Rn, (11) 1 4(3 − √ 8)(|u|2 A + |v|2 A) ≤ h(u, v) ≤ 1 4(3 + √ 8)(|u|2 A + |v|2 A), 1 4(3 − √ 8)λm(|u|2 + |v|2) ≤ h(u, v) ≤ 1 4(3 + √ 8)λM(|u|2 + |v|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Spatial domain and mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let d ≥ 1 and let Ω ⊂ Rd be a bounded polygonal (if d = 2) or polyhedral (if d ≥ 3) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We associate to this domain an admissible mesh, given by (i) a family T of open polygonal or polyhedral control volumes, which are also called cells, (ii) a family E of edges (or faces if d ≥ 3), and (iii) a family of points (xK)K∈T associated to the control volumes and satisfying [21, Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This definition implies that the straight line xKxL between two centers of neighboring cells is orthogonal to the edge (or face) σ = K|L between two cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For instance, triangular meshes with acute angles, Delaunay meshes, rectangular meshes, and Vorono¨ı meshes satisfy this condition [21, Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The size of the mesh is given by ∆x = maxK∈T diam(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The family of edges E is assumed to consist of interior edges Eint satisfying σ ⊂ Ω and boundary edges σ ∈ Eext satisfying σ ⊂ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For a given K ∈ T , EK denotes the set of edges of K with EK = Eint,K ∪ Eext,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For any σ ∈ E, there exists at least one cell K ∈ T such that σ ∈ EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For given σ ∈ E, we define the distance dσ = � d(xK, xL) if σ = K|L ∈ Eint,K, d(xK, σ) if σ ∈ Eext,K, where d is the Euclidean distance in Rd, and the transmissibility coefficient (12) τσ = m(σ) dσ , where m(σ) denotes the (d − 1)-dimensional Lebesgue measure of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We suppose the following mesh regularity condition: There exists 0 < ζ ≤ 1/2 such that for all K ∈ T and σ ∈ EK, (13) d(xK, σ) ≥ ζdσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This is equivalent to η ≤ d(xK, σ) d(xL, σ) ≤ 1 η for all σ = K|L, 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER where η = ζ/(1−ζ) ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The statement follows by observing that d(xK, σ)+d(xL, σ) = d(xK, xL) holds, which is a consequence of the orthogonality of σ = K|L and xKxL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Hence, the mesh regularity (13) means that the mesh is locally quasi-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A consequence of the mesh regularity is the following estimate (14) � σ∈Eint,K m(σ)dσ ≤ 1 ζ � σ∈Eint,K m(σ)d(xK, σ) = d ζ m(K) for K ∈ T , where we used in the last step the formula for the volume of a (hyper-)pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Given a triangulation T , let T > 0, NT ∈ N and introduce the time step size ∆t = T/NT and the time steps tk = k∆t for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We set ΩT = Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The space of piecewise constant functions is defined by VT = � v : Ω → R : ∃(vK)K∈T ⊂ R, v(x) = � K∈T vK1K(x) � , where 1K is the indicator function on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' To define a norm on this space, we define for K ∈ T , σ ∈ EK, vK,σ = � vL if σ = K|L ∈ Eint,K, vK if σ ∈ Eext,K, DK,σv := vK,σ − vK, Dσv := |DK,σv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let 1 ≤ q < ∞ and v ∈ VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The discrete W 1,q(Ω) norm on VT is given by ∥v∥1,q,T = � ∥v∥q 0,q,T + |v|q 1,q,T �1/q, where ∥v∥q 0,q,T = � K∈T m(K)|vK|q, |v|q 1,q,T = � σ∈Eint m(σ)dσ ���� Dσv dσ ���� q for v ∈ VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' When q = ∞, we define |v|1,∞,T = maxσ∈Eint |Dσv|/dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' If v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , vn) ∈ V n T is a vector-valued function, we write for notational convenience ∥v∥0,q,T = n � i=1 ∥vi∥0,q,T , ∥∇v∥0,q,T = n � i=1 ∥∇vi∥0,q,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We associate to the discrete W 1,q norm a dual norm with respect to the L2 inner product: ∥v∥−1,q,T = sup � � Ω vwdx : w ∈ VT , ∥w∥1,q,T = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finally, we introduce the space VT ,∆t of piecewise constant functions with values in VT , VT ,∆t = � v : ΩT → R : ∃(vk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=',NT ⊂ VT , v(x, t) = NT � k=1 vk(x)1[tk−1,tk)(t) � , BDF2 FINITE-VOLUME SCHEME 7 equipped with the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' H1(Ω)) norm ∥v∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='H1(Ω)) = � NT � k=1 ∆t∥vk∥2 1,2,T �1/2 for all v ∈ VT ,∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Discrete gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The discrete gradient is defined on a dual mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For this, we define the cell TK,σ of the dual mesh for K ∈ T and σ ∈ EK: “Diamond”: Let σ = K|L ∈ Eint,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then TK,σ is that cell whose vertices are given by xK, xL, and the end points of the edge σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In higher dimensions, they might be (double) (hyper-)pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' “Triangle”: Let σ ∈ Eext,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then TK,σ is that cell whose vertices are given by xK and the end points of the edge σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The union of all “diamonds” and “triangles” TK,σ equals the domain Ω (up to a set of measure zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The property that the straight line xKxL is orthogonal to the edge σ = K|L implies that m(σ)d(xK, xL) = d m(TK,σ) for all σ = K|L ∈ Eint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The approximate gradient of v ∈ VT ,∆t is then defined by ∇T v(x, t) = m(σ) m(TK,σ)DK,σ(vk)νK,σ for x ∈ TK,σ, t ∈ (tk−1, tk], where νK,σ is the unit vector that is normal to σ and points outwards of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The initial functions are approximated by their L2(Ω)-orthogo- nal projection on VT : (15) u0 i,K = 1 m(K) � K u0 i (x)dx for all K ∈ T , i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let uk−1 K = (uk−1 1,K , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , uk−1 n,K) for K ∈ T be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Since the BDF2 scheme is a two-step method, we need a first time step which is computed from the implicit Euler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The following time steps are determined from the BDF2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The finite-volume scheme reads as m(K) ∆t (u1 i,K − u0 i,K) + � σ∈EK F1 i,K,σ = 0, (16) m(K) ∆t �3 2uk i,K − 2uk−1 i,K + 1 2uk−2 i,K � + � σ∈EK Fk i,K,σ = 0, k ≥ 2, (17) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, K ∈ T , and the numerical fluxes are given by (18) Fk i,K,σ = −τσ � γDK,σuk i + (uk i,σ)+DK,σpi(uk) � , where τσ is defined in (12) and z+ = max{0, z} denotes the positive part of z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finally, the so-called mobility is given by (19) uk i,σ = M(uk i,K, uk i,L) for σ = K|L, uk i,σ = 0 else, 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER where M is a general mean function satisfying (i) M : [0, ∞)2 → [0, ∞) is Lipschitz continuous, satisfies M(u, u) = u (consistency), and has linear growth in the sense M(u, v) ≤ |u| + |v| for u, v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' (ii) There exists C > 0 such that |M(u, v) − u| ≤ C|u − v| for all u, v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Examples for M are M(u, v) = (u + v)/2 or M(u, v) = max{u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Note that we do not need logarithmic mean functions like in [27], since we do not use the chain rule in the cross-diffusion part, so that we can use simpler expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Remark 1 (Nonnegativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We truncate the mobility by (uk i,σ)+ in the numerical flux (18) to ensure the discrete Rao entropy inequality (see (21) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, when testing (17) with pi(uk), we need that the sum � σ∈Eint τσ(uk i,σ)+|DK,σpi(uk)|2 is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Un- fortunately, the quadratic Rao entropy does not allow us to prove the nonnegativity of the discrete solution, and standard maximum principle arguments do not apply here, so that the truncation cannot be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A positivity-preserving BDF2 finite-difference scheme was proposed in [13], but the proof relies on discrete L∞(Ω) bounds for uk−1, which are not available in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Also the Shannon entropy does not help (as in [27]), since it is not compatible with the BDF2 discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, when we wish to derive a discrete analog of (3), we need a finite continuous functional h(u, v) satisfying h(u, u) = �n i=1 ui(log ui−1) (consistency condition) such that n � i=1 �3 2uk i,K − 2uk−1 i,K + 1 2uk−2 i,K � log uk i,K ≥ h(uk K, uk−1 K ) − h(uk−1 K , uk−2 K ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' If uk i,K = uk−1 i,K → 0 and uk−2 i,K > 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n}, the previous inequality converges to −∞ ≥ −h(0, uk−2 K ), which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' At least, we obtain nonnegative solutions in the limit (∆x, ∆t) → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' see Theorem 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Remark 2 (Discrete integration by parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The fluxes Fk i,K,σ are consistent approximations of the exact fluxes through the edges if we impose the conservation Fi,K,σ + Fi,L,σ = 0 for all edges σ = K|L, requiring that they vanish on the Neumann boundary edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', Fi,K,σ = 0 for all σ ∈ Eext,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In particular, for v = (vK) ∈ VT , the following discrete integration-by-parts formulas hold: (20) � K∈T � σ∈EK Fi,K,σvK = − � σ∈Eint σ=K|L Fi,K,σDK,σv, � K∈T � σ∈EK τσ(DK,σv)vK = −|v|2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We impose the following hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' (H1) Data: Ω ⊂ Rd with d ≥ 1 is a bounded polygonal (d = 2) or polyhedral (d ≥ 3) domain, T > 0, and u0 ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We set ΩT = Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' (H2) Discretization: T is an admissible discretization of Ω satisfying (13) and tk = k∆t for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' (H3) Coefficients: Let γ > 0, and A = (aij) ∈ Rn×n is symmetric and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let λm > 0 and λM > 0 be the smallest and largest eigenvalue of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 9 The positivity of γ is not needed for the existence analysis but for the convergence result, where we need higher-order integrability that is deduced via the discrete Gagliardo– Nirenberg inequality from the gradient bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As mentioned in the introduction, the symmetry and positive definiteness of A can be replaced by the positivity of the real parts of the eigenvalues of A and the detailed-balance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Recall the discrete BDF2 Rao entropy (see (9)) H(u, v) = � K∈T m(K)h(uK, vK) = 1 4 � K∈T m(K) � 5|uK|2 A − 4(uK, vK)A + |vK|2 A � for u, v ∈ VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' If u = v, this expression reduces to the usual discrete Rao entropy, used for the implicit Euler scheme, H(u) := H(u, u) = 1 2 � K∈T m(K)|uK|2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Our first result is the existence of a discrete solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Theorem 3 (Existence and uniqueness of discrete solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let Hypotheses (H1)–(H3) hold, let k ∈ N, and let uk−1 ∈ V n T be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then there exists a solution uk = (uk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , uk n) ∈ V n T to scheme (15)–(19) satisfying the discrete entropy inequality H(uk, uk−1) + γ∆t|A1/2uk|2 1,2,T ≤ H(uk−1, uk−2) for k ≥ 2, (21) H(u1) + γ∆t|A1/2u1|2 1,2,T ≤ H(u0), (22) and the scheme preserves the mass, � K∈T m(K)uk i,K = � Ω u0 i (x)dx for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' These results also hold if γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, the solution is unique if γ > 0, minσ∈Eint dσ ≥ ξ∆x for some ξ > 0, and ∆t (∆x)d+2 < C(d, ξ, ζ)γλ2 m λ2 ML2H(u0) , where ζ is defined in (13) and L is the Lipschitz constant of the mean function M, defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The existence of a discrete solution is proved by a fixed-point argument using the Brouwer degree theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Uniform estimates are obtained from the discrete Rao entropy inequality (21), where the BDF2 time approximation is estimated according to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This inequality, which is the key of our analysis, is proved in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The uniqueness of solutions is proved by using the relative entropy method, which is equivalent to the energy method in the present case, since the Rao entropy is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In other words, we use the test function pi(uk) − pi(vk) in the difference of the equations (17) satisfied by two discrete solutions uk and vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The cross-diffusion part contains cubic ex- pressions, which turn into quadratic ones if |A1/2vk|1,∞,T is bounded (similar as in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' By an inverse inequality, this norm is bounded, up to some factor, by (∆x)−d/2−1∥A1/2vk∥0,2,T , and ∥A1/2vk∥0,2,T is bounded because of (21)–(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The remaining quadratic expression is estimated by using the gradient bounds (which requires γ > 0) and the discrete L2(Ω) bound coming from the time discretization (and introducing the factor ∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The condition minσ∈Eint dσ ≤ ξ∆x is discussed in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For the next result, we set ¯ui = m(Ω)−1 � Ω u0 i dx and recall the discrete Poincar´e– Wirtinger inequality ∥v − ¯v∥0,2,T ≤ CPζ−1/2|v|1,2,T for v ∈ VT [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then, 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER in view of (10), (23) ∥A1/2(v − ¯v)∥0,2,T ≤ CP � λM λmζ �1/2 |A1/2v|1,2,T for v ∈ VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Theorem 4 (Large-time behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let uk be a solution to scheme (15)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then, for k ≥ 2, ∥A1/2(uk − ¯u)∥0,2,T ≤ √ 2∥A1/2(u0 − ¯u)∥0,2,T (1 + κ∆t)−(k−2)/4, where κ = 4γλmζ/((3 + √ 8)C2 PλM) and CP > 0 is the constant of the Poincar´e–Wirtinger inequality (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The theorem states that uk converges exponentially fast to the constant steady state ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, setting λ∆t := log(1 + κ∆t)/(∆t) ↗ κ as ∆t → 0, we have ∥A1/2(uk − ¯u)∥0,2,T ≤ √ 2∥A1/2(u0 − ¯u)∥0,2,T exp(−λ∆ttk), k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The proof of Theorem 4 is based on the discrete entropy inequality for the discrete relative Rao entropy H(uk−¯u, uk−1−¯u), similar to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, by the discrete Poincar´e–Wirtinger inequality, the discrete gradient term is bounded from below by the discrete L2(Ω) norm of uk − ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As H(uk − ¯u, uk−1 − ¯u) can be estimated in terms of the discrete L2(Ω) norms of uk − ¯u and uk−1 − ¯u, we need to iterate the entropy inequality a second time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then, using (11), we arrive at the inequality H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−1H(uk−2 − ¯u, uk−3 − ¯u), and solving this recursion shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerial convergence of the scheme is proved in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' First, we show that the fully discrete solution uk m ∈ V n T , indexed with the space grid size ∆xm → 0 as m → ∞, converges, up to a subsequence, to a solution uk ∈ H1(Ω) to the semidiscrete system 1 ∆t(u1 i − u0 i ) = div(γi∇u1 i + u1 i ∇pi(u1)), 1 ∆t �3 2uk i − 2uk−1 i + 1 2uk−2 i � = div(γi∇uk i + (uk i )+∇pi(uk)) in Ω, with no-flux boundary conditions ∇uk i · ν = 0 on ∂Ω, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Second, we prove that a subsequence of the sequence of semidiscrete solutions converges to a weak solution to (1)–(2) as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Both steps may be summarized as follows (the precise convergence statements can be found in Propositions 9 and 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Theorem 5 (Convergence of the scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let Hypotheses (H1)–(H3) hold and let (Tm)m∈N be a sequence of admissible discretizations of Ω satisfying (13) uniformly in m and ∆xm → 0, ∆tm → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then the solution (um) to (15)–(19), constructed in Theorem 3, converges, up to a subsequence, as m → ∞ to a function u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , un) satisfying ui ≥ 0 in ΩT for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, ui ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' H1(Ω)), ∂tui ∈ L2d+4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,2d+4(Ω)′), and u is a weak solution to (1)–(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 11 The proof is based on suitable estimates uniform with respect to ∆xm and ∆tm, derived from the entropy inequality (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For the limit ∆xm → 0, we follow the strategy of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The compactness argument is different, since we still keep the time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The limit ∆tm → 0 is based on a higher-order integrability property derived from the Gagliardo– Nirenberg inequality and on the Aubin–Lions compactness lemma in the version of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We need the condition γ > 0 since the application of the discrete Gagliardo–Nirenberg inequality requires discrete gradient bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' However, the term involving pi(u) only pro- vides a bound for the discrete kinetic energy � σ∈Eint τσ(uk i,σ)+|DK,σpi(uk)|2, from which we are unable to conclude gradient bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For the Euler scheme, this issue can be overcome by using the Boltzmann entropy inequality, which provides bounds in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' H1(Ω)) and L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' L1(Ω)) (see (3)), and consequently in L2+2/d(ΩT), which is the required higher- order integrability bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As mentioned in the introduction, this entropy is not compatible with the BDF2 discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Therefore, the restriction γ > 0 seems to be unavoidable with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finally, we verify that the convergence of the semidiscrete system is of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Theorem 6 (Second-order convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let uk be a solution to (31) and assume that the solution to (1)–(2) satisfies u ∈ C3([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' L2(Ω)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,∞(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, let ε > 0 be arbitrary and assume that ∆t < 4(3 − √ 8)γλm λ3 M∥∇u∥2 L∞(ΩT ) + 4γλmε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then there exists C(ε) > 0, which is of order ε−1/2 as ε → 0 but independent of ∆t, such that max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=',NT ∥A1/2(uk i − ui(tk))∥L2(Ω) ≤ C(ε)(∆t)2 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We allow for the parameter ε > 0 to minimize the time step size constraint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' however, optimizing this constraint gives large constants C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The theorem is proved by analyzing the relative entropy H(u(tk) − uk, u(tk−1) − uk−1), using a Taylor expansion for ui up to order (∆t)3 (which requires a bound for ∂3 t ui), and iterating the entropy inequality once more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The resulting recursive inequality for the relative entropy can be solved, leading to the desired second-order bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof of Theorem 3 First, we make precise inequality (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Recall definition (8) of h(u, v) and let H(u, v) = � K∈T m(K)h(u, v) be the discrete Rao entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Lemma 7 (BDF2 inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It holds for u, v, w ∈ Rn that �3 2u − 2v + 1 2w �T Au = h(u, v) − h(v, w) + 1 4|u − 2v + w|2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER In particular, for uk, uk−1, uk−2 ∈ V n T , n � i,j=1 � K∈T m(K) �3 2uk i,K − 2uk−1 i,K + 1 2uk−2 i,K � aijuk j,K ≥ H(uk, uk−1) − H(uk−1, uk−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The proof follows by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Definition and continuity of the fixed-point operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We assume that k ≥ 2, since the existence of a solution u1 ∈ V n T to the Euler scheme (17) satisfying (22) follows from [26, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let uk−1 ∈ V n T be given and let R > 0, δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We set ZR = � w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , wn) ∈ V n T : ∥wi∥1,2,T < R for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n � , and let w ∈ ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We consider the linear regularized problem (24) ε � � σ∈EK τσDK,σwε i − m(K)wε i,K � = m(K) ∆t �3 2wi,K − 2uk−1 i,K + 1 2uk−2 i,K � + � σ∈EK F+ i,K,σ(w) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, K ∈ T , where F+ i,K,σ(w) = −τσ � γDK,σwi + w+ i,σDK,σpi(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The ε-regularization guarantees the coercivity of the associated bilinear form, while the truncation w+ i,σ is needed to obtain the nonnegativity of the entropy dissipation (see the estimate of I6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We claim that (24) has a unique solution wε ∈ V n T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, since the mapping g(wε) = ε(� σ∈EK τσDK,σwε i − m(K)wε i,K) is linear and acting on finite-dimensional spaces, we only need to verify its injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let wε be in the kernel of this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Multiplying g(wε) = 0 by wε i,K, summing over K ∈ T , and using the discrete integration-by-parts formula (20) gives 0 = � K∈T � σ∈EK τσ(DK,σwε i )wε i,K − � K∈T m(K)(wε i,K)2 = −∥wε i ∥2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This yields wε = 0 and proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Next, we show that the fixed-point mapping F : ZR → V n T , F(w) = wε, is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For this, we multiply (24) by −wε i,K, sum over K ∈ T , and use discrete integration by parts and the Cauchy–Schwarz inequality: ε∥wε i ∥2 1,2,T = − 1 ∆t � K∈T m(K) �3 2wi,K − 2uk−1 i,K + 1 2uk−2 i,K � wε i,K + � σ∈Eint σ=K|L F+ i,K,σ(w)DK,σwε i ≤ 1 ∆t ���� 3 2wi − 2uk−1 i + 1 2uk−2 i ���� 0,2,T ∥wε i ∥0,2,T + γ|wi|1,2,T |wε i |1,2,T (25) − � σ∈Eint σ=K|L τσ(wi,σ)+DK,σpi(w)DK,σwε i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 13 For the last term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' we use the Cauchy–Schwarz inequality and the fact that any norm on V n T is equivalent: − � σ∈Eint σ=K|L τσ(wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σpi(w)DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σwε i = − n � j=1 � σ∈Eint σ=K|L τσaijw+ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σDK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σwiDK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σwε i ≤ n � j=1 � � σ∈Eint σ=K|L τσ|Dσwε i |2 �1/2� � σ∈Eint σ=K|L τσa2 ij(w+ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)2|Dσwj|2 �1/2 ≤ C(A)∥w∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T n � j=1 |wε i |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T |wj|1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T ≤ C(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' R)∥wε i ∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' where we took into account the linear growth of w+ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ with respect to wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K and wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L (see (19)) and the definition of ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Inserting these estimates into (25) and dividing by ∥wε i ∥1,2,T , it follows that ε∥wε i ∥1,2,T ≤ C(A, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This bound allows us to verify the continuity of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, let wℓ → w as ℓ → ∞ and set wε,ℓ = F(wℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then (wε,ℓ)ℓ∈N is uniformly bounded in the discrete H1(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Therefore, there exists a subsequence, which is not relabeled, such that wε,ℓ → wε as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Passing to the limit ℓ → ∞ in scheme (24), we see that wε is a solution of the scheme and consequently wε = F(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Since the solution to the linear scheme (24) is unique, the entire sequence (wε,ℓ)ℓ∈N converges to wε, which shows the continuity of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Existence of a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' According to the Brouwer degree fixed-point theorem, it is sufficient to show that for all (wε, ρ) ∈ ZR × [0, 1] such that wε = ρF(wε), it holds that wε ̸∈ ∂ZR or, equivalently, ∥wε∥1,2,T < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We claim that this is true for sufficiently large R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, let wε be such a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It satisfies ε � � σ∈EK τσDK,σwε i − m(K)wε i,K � = ρ ∆t m(K) �3 2wε i,K − 2uk−1 i,K + 1 2uk−2 i,K � + ρ � σ∈EK F+ i,K,σ(wε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We multiply this equation by −(∆t)pi(wε) and sum over i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, K ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then 0 = I1 + I2 + I3, where I1 = −ε∆t n � i,j=1 � K∈T � � σ∈EK τσDK,σwε i − m(K)wε i,K � aijwε j,K, I2 = ρ n � i,j=1 � K∈T aij �3 2wε i,K − 2uk−1 i,K + 1 2uk−2 i,K � wε j,K, 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER I3 = −ρ∆t n � i=1 � σ∈Eint σ=K|L F+ i,K,σ(wε)DK,σpi(wε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' By discrete integration by parts, I1 = ε∆t n � i,j=1 � � σ∈Eint σ=K|L τσaijDK,σwε i DK,σwε j + � K∈T m(K)aijwε i,Kwε j,K � ≥ ελm∆t(|wε|2 1,2,T + ∥wε∥2 0,2,T ) = ελm∆t∥wε∥2 1,2,T , and by Lemma 7, I2 ≥ H(wε, uk−1) − H(uk−1, uk−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For the third term, we obtain I3 = ρ∆t n � i,j=1 � σ∈Eint σ=K|L τσγaijDK,σwε i DK,σwε j + ρ∆t n � i=1 � σ∈Eint σ=K|L τσ(wε i,σ)+ � n � j=1 aijDK,σwε j �2 ≥ γρ∆t � σ∈Eint σ=K|L τσ|A1/2DK,σwε|2 = γρ∆t|A1/2wε|2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Collecting these estimates gives (26) ε∆t∥wε∥2 1,2,T + H(wε, uk−1) + γ∆tρ|A1/2wε|2 1,2,T ≤ H(uk−1, uk−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Setting R = (ε∆t)−1/2H(uk−1, uk−2)1/2 + 1, we infer that ∥wε∥2 1,2,T ≤ (R − 1)2 < R2 and thus wε ̸∈ ∂ZR, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Hence, there exists a fixed point wε to F, which is a solution to (27) ε � � σ∈EK τσDK,σwε i −m(K)wε i,K � = m(K) ∆t �3 2wε i,K −2ui,K + 1 2uk−2 i,K � + � σ∈EK Fi,K,σ(wε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The solution wε to (27) satisfies the regularized entropy inequality (26) with ρ = 1, and the right-hand side is independent of ε and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It follows from the Bolzano–Weierstraß theorem that there exists a subsequence of wε, which is not relabeled, such that wε → w as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In particular, ε1/2wε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Since the problem is finite dimensional, we can pass to the limit ε → 0 in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Consequently, uk := w is a solution to (17)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The same limit in (26) with ρ = 1 leads to the discrete entropy inequality of Theorem 3, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Uniqueness of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let uk, vk ∈ V n T be two solutions to (15)–(19) with the same initial data u0 = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We take the difference of the equations satisfied by uk and vk, multiply the resulting equation by pi(uk K) − pi(vk K) = �n j=1 aij(uk j,K − vk j,K), sum over BDF2 FINITE-VOLUME SCHEME 15 i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, K ∈ T , and use discrete integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This leads to 0 = I4 + I5 + I6, where I4 = 3 2∆t n � i,j=1 � K∈T m(K)aij(uk i,K − vk i,K)(uk j,K − vk j,K) I5 = n � i,j=1 � σ∈Eint σ=K|L τσγaijDK,σ(uk i − vk i )DK,σ(uk j − vk j ) I6 = n � i,j,ℓ=1 � σ∈Eint σ=K|L τσaij � (uk i,σ)+DK,σuk j − (vk i,σ)+DK,σvk j � aiℓDK,σ(uk ℓ − vk ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' By the definition of the weighted norm, I4 = (3/(2∆t))∥A1/2(uk − vk)∥2 0,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, I5 ≥ γ � σ∈Eint σ=K|L τσ|(A1/2DK,σ(uk − vk))|2 = γ|A1/2(uk − vk)|2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We add and subtract the term (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σvk j in I6 and apply the Cauchy–Schwarz inequal- ity: I6 = n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='ℓ=1 � σ∈Eint σ=K|L τσaijaiℓ � (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk j − vk j ) + � (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ − (vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+� DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σvk j � DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk ℓ − vk ℓ ) = n � i=1 � σ∈Eint σ=K|L τσ(uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ � n � j=1 aijDK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk j − vk j ) �� n � ℓ=1 aiℓDK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk ℓ − vk ℓ ) � − � σ∈Eint σ=K|L τσ(ADK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σvk)T� diag � (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ − (vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+� A1/2� (A1/2DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk − vk)) ≥ − � � σ∈Eint σ=K|L τσ|ADK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σvk|2�� diag � (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ − (vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+� A1/2��2 �1/2 × � � σ∈Eint σ=K|L τσ|A1/2DK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ(uk − vk)|2 �1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' where diag((uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ − (vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+) denotes the diagonal matrix with the entries (uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ − (vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ)+ for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Together with |ADK,σvk| ≤ |A1/2||A1/2DK,σvk| ≤ λ1/2 M |DK,σvk|A and 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER �� diag � (uk i,σ)+ − (vk i,σ)+� A1/2�� ≤ �� diag � (uk i,σ)+ − (vk i,σ)+���|A1/2| ≤ λ1/2 M max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=',n |(uk i,σ)+ − (vk i,σ)+| ≤ λ1/2 M max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=',n |uk i,σ − vk i,σ|, we find that I6 ≥ −λM|A1/2vk|1,∞,T max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=',n � � σ∈Eint σ=K|L m(σ)dσ|uk i,σ − vk i,σ|2 �1/2 |A1/2(uk − vk)|1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It remains to estimate the term involving the difference |uk i,σ − vk i,σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' By the Lipschitz continuity of the mean function M(uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L) = uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ with Lipschitz constant L > 0 and the mesh regularity (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' � σ∈Eint σ=K|L m(σ)dσ|uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ|2 = 1 2 � K∈T � σ∈Eint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K m(σ)dσ|uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='σ|2 ≤ L2 2 � K∈T � σ∈Eint σ=K|L m(σ)dσ � |uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K| + |uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L| �2 ≤ 2L2 � K∈T � σ∈Eint σ=K|L m(σ)dσ|uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K|2 ≤ 2dL2 ζ � K∈T m(K)|uk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K − vk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='K|2 = 2dL2 ζ ∥uk − vk∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T ≤ 2dL2 λmζ ∥A1/2(uk − vk)∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This shows that I6 ≥ −λML λ1/2 m �2d ζ �1/2 |A1/2vk|1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T ∥A1/2(uk − vk)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T |A1/2(uk − vk)|1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Collecting the estimates for I4, I5, and I6 and using Young’s inequality gives 3 2∆t∥A1/2(uk − vk)∥2 0,2,T + γ|A1/2(uk − vk)|2 1,2,T (28) ≤ λML λ1/2 m �2d ζ �1/2 |A1/2vk|1,∞,T ∥A1/2(uk − vk)∥0,2,T |A1/2(uk − vk)|1,2,T ≤ 3 2∆t∥A1/2(uk − vk)∥2 0,2,T + ∆t 3 dλ2 ML2 λmζ |A1/2vk|2 1,∞,T |A1/2(uk − vk)|2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Now, the inverse inequality |A1/2vk|1,∞,T ≤ C′(d)(∆x)−d/2ζ−1/2|A1/2vk|1,2,T [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='10] and condition dσ ≥ ξ∆x imply that |A1/2vk|2 1,∞,T ≤ C′(d)2 (∆x)dζ � σ∈Eint σ=K|L m(σ) dσ |DK,σ(A1/2vk)|2 BDF2 FINITE-VOLUME SCHEME 17 ≤ C′(d)2 (∆x)dζ � σ∈Eint σ=K|L m(σ)dσ ξ2(∆x)2|DK,σ(A1/2vk)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It follows from (14) and |DK,σ(A1/2vk)|2 ≤ 2(|vk K|2 A + |vk L|2 A) that |A1/2vk|2 1,∞,T ≤ C′(d)2 (∆x)d+2ξ2ζ � K∈T d ζ m(K)|DK,σ(A1/2vk)|2 ≤ 2dC′(d)2 (∆x)d+2(ξζ)2 � K∈T m(K)|A1/2vk K|2 = C(d, ξ, ζ) (∆x)d+2 ∥A1/2vk∥2 0,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Using this inequality as well as the bound (3 − √ 8)∥A1/2vk∥2 0,2,T ≤ 4H(v1, v0) ≤ 2(3 + √ 8)(H(v1) + H(v0)) ≤ 4(3 + √ 8)H(u0), obtained from (21)–(22), we deduce from (28), for another constant C(d, ξ, ζ) that γ|A1/2(uk − vk)|2 1,2,T ≤ C(d, ξ, ζ)λ2 ML2 λm ∆t (∆x)d+2H(u0)|A1/2(uk − vk)|2 1,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then our smallness condition on ∆t/(∆x)d+2 implies that |A1/2(uk − vk)|1,2,T = 0 and consequently, uk = vk, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The quasi-uniform condition minσ∈Eint dσ ≥ ξ∆x > 0 implies condition (23) in [20], since the mesh regularity (13) gives minK∈T minσ∈EK d(xK, σ)/ diam(K) ≥ ζdσ/∆x ≥ ζξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It also implies the mesh regularity condition diam(K)/d(xK, σ) ≤ ξ0 in [20, (9)], since, because of (13) again, diam(K)/d(xK, σ) ≤ ∆x/(ζdσ) ≤ 1/(ξζ) =: ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It can be seen by considering quadratic cells that the quasi-uniform condition minσ∈Eint dσ ≥ ξ∆x > 0 generally does not imply the mesh regularity condition (13) and vice versa, so both conditions are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof of Theorem 4 We infer from mass conservation, � K∈T m(K)uk i,K = � K∈T m(K)u0 i,K = m(Ω)¯ui, that H(uk − ¯u, uk−1 − ¯u) = H(uk, uk−1) + 1 2 � K∈T m(K) � |¯u|2 A − 3(uk K, ¯u)A + (uk−1 K , ¯u)A � = H(uk, uk−1) − 1 2 m(Ω)|¯u|2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then the entropy inequality (21) shows that (29) H(uk − ¯u, uk−1 − ¯u) + γ∆t|A1/2uk|2 1,2,T ≤ H(uk−1 − ¯u, uk−2 − ¯u) for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Another iteration gives, for k ≥ 3, H(uk − ¯u, uk−1 − ¯u) + γ∆t � |A1/2uk|2 1,2,T + |A1/2uk−1|2 1,2,T � ≤ H(uk−2 − ¯u, uk−3 − ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER Hence, taking into account the discrete Poincar´e–Wirtinger inequality (23), H(uk − ¯u, uk−1 − ¯u) + γλmζ C2 PλM ∆t � ∥A1/2(uk − ¯u)∥2 0,2,T + ∥A1/2(uk−1 − ¯u)∥2 0,2,T � ≤ H(uk−2 − ¯u, uk−3 − ¯u), and the norm equivalence (10), H(uk − ¯u, uk−1 − ¯u) + 4γλmζ∆t (3 + √ 8)C2 PλM H(uk − ¯u, uk−1 − ¯u) ≤ H(uk−2 − ¯u, uk−3 − ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This can be written as H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−1H(uk−2 − ¯u, uk−3 − ¯u), where κ = 4γλmζ/((3 + √ 8)C2 PλM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Depending on whether k is odd or even, we resolve this iteration as follows: H(u2ℓ+1 − ¯u, u2ℓ − ¯u) ≤ (1 + κ∆t)−ℓH(u1 − ¯u, u0 − ¯u), H(u2ℓ+2 − ¯u, u2ℓ+1 − ¯u) ≤ (1 + κ∆t)−ℓH(u2 − ¯u, u1 − ¯u) ≤ (1 + κ∆t)−ℓH(u1 − ¯u, u0 − ¯u), where we used (29) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As in both cases ℓ ≥ (k − 2)/2, we conclude that (30) H(uk − ¯u, uk−1 − ¯u) ≤ (1 + κ∆t)−(k−2)/2H(u1 − ¯u, u0 − ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We want to express this inequality in terms of the ∥A1/2(·)∥0,2,T norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We observe that, by Young’s inequality, ∥A1/2(uk − ¯u)∥2 0,2,T ≤ 4H(uk − ¯u, uk−1 − ¯u) and, in view of (22), H(u1 − ¯u, u0 − ¯u) = H(u1) − H(¯u) ≤ H(u0) − H(¯u) = H(u0 − ¯u) = 1 2∥A1/2(u0 − ¯u)∥2 0,2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then we deduce from (30) that ∥A1/2(uk − ¯u)∥2 0,2,T ≤ 4H(uk − ¯u, uk−1 − ¯u) ≤ 4(1 + κ∆t)−(k−2)/2H(u1 − ¯u, u0 − ¯u) ≤ 2(1 + κ∆t)−(k−2)/2∥A1/2(u0 − ¯u)∥2 0,2,T , which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof of Theorem 5 We split the proof into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We first prove the convergence in the space variable and then the convergence in the time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' An alternative is to show the convergence in both variables simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Convergence in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We show the following result for ∆x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proposition 9 (Convergence in space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let the assumptions of Theorem 5 hold and let (uk m) be the sequence of solutions to (15)–(19) constructed in Theorem 3 associated to an admissible mesh Tm with mesh size ∆xm for m ∈ N satisfying ∆xm → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then there exists a subsequence which is not relabeled such that uk i,m → uk i strongly in L2(Ω) as m → ∞ and uk i solves for all φi ∈ W 1,max{2,d}(Ω), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n, (31) 1 ∆t � Ω �3 2uk i − 2uk−1 i + 1 2uk−2 i � φidx + � Ω � γ∇uk i + (uk i )+∇pi(uk) � ∇φidx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For fixed ∆t, the discrete entropy inequality in Theorem 3 provides a uniform bound for ∥uk m∥1,2,Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then, by the discrete Rellich–Kondrachov compactness theorem [21, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6], there exists a subsequence of (uk m) = (uk 1,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , uk n,m), which is not relabeled, such that uk m → uk strongly in L2(Ω) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Moreover, the sequence of discrete gradients (∇muk m) converges weakly in L2(Ω) to some function which can be identified by ∇uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' see [10, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let φi ∈ C2(Ω) and set φi,K := φi(xK) for K ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then the limit ∆xm → 0 in the BDF2 approximation becomes 1 ∆t � K∈T m(K) �3 2uk i,K − 2uk−1 i,K + 1 2uk−2 i,K � φi,K → 1 ∆t � Ω �3 2uk i − 2uk−1 i + 1 2uk i � φidx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Next, we set F m = F m 1 + F m 2 + F m 3 , where F m 1 = −γ � K∈T � σ∈EK τσDK,σuk i,mφi,K, F m 2 = − � K∈T � σ∈EK τσ(uk i,m,K)+DK,σpi(uk m)φi,K, F m 3 = − � K∈T � σ∈EK τσ � (uk i,m,σ)+ − (uk i,m,K)+� DK,σpi(uk m)φi,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We introduce the intermediate integral F m 0 = F m 01 + F m 02, where F m 01 = γ � Ω ∇muk m,i · ∇φidx, F m 02 = � Ω (uk i,m)+∇mpi(uk m) · ∇φidx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It follows from the weak convergence of the discrete gradients and the strong convergence in L2(Ω) that F m 0 → F as m → ∞, where F = γ � Ω ∇uk i · ∇φidx + � Ω (uk i )+∇pi(uk) · ∇φidx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Thus, if we can show that F m 0 − F m → 0, then |F m − F| ≤ |F m − F m 0 | + |F m 0 − F| → 0, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' By discrete integration by parts and the definition of the discrete gradient, F m 1 = γ � σ∈Eint σ=K|L τσDK,σuk i,mDK,σφi, 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER F m 01 = γ � σ∈Eint σ=K|L m(σ) m(TK,σ)DK,σuk i,m � TK,σ ∇φi · νK,σdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Using the Taylor expansion (here we need φi ∈ C2(Ω)) DK,σφi dσ = φi,L − φi,K d(xK, xL) = ∇φi · νK,σ + O(∆xm) for σ = K|L, where we have taken into account the property xK − xL = d(xK, xL)νK,σ, we obtain |F m 01 − F m 1 | ≤ γ � σ∈Eint σ=K|L m(σ)|DK,σuk i,m| ���� 1 m(TK,σ) � TK,σ ∇φi · νK,σdx − DK,σφi dσ ���� ≤ Cγ∆xm � σ∈Eint m(σ)|Dσuk i,m|, where C > 0 depends on the L∞ norm of D2φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We apply the Cauchy–Schwarz inequality and use the mesh property (14) to find that |F m 01 − F m 1 | ≤ Cγ∆xm � � σ∈Eint m(σ) dσ |Dσuk i,m|2 �1/2� � σ∈Eint m(σ)dσ �1/2 ≤ Cγ∆xm|uk i,m|1,2,Tm �d ζ m(Ω) �1/2 → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Similar arguments lead to |F m 02 − F m 2 | ≤ C∆xm � K∈Tm � σ∈Eint,K m(σ)(uk i,m,K)+|DK,σpi(uk m)| ≤ C∆xm � � K∈Tm |(uk i,m,K)+|2 � σ∈Eint,K m(σ)dσ �1/2 |pi(uk m)|1,2,Tm ≤ C∆xm �d ζ � K∈Tm m(K)|(uk i,m,K)+|2 �1/2 |pi(uk m)|1,2,Tm ≤ C(ζ)∆xm∥uk i,m∥0,2,Tm|pi(uk m)|1,2,Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The right-hand side converges to zero since |pi(uk m)|2 1,2,Tm = � σ∈Eint σ=K|L τσ � n � j=1 aijDK,σuk j,m �2 ≤ C(A)|uk m|2 1,2,Tm ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finally, using |DK,σφi| ≤ C(φi)∆xm and property (ii) of the mean function, |F m 3 | ≤ � σ∈Eint σ=K|L τσ|uk i,m,σ − uk i,m,K||DK,σpi(uk m)||DK,σφi| BDF2 FINITE-VOLUME SCHEME 21 ≤ C(φi)∆xm � σ∈Eint σ=K|L τσ|Dσuk i,m||DK,σpi(uk m)| ≤ C(φi, A)∆xm � � σ∈E τσ|Dσuk i,m|2 �1/2� n � j=1 � σ∈E τσ|Dσuk j,m|2 �1/2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This shows that F m 0 − F → 0 as m → ∞, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Convergence in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We wish to perform the limit ∆t → 0 in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For this, we need an estimate in a better space than L2(ΩT), provided by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Lemma 10 (Higher-order integrability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let (u(τ)) be a family of solutions to (31) associ- ated to the time step size τ := ∆t, constructed in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then there exists C > 0 independent of τ such that ∥u(τ)∥Lp(ΩT ) ≤ C for p = 2 + 4/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The lemma follows from the discrete entropy inequalities (21)–(22) and the Gagliar- do–Nirenberg inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Indeed, we infer from the entropy inequalities after summation over k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , NT that ∥u(τ)∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L2(Ω)) + ∥u(τ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='H1(Ω)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then it follows from the Gagliardo–Nirenberg inequality with θ = d/2 − d/p that ∥u(τ)∥p Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='Lp(Ω)) ≤ C � T 0 ∥u(τ)∥pθ H1(Ω)∥u(τ)∥p(1−θ) L2(Ω) dt ≤ C∥u(τ)∥p(1−θ) L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L2(Ω)) � T 0 ∥u(τ)∥2 H1(Ω)dt ≤ C, since pθ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' □ Proposition 11 (Convergence in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let (u(τ)) be a family of solutions to (31) with τ = ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then u(τ) converges to a weak solution u to (1)–(2) satisfying ui ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' H1(Ω)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' L2(Ω)), ∂tui ∈ Lr(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,2d+4(Ω)′), where r = (2d + 4)/(2d + 3) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We estimate the discrete time derivative Dτu(τ) i (t) := 3 2uk i − 2uk−1 i + 1 2uk−2 i for t ∈ [kτ, (k + 1)τ) for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let φi ∈ L2d+4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,2d+4(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then 1 τ � T 2τ ��⟨Dτu(τ) i , φi⟩W 1,d+2(Ω)′ ��rdt ≤ γrC � T 2τ � Ω |∇u(τ) i ∇φi|rdxdt + C � T 2τ � Ω |(u(τ) i )+∇pi(u(τ)) · ∇φi|rdxdt ≤ γrC∥∇u(τ) i ∥r L2(ΩT )∥∇φi∥r L2d+4(ΩT ) + C∥u(τ) i ∥r L(2d+4)/d(ΩT )∥∇pi(u(τ))∥r L2(ΩT )∥∇φi∥r L2d+4(ΩT ) 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER ≤ C∥φi∥r L2d+4(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)), where we used the fact that pi(u(τ)) is a linear combination of all u(τ) j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This implies the bound τ −1∥Dτu(τ) i ∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Let πτu(τ)(t) = u(τ)(t − τ) be a shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We relate the implicit Euler scheme and the BDF2 scheme by uk i − uk−1 i = 2 3 �3 2uk i − 2uk−1 i + 1 2uk−2 i � + 1 3(uk−1 i − uk−2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then ∥u(τ) − πτu(τ)∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) = ���2 3Dτu(τ) + 1 3πτ(u(τ) − πτu(τ)) ��� Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) ≤ 2 3∥Dτu(τ)∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) + 1 3∥u(τ) − πτu(τ)∥Lr(τ,T−τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Adding ∥u(τ) − πτu(τ)∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) ≤ C1 from the first Euler step (proved in a similar way as above) to the left-hand side and absorbing the last term on the right-hand side by the left-hand side, we find that 2 3τ ∥u(τ) − πτu(τ)∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) ≤ 2 3τ ∥Dτu(τ)∥Lr(2τ,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='W 1,2d+4(Ω)′) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Together with the uniform L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' H1(Ω)) bound for u(τ), we can apply the Aubin–Lions compactness lemma in the version of [16] to conclude that, up to a subsequence, as τ → 0, u(τ) → u strongly in L2(ΩT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In view of the higher-order estimate of Lemma 10, this convergence also holds in Lq(ΩT) for all q < 2 + 4/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, again up to a subsequence, Dτu(τ) ⇀ ∂tu weakly in Lr(2τ, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,2d+4(Ω)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' These convergences are sufficient to pass to the limit τ → 0 in (31) for test functions φi ∈ L2d+4(2τ, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' W 1,2d+4(Ω)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Second-order convergence As in the previous section, we set D∆tuk i = 3 2uk i − 2uk−1 i + 1 2uk−2 i and write (31) as (32) 1 ∆t � Ω D∆tuk i φidx + � Ω � γ∇uk i + (uk i )+∇pi(uk) � ∇φidx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A Taylor expansion shows that, for some ξk ∈ (0, T), D∆tui(tk) := 3 2ui(tk) − 2ui(tk−1) + 1 2ui(tk−2) = (∆t)∂tui(tk) − (∆t)3 3 ∂3 t ui(ξk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Then, using a test function φi ∈ H1(Ω) in (1), (33) 1 ∆t � Ω D∆tui(tk)φidx + � Ω (γ∇ui + ui∇pi(u))(tk) · ∇φidx = (∆t)2 3 � Ω ∂3 t ui(ξk)φidx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 23 We take the difference of (32) and (33), choose the test function φi = pi(u(tk)) − pi(uk) = (A(u(tk) − uk))i, and sum over i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , n: 1 ∆t � Ω D∆t(u(tk) − uk)TA(u(tk) − uk)dx = I7 + I8, where, (34) I7 = − n � i=1 � Ω � γ∇(ui(tk) − uk i ) + ui(tk)∇pi(u(tk)) − (uk i )+∇pi(uk) � ×∇(A(u(tk) − uk))idx, I8 = (∆t)2 3 n � i=1 � Ω ∂3 t ui(ξk)(A(u(tk) − uk))idx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Set vk := u(tk) − uk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' It follows from the BDF2 inequality in Lemma 7, applied to the left-hand side, that 1 ∆t � Ω D∆t(u(tk) − uk)TA(u(tk) − uk)dx ≥ 1 ∆t � H(vk, vk−1) − H(vk−1, vk−2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' For the terms I7 and I8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' we use the definition pi(uk) = (Auk)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' the Lipschitz continuity of z �→ z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' the nonnegativity of ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' and Young’s inequality: I7 = − n � i=1 � Ω γ∇(A1/2vk)i · ∇(A1/2vk)idx − n � i=1 � Ω � (ui(tk) − (uk i )+)∇(Au(tk))i + (uk i )+∇(A(u(tk) − uk))i � ∇(Avk)idx ≤ −γ∥∇(A1/2vk)∥2 L2(Ω) + λ−1/2 m ∥A1/2vk∥L2(Ω)λ3/2 M ∥∇u(tk)∥L∞(Ω)∥∇(A1/2vk)∥L2(Ω) − n � i=1 � Ω (uk i )+|∇(Avk)i|2dx ≤ λ3 M 4γλm ∥∇u∥2 L∞(ΩT )∥A1/2vk∥2 L2(Ω) and I8 ≤ (∆t)2 3λ1/2 m ∥∂3 t u∥L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L2(Ω))∥A1/2vk∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Summarizing, we obtain from (34) H(vk, vk−1) − H(vk−1, vk−2) ≤ C1∆t∥A1/2vk∥2 L2(Ω) + C2(∆t)3∥A1/2vk∥L2(Ω), (35) where C1 = λ3 M 4γλm ∥∇u∥2 L∞(ΩT ), C2 = 1 3λ1/2 m ∥∂3 t u∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We iterate this inequality once more and use the inequality a + b ≤ � 2(a2 + b2) as well as the norm equivalence (11): H(vk, vk−1) − H(vk−2, vk−3) ≤ C1∆t � ∥A1/2vk∥2 L2(Ω) + ∥A1/2vk−1∥2 L2(Ω) � + C2(∆t)3� ∥A1/2vk∥L2(Ω) + ∥A1/2vk−1∥L2(Ω) � ≤ C1∆t � ∥A1/2vk∥2 L2(Ω) + ∥A1/2vk−1∥2 L2(Ω) � 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER + √ 2C2(∆t)3� ∥A1/2vk∥2 L2(Ω) + ∥A1/2vk−1∥2 L2(Ω) �1/2 ≤ 4C1∆t 3 − √ 8H(vk, vk−1) + 4 √ 2C2(∆t)3 3 − √ 8 H(vk, vk−1)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We apply Young’s inequality for ε > 0: � 1 − 4(C1 + ε) 3 − √ 8 ∆t � H(vk, vk−1) ≤ H(vk−2, vk−3) + 2C2 2(∆t)5 (3 − √ 8)ε, and assume that ∆t < (3 − √ 8)/(4(C1 + ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This recursion is of the form ak ≤ bak−2 + bc(∆t)5, where ak = H(vk, vk−1) and b = � 1 − 4(C1 + ε) 3 − √ 8 ∆t �−1 , c = 2C2 2(∆t)5 (3 − √ 8)ε, and it can be resolved explicitly depending on whether k is odd or even: a2ℓ+1 ≤ bℓa1 + c(∆t)5 ℓ−1 � j=0 bj, a2ℓ+2 ≤ bℓa2 + c(∆t)5 ℓ−1 � j=0 bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The sum can be estimated according to ℓ−1 � j=0 bj = bℓ − 1 b − 1 ≤ � 1 − 4∆t 3 − √ 8(C1 + ε) �−ℓ+1 3 − √ 8 4∆t(C1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Since ℓ = tℓ/∆t ≤ T/∆t, the bracket approximates the exponential function and can be bounded by a constant depending only on C1 + ε and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This shows that there exist constants K1, K2 > 0 such that H(v2ℓ+1, v2ℓ) ≤ K1(C1, ε, T)H(v1, v0) + K2(C1, C2, ε−1, T)(∆t)4, H(v2ℓ+2, v2ℓ+1) ≤ K1(C1, ε, T)H(v2, v1) + K2(C1, C2, ε−1, T)(∆t)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Going back to inequality (35) for k = 2, we can argue in a similar way as before that H(v2, v1) is bounded by K3H(v1, v0) + K4(∆t)5 for some constants K3, K4 > 0, which are independent of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Furthermore, since v0 = 0, we have H(v1, v0) = (5/4)∥A1/2(u(t1) − u1)∥L2(Ω) ≤ K5(∆t)4 for some K5 > 0 independent of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This shows that H(vk, vk−1) ≤ K6(∆t)4, where K6 depends on C1, C2, ε−1, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Taking the square root and using (11) shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Numerical examples The finite-volume scheme (15)–(19) is implemented in Matlab, using the mobility M(u, v) = 1 2(u + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As the numerical scheme is implicit, we have solved the nonlinear system of equations at each time step by using the Matlab routine fsolve, based on Newton’s method with trust regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The optimality tolerance was chosen as 10−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' First example: one-dimensional domain, three species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We choose the domain Ω = (0, 1), the parameter γ = 1/2, as well as the positive definite matrix A and the initial data u0 according to A = � � 2 1 1/2 1 3 3/2 1/2 3/2 1 � � , u0(x) = � � cos(πx) + 2 2 − cos(2πx) 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical parameters are ∆x = 1/12 800 and ∆t = 1/128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical solution is illustrated in Figure 1 at various times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' All components converge to the constant steady state ¯u = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Interestingly, although initially equal to the steady state, the density u3 becomes nonconstant for positive times before it tends to the constant steady state for large times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Such a phenomenon is sometimes called uphill diffusion, which typically appears in thermodynamic multicomponent systems due to cross diffusion [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Densities u1(t) (darker blue line), u2(t) (lighter green line), u3(t) (dashed black line) at times t = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1 (from left to right) versus space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Second example: two-dimensional domain, two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We take Ω = (0, 1)2, ∆x = √ 2 · 2−5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='0044, ∆t = 1/256, γ = 1/2, and A = � 1 1/2 1/2 1 � , u0(x) = � 1(0,1/2)2(x) 1(1/2,1)2(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Figure 2 shows the evolution of u = (u1, u2) at various times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Although being discontinuous and segregated initially, the solution becomes smooth and mixes the densities for positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' This is not surprising, as full segregation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=', the supports of u1 and u2 do not intersect) is expected only when γ = 0 and det A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical scheme preserved the nonnegativity in all our experiments, even for the initial data of this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The numerical solutions are the same with or without the cutoff used in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Third example: exponential time decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We choose the one-dimensional domain Ω = (0, 1), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='1, ∆x = 2−7, ∆t = (10 · 27)−1, and A = � β 2 2 1 � , u0(x) = � 2 − cos(πx) 2 + cos(πx) � , 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='8 0 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='8 0 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='8 0 126 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨UNGEL AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Density u1(t) (upper row) and u2(t) (lower row) at times t = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 (from left to right) versus space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' where β > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The distance ∥A1/2(uk−¯u)∥L2(Ω) presented in Figure 3 for β = 5 and β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='01 shows that the time decay behaves exponentially, as predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' The decay rates (excluding the initial decay) are −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='37 for β = 5 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='03 for β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='01, and they decrease for smaller values of det A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We have also observed an exponential decay when γ = 0 with smaller decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Semilogarithmic plot for ∥A1/2(uk − ¯u)∥L2(Ω) versus time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Fourth example: Convergence rate in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' We choose the values for A and u0 as in the previous example as well as γ = 0, ∆x = 2−9, and ∆t = (10·2p)−1 with p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 0 Logarithm of distance 1 Reference line 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0 1 Time0 Logarithm of distance Reference line 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='2 0 1 Time0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 O 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='5 0BDF2 FINITE-VOLUME SCHEME 27 The reference solution uref is computed with the time step size ∆t = (10 · 29)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' As expected, the convergence rate at time T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='02, shown in Figure 4 for two different values of β, is about two, even in the case det A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Discrete L2(Ω) error ∥A1/2(u(∆t)−uref)(T)∥L2(Ω) versus time step size ∆t = (10 · 2p)−1 for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' , 8 for β = 5 (left) and β = 4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Amann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Schmeisser and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Triebel (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' ), Funct.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Carrillo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Sci.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 4 (2012), 137–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Bessemoulin-Chatard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Numer.' metadata={'source': 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+page_content=' On discrete functional inequalities for some finite volume schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 35 (2015), 1125–1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Calgaro and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Ezzoug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' L∞-stability of the IMEX-BDF2 finite volume scheme for convection- diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' In: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Canc`es and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Omnes (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' ), Finite Volumes for Complex Applications VIII, pp.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Finite volume scheme for multi-dimensional drift- diffusion equations and convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' ESAIM Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Model.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' VETTER [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Daus, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J¨ungel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Rigorous mean-field limit and cross diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 70 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 122, 21 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Chen and A.' metadata={'source': 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Positivity-preserving, energy stable numerical schemes for the Cahn–Hilliard equation with logarithmic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' X 3 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 1000031, 29 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' [14] W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Popul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' 21 (1982), 24–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content=' BDF2 FINITE-VOLUME SCHEME 29 Institute of Analysis and Scientific Computing, Technische Universit¨at Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address: juengel@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='at Institute of Analysis and Scientific Computing, Technische Universit¨at Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address: martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='vetter@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} +page_content='at' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfdwTr/content/2301.03200v1.pdf'} diff --git a/dNFKT4oBgHgl3EQfqS7x/vector_store/index.pkl b/dNFKT4oBgHgl3EQfqS7x/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f6eacba215baa44b20fc78304a4619e94671fb43 --- /dev/null +++ b/dNFKT4oBgHgl3EQfqS7x/vector_store/index.pkl @@ -0,0 +1,3 @@ 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Comprehensive Contact Interaction Analysis +R. J. Hern´andez-Pinto,1, ∗ L. X. Guti´errez-Guerrero,2, † A. Bashir,3, 4, ‡ M. A. Bedolla,5, § and I. M. Higuera-Angulo3, ¶ +1Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Sinaloa, +Ciudad Universitaria, Culiac´an, Sinaloa 80000, M´exico +2CONACyT-Mesoamerican Centre for Theoretical Physics, +Universidad Aut´onoma de Chiapas, Carretera Zapata Km. 4, +Real del Bosque (Ter´an), Tuxtla Guti´errez, Chiapas 29040, M´exico +3Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, +Edificio C-3, Ciudad Universitaria, Morelia, Michoac´an 58040, M´exico +4Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA +5Facultad de Ciencias en F´ısica y Matem´aticas, Universidad Aut´onoma de Chiapas, +Carretera Emiliano Zapata Km. 8, Rancho San Francisco, +Ciudad Universitaria Ter´an, Tuxtla Guti´errez, Chiapas 29040, M´exico +We carry out a comprehensive survey of electromagnetic form factors of all light, heavy and heavy- +light ground-state pseudoscalar and scalar mesons. Our analysis is based upon a Schwinger-Dyson +equations treatment of a vector × vector contact interaction. It incorporates confinement and en- +sures axial vector and vector Ward-Takahashi identities are satisfied along with the corresponding +corollaries such as the Goldberger-Treiman relations. The algebraic simplicity of the model allows +us to compute the form factors at arbitrarily large virtualities of the probing photon momentum +squared with relative ease. Wherever possible and insightful, we compare our results for the elec- +tromagnetic form factors and the charge radii with those obtained earlier through Schwinger-Dyson +equations, lattice and with experimental observations available. We also comment on the scope and +shortcomings of the model. +I. +INTRODUCTION +A major challenge in strong interaction physics is the +description of hadrons from first principles, i.e., by com- +mencing from the Lagrangian dynamics of elementary +degrees of freedom of quantum chromodynamics (QCD), +namely, quarks and gluons. The arduous task is then to +describe hadron properties by sewing together the Green +functions of dressed quarks through relativistic bound +state equations. In close analogy with the hydrogen atom +of electrodynamics, the simplest bound states of QCD are +the two-particle systems (mesons) composed of a quark +and an antiquark (q¯q′). Relativistic description of such +states through the Bethe-Salpeter equation (BSE) was +first formulated in Ref. [1]. +Solutions of this equation +presuppose the knowledge of the dressed quark propaga- +tor and the q¯q′ scattering kernel. The quark propagator +is obtained by solving the gap equation while the q¯q′ scat- +tering kernel is constructed by ensuring the axial vector +Ward-Takahashi identity is satisfied. +Several experimental facilities around the globe study +electromagnetic properties of mesons for a gradually in- +creasing interval of momentum squared (Q2) transferred +to the target by the incident probing photon. It enhances +the possibility of observing a gradual transition from non- +∗Electronic address: roger@uas.edu.mx +†Electronic address: lxgutierrez@conacyt.mx +‡Electronic address: adnan.bashir@umich.mx; abashir@jlab.org +§Electronic address: marco.bedolla@unach.mx +¶Electronic address: melany.higuera@umich.mx +perturbative QCD effects to its perturbative domain, fi- +nally settling onto its asymptotic predictions estimated +decades ago, all that in one single experiment. Resulting +elastic or electromagnetic form factors (EFFs) of mesons +thus provide us with an ideal platform to study numerous +uncanny facets of QCD, unfolding the complex structure +of these bound states at varying resolutions scales. +EFFs of pseudoscalar (PS) mesons, pion and kaon in +particular but, have been studied extensively, for exam- +ple, within the functional approach via Schwinger-Dyson +equations (SDEs) [2–4], lattice QCD [5–8], contact inter- +action (CI) [9–11], other models and formalisms, asymp- +totic QCD [12] and, of course, experimentally [13–15]. +π+, K+, K0, D+ and D0 have also been studied in a +hybrid model that combines the generalized Bertlmann- +Martin inequalities with smearing corrections due to rel- +ativistic effects [16]. Light and heavy PS mesons in the +light-front framework have been reported in [17] while +with a QCD potential model there are results for D+, +D0, D+ +s , B+, B0, B0 +s [18]. +PS mesons have additional and important relevance +as they contribute to the hadronic light-by-light (HLbL) +piece of the muon anomalous magnetic moment (AMM), +most dominantly through the single exchange of the light +mesons such as π, η and η′. Furthermore, there are also +loops with charged pions (π±) and kaons (K±). These +contributions have been computed with desirable accu- +racy within the SDE formalism [19–23]. On the other +hand, scalar (S) mesons have been less studied for tech- +nical hindrances and due to the fact their composition +is still debatable. However, similarly to the PS mesons, +they contribute to the AMM of the muon, see the review +arXiv:2301.11881v1 [hep-ph] 27 Jan 2023 + +2 +article [24] and references therein. +Additional overwhelming interest in studying mesons +arises from the fact that their BSE analysis provides an +important first step towards studying baryons in a quark- +diquark picture. +It is firmly established that the non- +pointlike diquark correlations play an important role in +baryons [25]. As a clear illustration, it has been demon- +strated that the quark-diquark picture of a nucleon pro- +duces its mass within 5% of what the Faddeev equation +of a three quark system [26] yields. With this realisa- +tion, it is useful to know that the BSE for diquarks is +exactly the same as that for corresponding mesons up to +a color and charge factor. The chiral partners form a set +of particles which transform into each other under chiral +transformation, like (σ, π) and (ρ, a1). Correspondingly, +there are diquark partners (0−, 0+) and (1+, 1−). The +Bethe-Salpeter amplitudes (BSAs) as well as the EFFs +for σ, π, ρ and a1 yield the corresponding description of +diquarks 0−, 0+, 1+ and 1−. The quark-diquark picture +has been successfully used to calculate EFFs and tran- +sition form factors (TFF) of baryons [27–33]. For com- +prehensive reviews in this connection, one can consult +Refs. [34, 35]. +We have already mentioned CI in the preceding discus- +sion. It is a symmetry preserving vector × vector inter- +action based on a momentum-independent gluon prop- +agator. It results in four quarks interacting at a point. +It was first proposed in [9] to calculate pion EFF. Subse- +quently, CI has extensively been employed to study EFFs +and TFFs of mesons in Refs. [10, 36–39] and of baryons +in Refs. [28–32]. It is a well-known realization that the +EFFs obtained from the CI are harder than the ones +obtained from full QCD predictions. However, the sim- +plicity of the model allows us to perform algebraic calcu- +lations. Moreover, the results obtained provide a bench- +mark to compare and contrast with refined QCD-based +SDE results in order to understand the correct pattern +of dynamical chiral symmetry breaking (DCSB) and the +large Q2 evolution of the EFFs which stems from asymp- +totic QCD where Q2 is much larger than any other mass +scale relevant to the problem. +In this work, we com- +pute EFFs using this momentum-independent interac- +tion, regularized in a symmetry-preserving manner for +a large number of PS and S mesons composed of light +quarks, heavy quarks, and the heavy-light combinations. +We must emphasize that the scalars like σ have a compli- +cated internal structure, possibly including a large com- +ponent of pion correlations. The σ in our article refers to +a quark-antiquark state alone, parity partner of the pion +and approximately twice as heavy as σ(600), [40]. +The article is organized as follows: in Sec. II we collect +the basic ingredients required to carry out the analysis in +the CI model: the dressed quark masses obtained through +the gap equation and the general expression for the BSAs +for PS and S mesons. We discuss the generalities of the +EFFs for PS and S mesons in Sec. III, i.e., the quark- +photon vertex and the triangle diagram which are the +two building blocks to calculate all the meson EFFs in +our formalism. Sec. IV is dedicated to computing EFFs +of the ground state PS mesons. It allows us to evaluate +their charge radii in the limit Q2 ≈ 0, and simultaneously +understand the asymptotic behaviour of the meson EFFs +at large Q2, i.e., Q2 → ∞. In Sec. V, we repeat our study +for the S mesons. A brief summary and perspectives for +future work are presented in Sec. VI. +II. +THE INGREDIENTS +Calculation of the meson EFFs presupposes the knowl- +edge of the dynamically generated dressed valence quark +masses, BSAs of the mesons as well as the quark-photon +interaction vertex at different probing momenta of the +incident photon. In this section, we provide a brief but +self contained introduction to the CI, its essential ingre- +dients and characteristics, namely, the gluon propagator, +the quark-gluon vertex and the set of parameters em- +ployed which, collectively, define the CI. This discussion +is followed by the solution of the gap equation to ob- +tain dynamically generated dressed quark masses. +We +then provide the general expressions of the BSAs for PS +and S mesons. +The corresponding BSE is set up con- +sistently with the gap equation. The numerical solution +is presented in the respective sections dedicated to the +analysis of these mesons. +A. +The Gap Equation +The starting point for our study is the dressed-quark +propagator for a quark of flavor f, which is obtained by +solving the gap equation, +S(p)−1 = iγ · p + mf + Σ(p) , +Σ(p) = 4 +3 +� +d4q +(2π)4 g2Dµν(p − q)γµS(q)Γν(q, p), (1) +where mf is the Lagrangian current-quark mass, Dµν(p) +is the gluon propagator and Γν(q, p) is the quark-gluon +vertex. It is a well-established fact by now that the Lan- +dau gauge gluon propagator saturates in the infrared and +a large effective mass scale is generated for the gluon, see +for example [41–46]. It also leads to the saturation of the +effective strong coupling at large distances. This modern +understanding of infrared QCD forms the defining ideas +of the CI proposed in [47]. We assume that the quarks in- +teract, not through massless vector-boson exchange but +via a CI. Thus the gluon propagator no longer runs with +a momentum scale but is frozen into a CI in keeping with +the infrared properties of QCD, see Fig. 1. Thus +g2Dµν(k) = 4πˆαIRδµν +(2) +where ˆαIR = αIR/m2 +g. The scale mg is for dimensional +reasons and is interpreted as the infrared gluon mass scale +generated dynamically within QCD [9, 48, 49]. We take + +3 +FIG. 1: Diagrammatic representation of the CI, employing +the simplified model of the gluon propagator in Eq. (2). +currently accepted value mg = 500 MeV [41, 50–52]. It is +clear that in the CI gap equation, the effective coupling +which appears is ˆαIR instead of αIR.We choose αIR/π to +be 0.36 so that ˆαIR has exactly the same value as in all +related previous works [9, 49, 53, 54]. The interaction +vertex is bare, i.e., Γν(q, p) = γν. +This constitutes an algebraically simple but useful and +predictive rainbow-ladder truncation of the SDE of the +quark propagator whose solution can readily be written +as follows: +S(q, Mf) ≡ −iγ · q σV (q, Mf) + σS(q, Mf) , +(3) +with +σV (q, Mf) = +1 +q2 + M 2 +f +, σS(q, Mf) = Mf σV (q, Mf) , (4) +where Mf, for the CI, is momentum-independent dynam- +ically generated dressed quark mass determined by +Mf = mf + Mf +4ˆαIR +3π +� ∞ +0 +ds s +1 +s + M 2 +f +. +(5) +Our regularization procedure follows Ref. [55]: +1 +s + M 2 +f += +� ∞ +0 +dτ e−τ(s+M 2 +f ) → +� τ 2 +IR +τ 2 +UV +dτ e−τ(s+M 2 +f ) += e−(s+M 2 +f )τ 2 +UV − e−(s+M 2 +f )τ 2 +IR +s + M 2 +f +, +(6) +where τIR,UV are, respectively, infrared and ultraviolet +regulators. It is apparent from Eq. (6) that a finite value +of τIR ≡ 1/ΛIR implements confinement by ensuring the +absence of quark production thresholds. +Since Eq. (5) +does not define a renormalisable theory, ΛUV ≡ 1/τUV +cannot be removed but instead plays a dynamical role, +setting the scale of all mass dimensioned quantities. Us- +ing Eq. (6), the gap equation becomes +Mf = mf + Mf +4ˆαIR +3π C(M 2 +f ) , +(7) +TABLE I: Ultraviolet regulator and coupling constant for dif- +ferent combinations of quarks in PS mesons. ˆαIR = ˆαIRL/ZH, +where ˆαIRL = 4.57 is extracted from the best-fit to data as +explained in Ref. [39]. ΛIR = 0.24 GeV is a fixed parameter. +quarks +ZH +ΛUV [GeV] +ˆαIR +u, d, s +1 +0.905 +4.57 +c, u, s +3.034 +1.322 +1.50 +c +13.122 +2.305 +0.35 +b, u +11.273 +3.222 +0.41 +b, s +17.537 +3.574 +0.26 +b, c +30.537 +3.886 +0.15 +b +129.513 +7.159 +0.035 +where +C(M 2) +M 2 += Γ(−1, M 2τ 2 +UV) − Γ(−1, M 2τ 2 +IR) +(8) +and Γ(α, x) is the incomplete gamma-function. +We report results for PS mesons using the parameter +values listed in Tables I, II, whose variation with quark +mass was dubbed as heavy parameters in Ref. [49]. In +this approach, the coupling constant and the ultraviolet +regulator vary as a function of the quark mass. +This +behavior was first suggested in Ref. [56] and later adopted +in several subsequent works [39, 49, 54, 57, 58]. Table II +presents the current quark masses mf used herein and +the dynamically generated dressed masses Mf of u, s, c +and b computed from the gap equation, Eq. (7). +TABLE II: Current (mf) and dressed masses (Mf) for quarks +in GeV, required as an input for the BSE and the EFFs. +mu = 0.007 +ms = 0.17 +mc = 1.08 +mb = 3.92 +Mu = 0.367 +Ms = 0.53 +Mc = 1.52 +Mb = 4.75 +A meson can consist of heavy (Q) or light (q) quarks. +We present the study of all heavy (Q ¯Q), heavy-light (Q¯q) +and (review) light (q¯q) mesons. We commence by setting +up the BSE for mesons by employing a kernel which is +consistent with that of the gap equation to obey axial vec- +tor Ward-Takahashi identity and low energy Goldberger- +Treiman relations, see Ref. [9] for details. The PS mesons +are JP C = 0−+ states while the S mesons are JP C = 0++ +states. The solution of the BSE yields BSAs whose gen- +eral form depends not only on the spin and parity of the +meson under consideration but also on the interaction +employed as explained in the next sub-section. + +000004 +B. +Bethe Salpeter Equation +The relativistic bound-state problem for hadrons char- +acterized by two valence-quarks may be studied using +the homogeneous BSE whose diagrammatic representa- +tion can be seen in Fig. 2. This equation is mathemati- +cally expressed as [1], +[Γ(k; P)]tu = +� +d4q +(2π)4 [χ(q; P)]srKrs +tu(q, k; P) , +(9) +where [Γ(k; P)]tu represents the bound-state’s BSA and +χ(q; P) = S(q+P)ΓS(q) is the BS wave-function; r, s, t, u +represent colour, flavor and spinor indices; and K is the +relevant quark-antiquark scattering kernel. +This equa- +tion possesses solutions on that discrete set of P 2-values +for which bound-states exist. +A general decomposition of the BSA for the PS and +the S mesons (f1 ¯f2) in the CI has the following form +ΓP S(P) = iγ5 EP S(P) + +1 +2MR +γ5γ · P FP S(P) , +ΓS(P) = ID ES(P) . +(10) +Note that Ei(P) and Fi(P) with i ∈ {PS, S} are known +as the BSAs of the meson under consideration, P is its +total momentum, ID is the identity matrix and MR = +Mf1M ¯ +f2/[Mf1 + M ¯ +f2] is the reduced mass of the system. +Eq. (9) has a solution when P 2 = −M 2 +M with MM being +the meson mass. After this initial and required set up of +the gap equation and the BSE, we now turn our attention +to the description of the EFFs of mesons. +III. +ELECTROMAGNETIC FORM FACTORS +The EFFs provide crucial information on the internal +structure of mesons. At low momenta, EFFs allow us to +unravel the complexities of non-perturbative QCD, i.e., +confinement, DCSB and the fully dressed quarks. At high +energies, we expect to confirm the validity of asymptotic +QCD for its realistic models while at intermediate ener- +gies, we observe a smooth transition from one facet of +FIG. 2: Diagrammatic representation of the BSE. Blue (solid) +circles represent dressed quark propagators S, red (solid) cir- +cle is the meson BSA Γ while the blue (solid) rectangle is the +dressed-quark-antiquark scattering kernel K. +strong interactions to the other, all in one single experi- +ment if we are able to chart out a wide range of momen- +tum transfer squared Q2 without breaking up the mesons +under study. While there are plenty of studies on the +pion EFFs, only a few are found about heavy-quarkonia +and practically none on heavy-light mesons. The process +involves an incident photon which probes mesons, inter- +acting with the electrically charged quarks making up +these two-particles bound states. Therefore, it is natural +to start this section by looking at the the structure of the +quark-photon vertex within the CI. +A. +The Quark-Photon Vertex +The quark-photon vertex, denoted by Γγ +µ(k+, k−, Mf1), +is related to the quark propagator through the following +vector Ward-Takahashi identity: +iPµΓγ +µ(k+, k−, Mf1) = S−1(k+, Mf1) − S−1(k−, Mf1) . +(11) +This identity is crucial for a sensible study of a bound- +state’s EFF. It is determined through the following inho- +mogeneous BSE, +Γγ +µ(Q, Mf1) = +γµ − 16πˆαIR +3 +� +d4q +(2π)4 γαχµ(q+, q, Mf1)γα , (12) +where χµ(q+, q, Mf1) = S(q + P, Mf1)Γµ(Q)S(q, Mf1). +Owing to the momentum-independent nature of the in- +teraction kernel, the general form of the solution is +Γγ +µ(Q, Mf1) = γL +µ (Q)PL(Q2, Mf1) + γT +µ (Q)PT (Q2, Mf1), +(13) +where γL +µ + γT +µ = γµ and +γT +µ (Q) = γµ − γ · Q +Q2 +Qµ . +(14) +Inserting this general form into Eq. (12), one readily ob- +tains (on simplifying notation) +PL = 1 , +PT = +1 +1 + Kγ(Q2, Mf1), +(15) +with +Kγ(Q2,Mf1) = 4ˆαIR +3π +� 1 +0 +dα α(1 − α)Q2 ¯C1(ω) , +(16) +where +¯C1(z) = − d +dz C(z) = Γ(0, z τ 2 +UV) − Γ(0, z τ 2 +IR) +(17) + +is +K +is5 +0 +2 +4 +6 +8 +10 +Q +2(GeV +2) +0 +0.5 +1 +1.5 +2 +PT(Q +2) +FIG. 3: Dressing function of the transverse quark-photon ver- +tex, PT (Q2), in Eq. (15). +and +ω = ω(M 2 +f1, α, Q2) = M 2 +f1 + α(1 − α)Q2 . +(18) +One can clearly observe from Fig. 3 that PT (Q2) → 1 +when Q2 → ∞, yielding the perturbative bare vertex +γµ as expected. This quark-photon vertex provides us +with the required electromagnetic interaction capable of +probing the EFFs of mesons through a triangle diagram +which keeps the identity of the meson bound state intact. +B. +The Triangle Diagram +Let us start from the general considerations for the +electromagnetic interactions of mesons. In the impulse +approximation, the MγM vertex, which describes the +interaction between a meson (f1 ¯f2) and a photon, reads +ΛM,f1 = Nc +� +d4ℓ +(2π)4 Tr GM,f1 , +(19) +where +GM,f1 = iΓM(kf) S(ℓ + ki, Mf1) iΓλ(Q, Mf1) +× S(ℓ + kf, Mf1)i¯ΓM(−ki) S(ℓ, M ¯ +f2) . +The notation assumes that it is the quark f1 which in- +teracts with the photon while the antiquark ¯f2 remains +a spectator. We define ΛM, ¯ +f2 similarly. Furthermore, we +denote the incoming photon momentum by Q while the +incoming and outgoing momenta of M by: ki = k − Q/2 +and kf = k + Q/2, respectively. The assignments of mo- +menta are shown in the triangle diagram of Fig. 4. +ΛM,f corresponds to the EFFs of different mesons un- +der study. The contribution from the interaction of the +photon with quark f1 can be represented as F M,f1(Q2) +(stemming from ΛM,f1) while the contribution arising +from its interaction with quark ¯f2 can be represented as +F M, ¯ +f2(Q2) (coming from ΛM, ¯ +f2). The total form factor +F M(Q2) is defined as follows [59]: +F M(Q2) = ef1F M,f1(Q2) + e ¯ +f2F M, ¯ +f2(Q2) , +(20) +where ef1 and e ¯ +f2 are the quark and the antiquark electric +charges, respectively 1. Both for PS and S mesons, F M,f1 +is straightforwardly related to ΛM,f1: +ΛS,f1 = −2kλF S,f1 , ΛP S,f1 = −2kλF P S,f1 . +(21) +All information necessary for the calculation of the EFFs +is now complete. We can employ numerical values of the +parameters listed in Tables I and II and proceed to com- +pute the EFFs. Our evaluated analytical expressions and +numerical results for PS and S mesons occupy the details +of the next two sections. Keeping in mind that the pairs +of (PS, S) mesons can be considered as parity partners, +we embark upon their treatment in the following sections +in that order. +IV. +PSEUDOSCALAR MESONS +We start with a detailed discussion and results on the +ground state PS mesons. These are negative parity, zero +angular momentum 0−+ states and occupy a special role +in hadron physics. +Simultaneously these are the sim- +plest bound states of a quark and antiquark and also +emerge as Goldstone bosons associated with DCSB. Pi- +ons are the lightest hadrons and are produced copiously +in collider machines at all energies. The pion cloud effect +substantially contributes to several static and dynami- +FIG. 4: +The triangle diagram for the impulse approximation +to the MγM vertex. +1 For neutral mesons composed of same flavored quarks, the total +EFF is simply F M = F M,f1. + +e + kf +e + ki +k, +k; +M +CM +e6 +0 +2 +4 +6 +8 +Q2[GeV2] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fem +PS (Q2) +cb +cs +ud +ub +us +0 +2 +4 +6 +8 +Q2[GeV2] +bb +ss +cc +0 +2 +4 +6 +8 +10 +Q2[GeV2] +sb +cu +FIG. 5: +EFFs of PS mesons in a CI model. Left panel: electrically charged mesons composed of quarks of different flavors. +Central panel: quarkonia including a hypothetical ground state strangeonium (s¯s). Right panel: electrically neutral mesons +composed of quarks of different flavors. +cal hadron properties. +Therefore, understanding their +internal structure has been of great interest both for ex- +perimenters and theoreticians. The study of PS mesons +is crucial in understanding the capabilities and limita- +tions of the CI model employed in this work to repro- +duce and predict phenomenological results. +Being the +Goldstone bosons associated with DCSB, their analysis +requires care in treating the associated subtleties. From +Eqs. (10), we can see that the BSA of PS mesons is the +only one to be composed of two terms, necessary to en- +sure the axial vector Ward-Takahashi identity and the +Goldberger-Treiman relations are exactly satisfied. +In +this article, we extend and expand the work presented +in [9, 39, 57] and compute the EFFs of a larger number +of PS mesons composed of qq, qQ and QQ quarks. With +TABLE III: Calculated values for the BSAs and masses for +PS mesons in CI model using the parameters in Tables I and II +(compare the parameters with the ones in Ref. [49]). +Mass[GeV] EP S FP S +mexp +P S [GeV] error [%] +u¯d +0.139 +3.59 0.47 +0.139 +0.008 +u¯s +0.499 +3.81 0.59 +0.493 +1.162 +s¯s +0.701 +4.04 0.75 +— +— +c¯u +1.855 +3.03 0.37 +1.864 +0.494 +c¯s +1.945 +3.24 0.51 +1.986 +1.183 +u¯b +5.082 +3.72 0.21 +5.279 +3.735 +s¯b +5.281 +2.85 0.21 +5.366 +1.586 +c¯b +6.138 +2.58 0.39 +6.274 +2.166 +c¯c +2.952 +2.15 0.40 +2.983 +1.053 +b¯b +9.280 +2.04 0.39 +9.398 +1.262 +straightforward algebraic manipulations: +F P S,f1 = PT (Q2) +� +E2 +P ST P S +EE(Q2) + EP SFP ST P S +EF (Q2) ++F 2 +P ST P S +F F (Q2) +� +, +(22) +where +T P S +EE(Q2) = +3 +4π2 +� � 1 +0 +dα C1(ω1) ++ 2 +� 1 +0 +dα dβ α AP S +EE C2(ω2) +� +, +T P S +EF (Q2) = − 3 +2π2 +1 +MR +� 1 +0 +dα dβ α +� +A(1) +EF C1(ω2) ++ (A(2) +EF − ω2A(1) +EF ) C2(ω2) +� +, +T P S +F F (Q2) = +3 +4π2 +1 +M 2 +R +� 1 +0 +dα dβ α +� +A(1) +F F C1(ω2) ++ (A(2) +F F − ω2A(1) +F F ) C2(ω2) +� +, +and +ω1 = ω1(Mf1, α, Q2) = M 2 +f1 + α Q2(1 − α) , +ω2 = ω2(Mf1, M ¯ +f2, α, β, MM) = α M 2 +f1 + (1 − α)M 2 +¯ +f2 +− α(1 − α) M 2 +M + α2 β (1 − β) Q2 , +¯C2(z) = (exp(−z τuv) − exp(−z τir))/(2z) . +(23) + +7 +The coefficients Ai are given by the following expressions: +AP S +EE = α(M 2 +f1+M 2 +M) + 2(1 − α)Mf1M ¯ +f2 + (α − 2)M 2 +¯ +f2 , +A(1) +EF = Mf1 + M ¯ +f2 , +A(2) +EF = 2M 2 +f1M ¯ +f2 − αMf1(4(α − 1)M 2 +M + αQ2) ++M ¯ +f2(2(α − 1)2M 2 +M + αQ2(2α(β − 1)β+α − 1)) , +A(1) +F F = (3α − 2)M 2 +M + αQ2 , +A(2) +F F = 2α((α − 1)2M 4 +M ++ αM 2 +MQ2(3αβ2 − 3αβ + α − 2β2 + 2β − 1)) ++2αM 2 +MM 2 +f1 − 2Mf1M ¯ +f2(2(α − 1)M 2 +M + αQ2) . +The resulting EFFs for charged as well as neutral mesons +are shown in Fig. 5. For the practical utility and intuitive +understanding of their low and large Q2 behavior, we +perform an interpolation for PS mesons EFF in the range +Q2 ∈ [0, 8M 2 +M]. We adopt the following functional form: +F P S(Q2) = eM + aP S Q2 + bP S Q4 +1 + cP S Q2 + dP S Q4 , +(24) +where eM := F P S(Q2 = 0) is the electric charge of the +meson and aP S, bP S, cP S, dP S are the fitted coefficients. +The best fit corresponds to the values listed in Table IV. +The fit of Eq. (24) resonates with our observation that +the EFFs of PS mesons tend to constant values for large +Q2 when it becomes by far the largest energy scale in the +problem. It is a straightforward consequence of CI treat- +ment, and it is characteristic of a point-like interaction +which leads to harder EFF. However, the heavy as well +as heavy-light mesons approach a constant value much +slower than the light ones. This comparative large Q2 +behavior of EFFs owes itself to the fact that Q2 becomes +larger than all other energy scales at much higher values. +TABLE IV: Parameters from the fit of Eq.(24) for PS mesons +aP S +bP S +cP S +dP S +u¯d +0.330 +0.029 +1.190 +0.068 +u¯s +0.335 +0.029 +1.092 +0.065 +s¯s +0.328 +0.040 +0.874 +0.092 +c¯u +0.616 +−0.001 +1.370 +0.109 +c¯s +0.615 +0.028 +0.897 +0.111 +u¯b +1.143 +0.033 +1.921 +0.146 +s¯b +0.218 +0.000 +0.840 +0.009 +c¯b +0.333 +0.003 +0.493 +0.021 +c¯c +1.778 +0.057 +1.994 +0.334 +b¯b +0.099 +0.000 +0.127 +0.002 +The behavior of the form factors at the other extreme, +Q2 ≃ 0 allows us to extract charge radii: +r2 +M = −6 dFM(Q2) +dQ2 +���� +Q2=0 +. +(25) +For c¯u and s¯b states, which are normalized to F M(0) = 0, +we define r2 +M with a positive sign in the above equation. +The charge radii set the trend for the subsequent evolu- +tion of the form factors as a function of Q2, specially for +its small and intermediate values. Fig. 6 depicts charge +radii for all the PS mesons studied, allowing for a 5% +variation around the central value. With this permitted +spread in the charge radii, one can obtain a band for the +Q2 evolution of the EFFs. To avoid over-crowding, we +have avoided depicting such a band for each EFF. How- +ever, Fig. 7 shows a representative plot for the pion per- +mitting a 5% variation in its charge radius in conjunction +with the available experimental results. +Finally we list the central values of all ground state PS +mesons charge radii in Table V, along with a direct com- +parison with available experimental observations, lattice +results and the SDE findings. Moreover, we also report +the transition charge radii of light PS mesons and flavor- +less neutral heavy PS mesons to two photons invoking +the following analytical parameter fit [60]: +rt +M = +r0 +1 + (MM/mt) ln(1 + MM/mt) , +(26) +where r0 = 0.67 fm and mt = 1.01 GeV. It also yields +reasonable results for the π point (mass = 0.139 GeV) +and the K point (mass = 0.493 GeV) as they are made +of light quarks. But we cannot expect it to serve exactly +as it is for mesons with vastly off-balanced quark masses. +However, if CI results were to follow this formula, we +would only need to assign r0 = 0.458 fm. The last row +of Table V lists the resulting values which we denote as +rt +M(CI). Let us now summarize our findings and make +explicit comparisons with related works: +• As desired, pion EFF and its charge radius agree +with the first results employing the CI [9]. As an +add-on, in this article we allow for a 5% variation of +the pion charge radius to see its effect on the evolu- +tion of the EFF as a function of Q2, Fig. 7. A small +variation of the initial slope of the curve Q2 ≃ 0 +opens a noticeable spread for large Q2 but keeps +the qualitative and quantitative behaviour fully in- +tact. +• In Fig. 8, we draw kaon EFF over the range of +Q2 values where (relatively poor) experimental ob- +servations are available. Although large error bars +prevent us from commenting decisively on the va- +lidity of the CI but we expect it will yield harder +results as compared to precise experimental mea- +surements whenever these results will become avail- +able. Our reported value of its charge radius is an +indication of this behavior. + +8 +TABLE V: +The charge radii of PS meson systems, calculated with the CI model, refined SDE studies, lattice QCD and +extraction from data, in a hybrid model (HM), light-front framework (LFF) and a QCD potential model (PM). In the two rows +after the experimental results, we also provide the best fit results for the transition charge radii of PS mesons to γγ∗, Eq. (26) +and the same fit adapted to the CI. All results are presented in fm. +u¯d +u¯s +s¯s +c¯u +c¯s +u¯b +s¯b +c¯b +c¯c +b¯b +Our Result +0.45 +0.42 +0.36 +0.36 +0.26 +0.34 +0.24 +0.17 +0.20 +0.07 +SDE [4, 23] +0.676 ± 0.002 +0.593 ± 0.002 +- +- +- +- +- +- +0.24 +0.09 +Lattice [7, 8, 63] +0.648 ± 0.141 +0.566 (extracted) +- +- +- +- +- +- +0.25 +- +Exp. [64] +0.659 ± 0.004 +0.560 ± 0.031 +- +- +- +- +- +- +- +- +rt +M [60] +0.658 +0.568 +- +- +- +- +- +- +0.13 +0.03 +rt +M(CI) +0.45 +0.38 +0.33 +- +- +- +- +- +0.09 +0.02 +HM [16] +0.66 +0.65 +- +0.47 +0.50 +- +- +- +- +- +LFF [17] +0.66 +0.58 +- +0.55 +0.35 +0.61 +0.34 +0.20 +- +- +PM [18] +- +- +- +0.67 +0.46 +0.73 +0.46 +- +- +- ++ +K + +h0 +D0 +B + +D + +s +B0 +s +c +B + +c +b +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +charge radii [fm] +0.45 +0.42 +0.36 +0.36 +0.34 +0.26 +0.24 +0.20 +0.17 +0.07 +FIG. 6: Charge radii of ground state PS mesons in the CI. +• As depicted in Table V, pion and kaon charge +radii [4, 7, 8, 23, 63, 64] are known experimentally +and through lattice and SDE studies. As CI EFFs +come out to be harder than full QCD predictions, +we expect our PS mesons charge radii to under- +shoot the exact results. This is precisely what we +observe for the pion and the kaon. The percentage +relative difference between the experimental value +and our calculation for the pion charge radius is ap- +proximately 32%, while for the kaon charge radius +is slightly less, 25%. Similar difference between the +SDE and the CI results for heavy quarkonia is ob- +served: For ηc, it is 20% while for ηb is is 22%, not +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Q2[GeV2] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Fem(Q2) +Contact Interaction +JLab (W=2.20 GeV) +JLab (W=1.95 GeV) +Amendolia et al. +FIG. 7: EFF for π-meson. +The central curve is obtained +using the τUV value from the Table I. The filled band allows +for a 5% variation in the charge radius. Dots represent the +experimental data from Refs. [14, 15, 61]. +too dissimilar. This comparatively analogous be- +havior augments our expectation that we will be in +the same ballpark for the PS mesons whose charge +radii are neither known experimentally as yet nor +lattice offers any results. +• There are no experimental or lattice (to the best of +our knowledge) results available for c¯u, c¯s, u¯b, s¯b +and c¯b mesons for comparison. However, the gen- +eral trend of decreasing charge radii with increasing +constituent quark mass seems reassuring, e.g., the + +9 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +Q2[GeV2] +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +Fem +K (Q2) +Contact Interaction +Dally et al. +Amendolia et al. +FIG. 8: EFF for K-meson. The central curve of the (blue) +band is obtained by using the ΛUV value from Table I. The +filled (blue) band allows for a 5% variation in the charge ra- +dius. The experimental data is from Ref. [61]. +0 +1 +2 +3 +4 +5 +Q2[GeV2] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Fem +c (Q2) +Contact Interaction +Dudek at al. +FIG. 9: EFF for ηc-meson. The lower (green) solid curve is +the lattice result, Ref. [62]. The central curve of the (blue) +band is obtained using the ΛUV value from Table I. The filled +(blue) band allows for a 5% variation in the charge radius. +following hierarchies are noticeable: +ru¯d > ru¯s > rc¯u > ru¯b , +ru¯s > rs¯s > rc¯s > rs¯b , +rc¯u > rc¯s > rc¯c > rc¯b , +ru¯u > rs¯s > rc¯c > rb¯b . +We must emphasize that the CI is only a simple +‘ +FIG. 10: The S meson, e.g., σ is viewed as the parity partner +of the pion π. Note that the scalars in this article only refer +to their quark-antiquark content. +model. Refined QCD calculations are required to +confirm or refute these findings. +This concludes our detailed analysis of all the ground +state PS heavy (Q ¯Q), heavy-light (Q¯q) as well as light +(q¯q) mesons. +We now turn our attention to a similar +analysis of the scalar mesons. +V. +SCALAR MESONS +Recall that an S meson is a 0++ state. It can be con- +sidered as the chiral partner of the PS meson Fig. 10. +We work under the assumption that all states are purely +quark-antiquark states. Then, for example, the states π +and σ get transformed into each other through the fol- +lowing chiral transformation: +q → e−iγ5 τ +2 ·θq . +(27) +The explicit expression for the EFFs for S mesons with +mass MM constituted from a quark f1 and an antiquark +¯f2 is given by Eq. (20) with +F S,f1 = PT (Q2)E2 +ST S +EE(Q2) , +(28) +where +T S +EE(Q2) = − 3 +4π2 +� � 1 +0 +dα C1(ω1) ++ 2 +� 1 +0 +dα dβ α AS +EE C2(ω2) +� +, +(29) +with +AS +EE = αMf1 − 2(1 − α)Mf1M ¯ +f2 ++ (α − 2)M 2 +¯ +f2 + αM 2 +M . +(30) +Note the close resemblance between T S +EE and T P S +EE. As +expected, there are only sign differences between the two +due to the presence, or absence, of the γ5 matrix. +In +Table VI, we present the parameters used for S mesons + +JPC = 0-+ +JPC = 0++ +b +T +0 +q +q +q +q +iq5q +bb10 +0 +2 +4 +6 +8 +Q2[GeV2] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fem +S (Q2) +cb +ub +cs +us +ud +0 +2 +4 +6 +8 +Q2[GeV2] +ss +bb +cc +0 +2 +4 +6 +8 +10 +Q2[GeV2] +sb +cu +FIG. 11: EFFs for S mesons in the CI model. Left panel: electrically charges mesons composed of quarks of different flavors. +Central panel: quarkonia including a hypothetical ground state strangeonium (s¯s). Right panel: electrically neutral mesons +composed of quarks of different flavors. EFFs of electrically neutral but flavored mesons have been normalized to F S(0) = 0. +TABLE VI: Ultraviolet regulator and the coupling constant +for different combinations of quarks in S mesons. As before, +ˆαIR = ˆαIRL/ZH, where ˆαIRL = 4.57 is extracted from a best- +fit to data as explained in Ref. [39]. ΛIR = 0.24 GeV. +quarks +ZH +ΛUV [GeV] +ˆαIR +u, d, s +1 +0.905 +4.57 +c, u +3.034 +1.322 +1.50 +c, s +3.034 +2.222 +1.50 +c +13.122 +2.305 +0.35 +b, u +18.473 +10.670 +0.25 +b, s +29.537 +11.064 +0.15 +b, c +34.216 +14.328 +0.13 +b +127.013 +26.873 +0.036 +in order to compute the masses, amplitudes and charge +radii. +We enlist the masses and BSAs of S mesons in +Table VII while the EFFs are depicted in Fig. 11. On the +right and central panels we present the results for neutral +mesons while the left panel displays the EFFs of charged +mesons. We emphasize that for electrically neutral but +flavored S mesons, we normalize the EFFs to zero at +Q2 = 0, while for flavorless mesons, the normalization is +F S(0) = 1 to be consistent with the definition employed +in Eq. (20). We again perform a fit in the range Q2 ∈ +[0, 8M 2 +M], where MM is the mass of the S meson. All the +TABLE VII: Computed values of the S mesons masses and +BSAs in the CI model, see [49] for comparison, using the +parameters listed in Tables II and VI. +Mass [GeV] +ES +mexp +S +[GeV] +error [%] +u¯d +1.22 +0.66 +— +— +u¯s +1.38 +0.65 +— +— +s¯s +1.46 +0.64 +— +— +c¯u +2.31 +0.39 +2.30 +0.19 +c¯s +2.42 +0.42 +2.32 +3.54 +u¯b +5.30 +1.53 +— +— +s¯b +5.64 +0.26 +— +— +c¯b +6.36 +1.23 +6.71 +5.26 +c¯c +3.33 +0.16 +3.42 +2.73 +b¯b +9.57 +0.69 +9.86 +2.95 +curves are faithfully reproduced by the following choice: +F S(Q2) = eM + aS Q2 + bS Q4 +1 + cS Q2 + dS Q4 , +(31) +where eM := F S(Q2 = 0) is the electric charge of the +meson and aS, bS, cS, dS are the parameters of the fit. +These values for S mesons are listed in Table IX. Based +on these numbers, we can immediately infer the large Q2 +behavior of these EFFs. The coefficient bS ≈ 0 for all S +mesons under consideration. Therefore, the EFFs for S +mesons fall as 1/Q2 for large Q2. +We present the numerical values of the charge radii + +11 +TABLE IX: Parameters for the fit in Eq. (31) for S mesons. +aS +bS +cS +dS +u¯d +0.286 +0.003 +1.543 +0.617 +u¯s +0.266 +0.002 +1.486 +0.629 +s¯s +0.217 +0.001 +1.271 +0.542 +c¯u +0.759 +−0.005 +0.680 +0.641 +c¯s +0.004 +0.001 +0.783 +0.047 +u¯b +0.984 +0.001 +1.619 +0.087 +s¯b +0.210 +0.001 +0.175 +0.115 +c¯b +0.289 +0.001 +0.743 +0.026 +c¯c +0.217 +0.001 +0.860 +0.673 +b¯b +0.269 +0.000 +1.607 +0.020 +0 +2 +4 +6 +8 +10 +Q2[GeV2] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fem(Q2) +FIG. 12: +EFF for σ-meson. The central curve is obtained +using the ΛUV value from Table VI while the band represents +a 5% variation in the charge radius. +for S mesons in Table X. We must reiterate that for the +S mesons there are no reported measurements of their +charge radii. Theoretical results are also scarce for any +direct and meaningful comparison. It is worth mention- +ing again that the internal structure of scalar mesons is +not well-established. Our results are based on consider- +ing them as effective quark-antiquark states. +We would like to remind the reader that we again allow +for a 5% variation in the charge radii of S mesons. How- +ever, in Fig. 11, we present the EFFs only for their cen- +tral values for visual clarity, refraining from showing the +corresponding band to avoid possible overlapping. How- +ever, in Fig. 12, we depict a representative plot with a 5% +variation in the charge radius for the lightest scalar me- +son, σ, alone. Other mesons have similar bands. Finally, +in Fig. 13, we plot the charge radii, extracted from the +EFFs, as a function of the S meson mass. In general, the +charge radii decrease when the S meson masses increase +just as we observed for the PS mesons. +K * +0 +f0 +D * +0 +D * +s0 +c0 +B * +0 +B * +s0 +B * +c0 +b0 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +charge radii [fm] +0.55 +0.54 +0.50 +0.47 +0.44 +0.43 +0.42 +0.41 +0.40 +0.39 +FIG. 13: Charge radii of S mesons. +VI. +CONCLUSIONS +In this work we present an exhaustive computation of +EFFs employing the CI model for twenty ground state PS +and S mesons. Note that the CI findings for light mesons +and heavy quarkonia are already found in the literature +as mentioned before [9, 39, 56, 57]. We include these re- +sults for the sake of completeness and as a guide to pin +down the best parameters in order to explore heavy-light +systems. We thus report first results on the latter mesons +within this model/formalism. We expect these new EFFs +to be harder than the exact QCD predictions, especially +for the PS mesons due to the necessary inclusion of the +F-amplitude. We also anticipate the charge radii to be +in the ballpark of a (20-25)% error in light of the results +where comparison with realistic studies and/or experi- +ment has been possible. +Furthermore, we analyze the sensitivity of the evolu- +tion of the EFFs by a change in appropriate parameters +to allow for a 5% variation in the charge radii of the +corresponding mesons. +The evolution band has been +shown explicitly for π, K, σ and ηc alone to avoid +over-crowding in other collective plots. +However, it is +worth mentioning that the corresponding bands in other +EFFs are almost identical. Interpolations have also been +provided in Eqs. (24, 31) and Tables IV and IX which +allow for a convenient algebraic analysis of the behavior +of the EFFs in the momentum range that we mentioned + +12 +TABLE X: The charge radii for S mesons. All quantities are reported in fm. +u¯d +u¯s +s¯s +c¯u +c¯s +u¯b +s¯b +c¯b +c¯c +b¯b +Our Result +0.55 +0.54 +0.50 +0.47 +0.44 +0.42 +0.41 +0.40 +0.43 +0.39 +above and for any application the reader may deem +useful. We plan to recalculate these EFFs for vector and +axial vector mesons followed by the same computation +within a more realistic algebraic model and in the long +run for truncations more akin to full non-perturbative +QCD. +It +is +also +straightforward +to +generalise +our +analysis to study diquarks EFFs which are crucial +in the subsequent computation of baryons EFFs such +as the ones reported recently in [32]. All this is for future. +Acknowledgments +L. X. Guti´errez-Guerrero wishes to thank the support +from C´atedras CONACyT program of Mexico. The work +of R. J. Hern´andez-Pinto is supported by CONACyT +(Mexico) Project No. 320856 (Paradigmas y Controver- +sias de la Ciencia 2022), Ciencia de Frontera 2021-2042 +and Sistema Nacional de Investigadores as well as by +PROFAPI 2022 Grant No. +PRO A1 024 (Universidad +Aut´onoma de Sinaloa). The work of A. Bashir is sup- +ported in part by the US Department of Energy (DOE) +Contract No. +DE-AC05-06OR23177, under which Jef- +ferson Science Associates, LLC operates Jefferson Lab. +A. Bashir also acknowledges Coordinaci´on de la Investi- +gaci´on Cient´ıfica of the Universidad Michoacana de San +Nicol´as de Hidalgo grant 4.10 and the Fulbright-Garc´ıa +Robles scholarship for his stay as a visiting scientist at the +Thomas Jefferson National Accelerator Facility, Newport +News, Virginia, USA. We thank Jozef Dudek and Chris- +tine Davies for helpful communication on lattice results +on EFFs and charge radii. +[1] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 +(1951). +[2] P. Maris and P. C. 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(Particle Data Group), PTEP 2020, +083C01 (2020). + diff --git a/e9FKT4oBgHgl3EQfsC7L/content/tmp_files/load_file.txt b/e9FKT4oBgHgl3EQfsC7L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e74bf498810854667b32c071cb69b85b1f5fe78a --- /dev/null +++ b/e9FKT4oBgHgl3EQfsC7L/content/tmp_files/load_file.txt @@ -0,0 +1,1384 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf,len=1383 +page_content='Electromagnetic Form Factors and Charge Radii of Pseudoscalar and Scalar Mesons: A Comprehensive Contact Interaction Analysis R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Hern´andez-Pinto,1, ∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Guti´errez-Guerrero,2, † A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir,3, 4, ‡ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bedolla,5, § and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Higuera-Angulo3, ¶ 1Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Sinaloa, Ciudad Universitaria, Culiac´an, Sinaloa 80000, M´exico 2CONACyT-Mesoamerican Centre for Theoretical Physics, Universidad Aut´onoma de Chiapas, Carretera Zapata Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 4, Real del Bosque (Ter´an), Tuxtla Guti´errez, Chiapas 29040, M´exico 3Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio C-3, Ciudad Universitaria, Morelia, Michoac´an 58040, M´exico 4Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA 5Facultad de Ciencias en F´ısica y Matem´aticas, Universidad Aut´onoma de Chiapas, Carretera Emiliano Zapata Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 8, Rancho San Francisco, Ciudad Universitaria Ter´an, Tuxtla Guti´errez, Chiapas 29040, M´exico We carry out a comprehensive survey of electromagnetic form factors of all light, heavy and heavy- light ground-state pseudoscalar and scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Our analysis is based upon a Schwinger-Dyson equations treatment of a vector × vector contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It incorporates confinement and en- sures axial vector and vector Ward-Takahashi identities are satisfied along with the corresponding corollaries such as the Goldberger-Treiman relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The algebraic simplicity of the model allows us to compute the form factors at arbitrarily large virtualities of the probing photon momentum squared with relative ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Wherever possible and insightful, we compare our results for the elec- tromagnetic form factors and the charge radii with those obtained earlier through Schwinger-Dyson equations, lattice and with experimental observations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We also comment on the scope and shortcomings of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' INTRODUCTION A major challenge in strong interaction physics is the description of hadrons from first principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', by com- mencing from the Lagrangian dynamics of elementary degrees of freedom of quantum chromodynamics (QCD), namely, quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The arduous task is then to describe hadron properties by sewing together the Green functions of dressed quarks through relativistic bound state equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In close analogy with the hydrogen atom of electrodynamics, the simplest bound states of QCD are the two-particle systems (mesons) composed of a quark and an antiquark (q¯q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Relativistic description of such states through the Bethe-Salpeter equation (BSE) was first formulated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Solutions of this equation presuppose the knowledge of the dressed quark propaga- tor and the q¯q′ scattering kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The quark propagator is obtained by solving the gap equation while the q¯q′ scat- tering kernel is constructed by ensuring the axial vector Ward-Takahashi identity is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Several experimental facilities around the globe study electromagnetic properties of mesons for a gradually in- creasing interval of momentum squared (Q2) transferred to the target by the incident probing photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It enhances the possibility of observing a gradual transition from non- ∗Electronic address: roger@uas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='mx †Electronic address: lxgutierrez@conacyt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='mx ‡Electronic address: adnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='bashir@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='mx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' abashir@jlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='org §Electronic address: marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='bedolla@unach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='mx ¶Electronic address: melany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='higuera@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='mx perturbative QCD effects to its perturbative domain, fi- nally settling onto its asymptotic predictions estimated decades ago, all that in one single experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Resulting elastic or electromagnetic form factors (EFFs) of mesons thus provide us with an ideal platform to study numerous uncanny facets of QCD, unfolding the complex structure of these bound states at varying resolutions scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' EFFs of pseudoscalar (PS) mesons, pion and kaon in particular but, have been studied extensively, for exam- ple, within the functional approach via Schwinger-Dyson equations (SDEs) [2–4], lattice QCD [5–8], contact inter- action (CI) [9–11], other models and formalisms, asymp- totic QCD [12] and, of course, experimentally [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' π+, K+, K0, D+ and D0 have also been studied in a hybrid model that combines the generalized Bertlmann- Martin inequalities with smearing corrections due to rel- ativistic effects [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Light and heavy PS mesons in the light-front framework have been reported in [17] while with a QCD potential model there are results for D+, D0, D+ s , B+, B0, B0 s [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' PS mesons have additional and important relevance as they contribute to the hadronic light-by-light (HLbL) piece of the muon anomalous magnetic moment (AMM), most dominantly through the single exchange of the light mesons such as π, η and η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Furthermore, there are also loops with charged pions (π±) and kaons (K±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' These contributions have been computed with desirable accu- racy within the SDE formalism [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' On the other hand, scalar (S) mesons have been less studied for tech- nical hindrances and due to the fact their composition is still debatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, similarly to the PS mesons, they contribute to the AMM of the muon, see the review arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='11881v1 [hep-ph] 27 Jan 2023 2 article [24] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Additional overwhelming interest in studying mesons arises from the fact that their BSE analysis provides an important first step towards studying baryons in a quark- diquark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is firmly established that the non- pointlike diquark correlations play an important role in baryons [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As a clear illustration, it has been demon- strated that the quark-diquark picture of a nucleon pro- duces its mass within 5% of what the Faddeev equation of a three quark system [26] yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' With this realisa- tion, it is useful to know that the BSE for diquarks is exactly the same as that for corresponding mesons up to a color and charge factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The chiral partners form a set of particles which transform into each other under chiral transformation, like (σ, π) and (ρ, a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Correspondingly, there are diquark partners (0−, 0+) and (1+, 1−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The Bethe-Salpeter amplitudes (BSAs) as well as the EFFs for σ, π, ρ and a1 yield the corresponding description of diquarks 0−, 0+, 1+ and 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The quark-diquark picture has been successfully used to calculate EFFs and tran- sition form factors (TFF) of baryons [27–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' For com- prehensive reviews in this connection, one can consult Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We have already mentioned CI in the preceding discus- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is a symmetry preserving vector × vector inter- action based on a momentum-independent gluon prop- agator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It results in four quarks interacting at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It was first proposed in [9] to calculate pion EFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Subse- quently, CI has extensively been employed to study EFFs and TFFs of mesons in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [10, 36–39] and of baryons in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is a well-known realization that the EFFs obtained from the CI are harder than the ones obtained from full QCD predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, the sim- plicity of the model allows us to perform algebraic calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Moreover, the results obtained provide a bench- mark to compare and contrast with refined QCD-based SDE results in order to understand the correct pattern of dynamical chiral symmetry breaking (DCSB) and the large Q2 evolution of the EFFs which stems from asymp- totic QCD where Q2 is much larger than any other mass scale relevant to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In this work, we com- pute EFFs using this momentum-independent interac- tion, regularized in a symmetry-preserving manner for a large number of PS and S mesons composed of light quarks, heavy quarks, and the heavy-light combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We must emphasize that the scalars like σ have a compli- cated internal structure, possibly including a large com- ponent of pion correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The σ in our article refers to a quark-antiquark state alone, parity partner of the pion and approximately twice as heavy as σ(600), [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The article is organized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' II we collect the basic ingredients required to carry out the analysis in the CI model: the dressed quark masses obtained through the gap equation and the general expression for the BSAs for PS and S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We discuss the generalities of the EFFs for PS and S mesons in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' III, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', the quark- photon vertex and the triangle diagram which are the two building blocks to calculate all the meson EFFs in our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' IV is dedicated to computing EFFs of the ground state PS mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It allows us to evaluate their charge radii in the limit Q2 ≈ 0, and simultaneously understand the asymptotic behaviour of the meson EFFs at large Q2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', Q2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' V, we repeat our study for the S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A brief summary and perspectives for future work are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' THE INGREDIENTS Calculation of the meson EFFs presupposes the knowl- edge of the dynamically generated dressed valence quark masses, BSAs of the mesons as well as the quark-photon interaction vertex at different probing momenta of the incident photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In this section, we provide a brief but self contained introduction to the CI, its essential ingre- dients and characteristics, namely, the gluon propagator, the quark-gluon vertex and the set of parameters em- ployed which, collectively, define the CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This discussion is followed by the solution of the gap equation to ob- tain dynamically generated dressed quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We then provide the general expressions of the BSAs for PS and S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The corresponding BSE is set up con- sistently with the gap equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The numerical solution is presented in the respective sections dedicated to the analysis of these mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The Gap Equation The starting point for our study is the dressed-quark propagator for a quark of flavor f, which is obtained by solving the gap equation, S(p)−1 = iγ · p + mf + Σ(p) , Σ(p) = 4 3 � d4q (2π)4 g2Dµν(p − q)γµS(q)Γν(q, p), (1) where mf is the Lagrangian current-quark mass, Dµν(p) is the gluon propagator and Γν(q, p) is the quark-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is a well-established fact by now that the Lan- dau gauge gluon propagator saturates in the infrared and a large effective mass scale is generated for the gluon, see for example [41–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It also leads to the saturation of the effective strong coupling at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This modern understanding of infrared QCD forms the defining ideas of the CI proposed in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We assume that the quarks in- teract, not through massless vector-boson exchange but via a CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Thus the gluon propagator no longer runs with a momentum scale but is frozen into a CI in keeping with the infrared properties of QCD, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Thus g2Dµν(k) = 4πˆαIRδµν (2) where ˆαIR = αIR/m2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The scale mg is for dimensional reasons and is interpreted as the infrared gluon mass scale generated dynamically within QCD [9, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We take 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 1: Diagrammatic representation of the CI, employing the simplified model of the gluon propagator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' currently accepted value mg = 500 MeV [41, 50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is clear that in the CI gap equation, the effective coupling which appears is ˆαIR instead of αIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='We choose αIR/π to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 so that ˆαIR has exactly the same value as in all related previous works [9, 49, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The interaction vertex is bare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', Γν(q, p) = γν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This constitutes an algebraically simple but useful and predictive rainbow-ladder truncation of the SDE of the quark propagator whose solution can readily be written as follows: S(q, Mf) ≡ −iγ · q σV (q, Mf) + σS(q, Mf) , (3) with σV (q, Mf) = 1 q2 + M 2 f , σS(q, Mf) = Mf σV (q, Mf) , (4) where Mf, for the CI, is momentum-independent dynam- ically generated dressed quark mass determined by Mf = mf + Mf 4ˆαIR 3π � ∞ 0 ds s 1 s + M 2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (5) Our regularization procedure follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [55]: 1 s + M 2 f = � ∞ 0 dτ e−τ(s+M 2 f ) → � τ 2 IR τ 2 UV dτ e−τ(s+M 2 f ) = e−(s+M 2 f )τ 2 UV − e−(s+M 2 f )τ 2 IR s + M 2 f , (6) where τIR,UV are, respectively, infrared and ultraviolet regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is apparent from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (6) that a finite value of τIR ≡ 1/ΛIR implements confinement by ensuring the absence of quark production thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (5) does not define a renormalisable theory, ΛUV ≡ 1/τUV cannot be removed but instead plays a dynamical role, setting the scale of all mass dimensioned quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (6), the gap equation becomes Mf = mf + Mf 4ˆαIR 3π C(M 2 f ) , (7) TABLE I: Ultraviolet regulator and coupling constant for dif- ferent combinations of quarks in PS mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ˆαIR = ˆαIRL/ZH, where ˆαIRL = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='57 is extracted from the best-fit to data as explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ΛIR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 GeV is a fixed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' quarks ZH ΛUV [GeV] ˆαIR u, d, s 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='905 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='57 c, u, s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='034 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='322 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 c 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='122 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='35 b, u 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='273 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='41 b, s 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='537 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='26 b, c 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='537 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='15 b 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='513 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='035 where C(M 2) M 2 = Γ(−1, M 2τ 2 UV) − Γ(−1, M 2τ 2 IR) (8) and Γ(α, x) is the incomplete gamma-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We report results for PS mesons using the parameter values listed in Tables I, II, whose variation with quark mass was dubbed as heavy parameters in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In this approach, the coupling constant and the ultraviolet regulator vary as a function of the quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This behavior was first suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [56] and later adopted in several subsequent works [39, 49, 54, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Table II presents the current quark masses mf used herein and the dynamically generated dressed masses Mf of u, s, c and b computed from the gap equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' TABLE II: Current (mf) and dressed masses (Mf) for quarks in GeV, required as an input for the BSE and the EFFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' mu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='007 ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='17 mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='08 mb = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='92 Mu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='367 Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='53 Mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='52 Mb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='75 A meson can consist of heavy (Q) or light (q) quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We present the study of all heavy (Q ¯Q), heavy-light (Q¯q) and (review) light (q¯q) mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We commence by setting up the BSE for mesons by employing a kernel which is consistent with that of the gap equation to obey axial vec- tor Ward-Takahashi identity and low energy Goldberger- Treiman relations, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [9] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The PS mesons are JP C = 0−+ states while the S mesons are JP C = 0++ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The solution of the BSE yields BSAs whose gen- eral form depends not only on the spin and parity of the meson under consideration but also on the interaction employed as explained in the next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 000004 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bethe Salpeter Equation The relativistic bound-state problem for hadrons char- acterized by two valence-quarks may be studied using the homogeneous BSE whose diagrammatic representa- tion can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This equation is mathemati- cally expressed as [1], [Γ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P)]tu = � d4q (2π)4 [χ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P)]srKrs tu(q, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P) , (9) where [Γ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P)]tu represents the bound-state’s BSA and χ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P) = S(q+P)ΓS(q) is the BS wave-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' r, s, t, u represent colour, flavor and spinor indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' and K is the relevant quark-antiquark scattering kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This equa- tion possesses solutions on that discrete set of P 2-values for which bound-states exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A general decomposition of the BSA for the PS and the S mesons (f1 ¯f2) in the CI has the following form ΓP S(P) = iγ5 EP S(P) + 1 2MR γ5γ · P FP S(P) , ΓS(P) = ID ES(P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (10) Note that Ei(P) and Fi(P) with i ∈ {PS, S} are known as the BSAs of the meson under consideration, P is its total momentum, ID is the identity matrix and MR = Mf1M ¯ f2/[Mf1 + M ¯ f2] is the reduced mass of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (9) has a solution when P 2 = −M 2 M with MM being the meson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' After this initial and required set up of the gap equation and the BSE, we now turn our attention to the description of the EFFs of mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ELECTROMAGNETIC FORM FACTORS The EFFs provide crucial information on the internal structure of mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' At low momenta, EFFs allow us to unravel the complexities of non-perturbative QCD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', confinement, DCSB and the fully dressed quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' At high energies, we expect to confirm the validity of asymptotic QCD for its realistic models while at intermediate ener- gies, we observe a smooth transition from one facet of FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 2: Diagrammatic representation of the BSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Blue (solid) circles represent dressed quark propagators S, red (solid) cir- cle is the meson BSA Γ while the blue (solid) rectangle is the dressed-quark-antiquark scattering kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' strong interactions to the other, all in one single experi- ment if we are able to chart out a wide range of momen- tum transfer squared Q2 without breaking up the mesons under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' While there are plenty of studies on the pion EFFs, only a few are found about heavy-quarkonia and practically none on heavy-light mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The process involves an incident photon which probes mesons, inter- acting with the electrically charged quarks making up these two-particles bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Therefore, it is natural to start this section by looking at the the structure of the quark-photon vertex within the CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The Quark-Photon Vertex The quark-photon vertex, denoted by Γγ µ(k+, k−, Mf1), is related to the quark propagator through the following vector Ward-Takahashi identity: iPµΓγ µ(k+, k−, Mf1) = S−1(k+, Mf1) − S−1(k−, Mf1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (11) This identity is crucial for a sensible study of a bound- state’s EFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is determined through the following inho- mogeneous BSE, Γγ µ(Q, Mf1) = γµ − 16πˆαIR 3 � d4q (2π)4 γαχµ(q+, q, Mf1)γα , (12) where χµ(q+, q, Mf1) = S(q + P, Mf1)Γµ(Q)S(q, Mf1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Owing to the momentum-independent nature of the in- teraction kernel, the general form of the solution is Γγ µ(Q, Mf1) = γL µ (Q)PL(Q2, Mf1) + γT µ (Q)PT (Q2, Mf1), (13) where γL µ + γT µ = γµ and γT µ (Q) = γµ − γ · Q Q2 Qµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (14) Inserting this general form into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (12), one readily ob- tains (on simplifying notation) PL = 1 , PT = 1 1 + Kγ(Q2, Mf1), (15) with Kγ(Q2,Mf1) = 4ˆαIR 3π � 1 0 dα α(1 − α)Q2 ¯C1(ω) , (16) where ¯C1(z) = − d dz C(z) = Γ(0, z τ 2 UV) − Γ(0, z τ 2 IR) (17) is K is5 0 2 4 6 8 10 Q 2(GeV 2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 2 PT(Q 2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 3: Dressing function of the transverse quark-photon ver- tex, PT (Q2), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' and ω = ω(M 2 f1, α, Q2) = M 2 f1 + α(1 − α)Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (18) One can clearly observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 3 that PT (Q2) → 1 when Q2 → ∞, yielding the perturbative bare vertex γµ as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This quark-photon vertex provides us with the required electromagnetic interaction capable of probing the EFFs of mesons through a triangle diagram which keeps the identity of the meson bound state intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The Triangle Diagram Let us start from the general considerations for the electromagnetic interactions of mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In the impulse approximation, the MγM vertex, which describes the interaction between a meson (f1 ¯f2) and a photon, reads ΛM,f1 = Nc � d4ℓ (2π)4 Tr GM,f1 , (19) where GM,f1 = iΓM(kf) S(ℓ + ki, Mf1) iΓλ(Q, Mf1) × S(ℓ + kf, Mf1)i¯ΓM(−ki) S(ℓ, M ¯ f2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The notation assumes that it is the quark f1 which in- teracts with the photon while the antiquark ¯f2 remains a spectator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We define ΛM, ¯ f2 similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Furthermore, we denote the incoming photon momentum by Q while the incoming and outgoing momenta of M by: ki = k − Q/2 and kf = k + Q/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The assignments of mo- menta are shown in the triangle diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ΛM,f corresponds to the EFFs of different mesons un- der study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The contribution from the interaction of the photon with quark f1 can be represented as F M,f1(Q2) (stemming from ΛM,f1) while the contribution arising from its interaction with quark ¯f2 can be represented as F M, ¯ f2(Q2) (coming from ΛM, ¯ f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The total form factor F M(Q2) is defined as follows [59]: F M(Q2) = ef1F M,f1(Q2) + e ¯ f2F M, ¯ f2(Q2) , (20) where ef1 and e ¯ f2 are the quark and the antiquark electric charges, respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Both for PS and S mesons, F M,f1 is straightforwardly related to ΛM,f1: ΛS,f1 = −2kλF S,f1 , ΛP S,f1 = −2kλF P S,f1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (21) All information necessary for the calculation of the EFFs is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We can employ numerical values of the parameters listed in Tables I and II and proceed to com- pute the EFFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Our evaluated analytical expressions and numerical results for PS and S mesons occupy the details of the next two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Keeping in mind that the pairs of (PS, S) mesons can be considered as parity partners, we embark upon their treatment in the following sections in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' PSEUDOSCALAR MESONS We start with a detailed discussion and results on the ground state PS mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' These are negative parity, zero angular momentum 0−+ states and occupy a special role in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Simultaneously these are the sim- plest bound states of a quark and antiquark and also emerge as Goldstone bosons associated with DCSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Pi- ons are the lightest hadrons and are produced copiously in collider machines at all energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The pion cloud effect substantially contributes to several static and dynami- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 4: The triangle diagram for the impulse approximation to the MγM vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 1 For neutral mesons composed of same flavored quarks, the total EFF is simply F M = F M,f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' e + kf e + ki k, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' M CM e6 0 2 4 6 8 Q2[GeV2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 Fem PS (Q2) cb cs ud ub us 0 2 4 6 8 Q2[GeV2] bb ss cc 0 2 4 6 8 10 Q2[GeV2] sb cu FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 5: EFFs of PS mesons in a CI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Left panel: electrically charged mesons composed of quarks of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Central panel: quarkonia including a hypothetical ground state strangeonium (s¯s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Right panel: electrically neutral mesons composed of quarks of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' cal hadron properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Therefore, understanding their internal structure has been of great interest both for ex- perimenters and theoreticians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The study of PS mesons is crucial in understanding the capabilities and limita- tions of the CI model employed in this work to repro- duce and predict phenomenological results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Being the Goldstone bosons associated with DCSB, their analysis requires care in treating the associated subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (10), we can see that the BSA of PS mesons is the only one to be composed of two terms, necessary to en- sure the axial vector Ward-Takahashi identity and the Goldberger-Treiman relations are exactly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In this article, we extend and expand the work presented in [9, 39, 57] and compute the EFFs of a larger number of PS mesons composed of qq, qQ and QQ quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' With TABLE III: Calculated values for the BSAs and masses for PS mesons in CI model using the parameters in Tables I and II (compare the parameters with the ones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Mass[GeV] EP S FP S mexp P S [GeV] error [%] u¯d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='139 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='008 u¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='499 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='493 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='162 s¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='701 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='75 — — c¯u 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='855 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='494 c¯s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='945 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='986 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='183 u¯b 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='082 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='279 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='735 s¯b 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='281 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='366 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='586 c¯b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='138 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='39 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='274 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='166 c¯c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='952 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='983 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='053 b¯b 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='280 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='398 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='262 straightforward algebraic manipulations: F P S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='f1 = PT (Q2) � E2 P ST P S EE(Q2) + EP SFP ST P S EF (Q2) +F 2 P ST P S F F (Q2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (22) where T P S EE(Q2) = 3 4π2 � � 1 0 dα C1(ω1) + 2 � 1 0 dα dβ α AP S EE C2(ω2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' T P S EF (Q2) = − 3 2π2 1 MR � 1 0 dα dβ α � A(1) EF C1(ω2) + (A(2) EF − ω2A(1) EF ) C2(ω2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' T P S F F (Q2) = 3 4π2 1 M 2 R � 1 0 dα dβ α � A(1) F F C1(ω2) + (A(2) F F − ω2A(1) F F ) C2(ω2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' and ω1 = ω1(Mf1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Q2) = M 2 f1 + α Q2(1 − α) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ω2 = ω2(Mf1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' M ¯ f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' MM) = α M 2 f1 + (1 − α)M 2 ¯ f2 − α(1 − α) M 2 M + α2 β (1 − β) Q2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ¯C2(z) = (exp(−z τuv) − exp(−z τir))/(2z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (23) 7 The coefficients Ai are given by the following expressions: AP S EE = α(M 2 f1+M 2 M) + 2(1 − α)Mf1M ¯ f2 + (α − 2)M 2 ¯ f2 , A(1) EF = Mf1 + M ¯ f2 , A(2) EF = 2M 2 f1M ¯ f2 − αMf1(4(α − 1)M 2 M + αQ2) +M ¯ f2(2(α − 1)2M 2 M + αQ2(2α(β − 1)β+α − 1)) , A(1) F F = (3α − 2)M 2 M + αQ2 , A(2) F F = 2α((α − 1)2M 4 M + αM 2 MQ2(3αβ2 − 3αβ + α − 2β2 + 2β − 1)) +2αM 2 MM 2 f1 − 2Mf1M ¯ f2(2(α − 1)M 2 M + αQ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The resulting EFFs for charged as well as neutral mesons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' For the practical utility and intuitive understanding of their low and large Q2 behavior, we perform an interpolation for PS mesons EFF in the range Q2 ∈ [0, 8M 2 M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We adopt the following functional form: F P S(Q2) = eM + aP S Q2 + bP S Q4 1 + cP S Q2 + dP S Q4 , (24) where eM := F P S(Q2 = 0) is the electric charge of the meson and aP S, bP S, cP S, dP S are the fitted coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The best fit corresponds to the values listed in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (24) resonates with our observation that the EFFs of PS mesons tend to constant values for large Q2 when it becomes by far the largest energy scale in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is a straightforward consequence of CI treat- ment, and it is characteristic of a point-like interaction which leads to harder EFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, the heavy as well as heavy-light mesons approach a constant value much slower than the light ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This comparative large Q2 behavior of EFFs owes itself to the fact that Q2 becomes larger than all other energy scales at much higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' TABLE IV: Parameters from the fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (24) for PS mesons aP S bP S cP S dP S u¯d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='029 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='068 u¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='029 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='065 s¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='092 c¯u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='616 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='109 c¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='111 u¯b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='033 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='146 s¯b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='009 c¯b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='021 c¯c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='057 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='334 b¯b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='002 The behavior of the form factors at the other extreme, Q2 ≃ 0 allows us to extract charge radii: r2 M = −6 dFM(Q2) dQ2 ���� Q2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (25) For c¯u and s¯b states, which are normalized to F M(0) = 0, we define r2 M with a positive sign in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The charge radii set the trend for the subsequent evolu- tion of the form factors as a function of Q2, specially for its small and intermediate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 6 depicts charge radii for all the PS mesons studied, allowing for a 5% variation around the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' With this permitted spread in the charge radii, one can obtain a band for the Q2 evolution of the EFFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' To avoid over-crowding, we have avoided depicting such a band for each EFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' How- ever, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 7 shows a representative plot for the pion per- mitting a 5% variation in its charge radius in conjunction with the available experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Finally we list the central values of all ground state PS mesons charge radii in Table V, along with a direct com- parison with available experimental observations, lattice results and the SDE findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Moreover, we also report the transition charge radii of light PS mesons and flavor- less neutral heavy PS mesons to two photons invoking the following analytical parameter fit [60]: rt M = r0 1 + (MM/mt) ln(1 + MM/mt) , (26) where r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='67 fm and mt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='01 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It also yields reasonable results for the π point (mass = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='139 GeV) and the K point (mass = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='493 GeV) as they are made of light quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' But we cannot expect it to serve exactly as it is for mesons with vastly off-balanced quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, if CI results were to follow this formula, we would only need to assign r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='458 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The last row of Table V lists the resulting values which we denote as rt M(CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Let us now summarize our findings and make explicit comparisons with related works: As desired, pion EFF and its charge radius agree with the first results employing the CI [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As an add-on, in this article we allow for a 5% variation of the pion charge radius to see its effect on the evolu- tion of the EFF as a function of Q2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A small variation of the initial slope of the curve Q2 ≃ 0 opens a noticeable spread for large Q2 but keeps the qualitative and quantitative behaviour fully in- tact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 8, we draw kaon EFF over the range of Q2 values where (relatively poor) experimental ob- servations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Although large error bars prevent us from commenting decisively on the va- lidity of the CI but we expect it will yield harder results as compared to precise experimental mea- surements whenever these results will become avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Our reported value of its charge radius is an indication of this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 8 TABLE V: The charge radii of PS meson systems, calculated with the CI model, refined SDE studies, lattice QCD and extraction from data, in a hybrid model (HM), light-front framework (LFF) and a QCD potential model (PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In the two rows after the experimental results, we also provide the best fit results for the transition charge radii of PS mesons to γγ∗, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (26) and the same fit adapted to the CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' All results are presented in fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' u¯d u¯s s¯s c¯u c¯s u¯b s¯b c¯b c¯c b¯b Our Result 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='07 SDE [4, 23] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='676 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='593 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='09 Lattice [7, 8, 63] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='648 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='566 (extracted) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='25 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='659 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='560 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='031 rt M [60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='03 rt M(CI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='02 HM [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 LFF [17] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='20 PM [18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='46 + K + h0 D0 B + D + s B0 s c B + c b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 charge radii [fm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='07 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 6: Charge radii of ground state PS mesons in the CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As depicted in Table V, pion and kaon charge radii [4, 7, 8, 23, 63, 64] are known experimentally and through lattice and SDE studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As CI EFFs come out to be harder than full QCD predictions, we expect our PS mesons charge radii to under- shoot the exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This is precisely what we observe for the pion and the kaon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The percentage relative difference between the experimental value and our calculation for the pion charge radius is ap- proximately 32%, while for the kaon charge radius is slightly less, 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Similar difference between the SDE and the CI results for heavy quarkonia is ob- served: For ηc, it is 20% while for ηb is is 22%, not 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 Q2[GeV2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 Fem(Q2) Contact Interaction JLab (W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='20 GeV) JLab (W=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='95 GeV) Amendolia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 7: EFF for π-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The central curve is obtained using the τUV value from the Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The filled band allows for a 5% variation in the charge radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Dots represent the experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [14, 15, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' too dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This comparatively analogous be- havior augments our expectation that we will be in the same ballpark for the PS mesons whose charge radii are neither known experimentally as yet nor lattice offers any results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' There are no experimental or lattice (to the best of our knowledge) results available for c¯u, c¯s, u¯b, s¯b and c¯b mesons for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, the gen- eral trend of decreasing charge radii with increasing constituent quark mass seems reassuring, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', the 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='150 Q2[GeV2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 Fem K (Q2) Contact Interaction Dally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Amendolia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 8: EFF for K-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The central curve of the (blue) band is obtained by using the ΛUV value from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The filled (blue) band allows for a 5% variation in the charge ra- dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The experimental data is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 0 1 2 3 4 5 Q2[GeV2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 Fem c (Q2) Contact Interaction Dudek at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 9: EFF for ηc-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The lower (green) solid curve is the lattice result, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The central curve of the (blue) band is obtained using the ΛUV value from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The filled (blue) band allows for a 5% variation in the charge radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' following hierarchies are noticeable: ru¯d > ru¯s > rc¯u > ru¯b , ru¯s > rs¯s > rc¯s > rs¯b , rc¯u > rc¯s > rc¯c > rc¯b , ru¯u > rs¯s > rc¯c > rb¯b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We must emphasize that the CI is only a simple ‘ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 10: The S meson, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', σ is viewed as the parity partner of the pion π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Note that the scalars in this article only refer to their quark-antiquark content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Refined QCD calculations are required to confirm or refute these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' This concludes our detailed analysis of all the ground state PS heavy (Q ¯Q), heavy-light (Q¯q) as well as light (q¯q) mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We now turn our attention to a similar analysis of the scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' SCALAR MESONS Recall that an S meson is a 0++ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It can be con- sidered as the chiral partner of the PS meson Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We work under the assumption that all states are purely quark-antiquark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Then, for example, the states π and σ get transformed into each other through the fol- lowing chiral transformation: q → e−iγ5 τ 2 ·θq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (27) The explicit expression for the EFFs for S mesons with mass MM constituted from a quark f1 and an antiquark ¯f2 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (20) with F S,f1 = PT (Q2)E2 ST S EE(Q2) , (28) where T S EE(Q2) = − 3 4π2 � � 1 0 dα C1(ω1) + 2 � 1 0 dα dβ α AS EE C2(ω2) � , (29) with AS EE = αMf1 − 2(1 − α)Mf1M ¯ f2 + (α − 2)M 2 ¯ f2 + αM 2 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (30) Note the close resemblance between T S EE and T P S EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As expected, there are only sign differences between the two due to the presence, or absence, of the γ5 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In Table VI, we present the parameters used for S mesons JPC = 0-+ JPC = 0++ b T 0 q q q q iq5q bb10 0 2 4 6 8 Q2[GeV2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 Fem S (Q2) cb ub cs us ud 0 2 4 6 8 Q2[GeV2] ss bb cc 0 2 4 6 8 10 Q2[GeV2] sb cu FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 11: EFFs for S mesons in the CI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Left panel: electrically charges mesons composed of quarks of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Central panel: quarkonia including a hypothetical ground state strangeonium (s¯s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Right panel: electrically neutral mesons composed of quarks of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' EFFs of electrically neutral but flavored mesons have been normalized to F S(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' TABLE VI: Ultraviolet regulator and the coupling constant for different combinations of quarks in S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' As before, ˆαIR = ˆαIRL/ZH, where ˆαIRL = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='57 is extracted from a best- fit to data as explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' ΛIR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='24 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' quarks ZH ΛUV [GeV] ˆαIR u, d, s 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='905 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='57 c, u 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='034 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='322 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 c, s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='034 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='222 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 c 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='122 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='35 b, u 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='473 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='25 b, s 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='537 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='15 b, c 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='216 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='13 b 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='013 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='036 in order to compute the masses, amplitudes and charge radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We enlist the masses and BSAs of S mesons in Table VII while the EFFs are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' On the right and central panels we present the results for neutral mesons while the left panel displays the EFFs of charged mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We emphasize that for electrically neutral but flavored S mesons, we normalize the EFFs to zero at Q2 = 0, while for flavorless mesons, the normalization is F S(0) = 1 to be consistent with the definition employed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We again perform a fit in the range Q2 ∈ [0, 8M 2 M], where MM is the mass of the S meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' All the TABLE VII: Computed values of the S mesons masses and BSAs in the CI model, see [49] for comparison, using the parameters listed in Tables II and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Mass [GeV] ES mexp S [GeV] error [%] u¯d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='66 — — u¯s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='65 — — s¯s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='64 — — c¯u 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='19 c¯s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='54 u¯b 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='53 — — s¯b 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='26 — — c¯b 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='26 c¯c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='73 b¯b 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='69 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='95 curves are faithfully reproduced by the following choice: F S(Q2) = eM + aS Q2 + bS Q4 1 + cS Q2 + dS Q4 , (31) where eM := F S(Q2 = 0) is the electric charge of the meson and aS, bS, cS, dS are the parameters of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' These values for S mesons are listed in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Based on these numbers, we can immediately infer the large Q2 behavior of these EFFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The coefficient bS ≈ 0 for all S mesons under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Therefore, the EFFs for S mesons fall as 1/Q2 for large Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We present the numerical values of the charge radii 11 TABLE IX: Parameters for the fit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (31) for S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' aS bS cS dS u¯d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='617 u¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='629 s¯s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='542 c¯u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='759 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='680 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0 Fem(Q2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 12: EFF for σ-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The central curve is obtained using the ΛUV value from Table VI while the band represents a 5% variation in the charge radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' for S mesons in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We must reiterate that for the S mesons there are no reported measurements of their charge radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Theoretical results are also scarce for any direct and meaningful comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is worth mention- ing again that the internal structure of scalar mesons is not well-established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Our results are based on consider- ing them as effective quark-antiquark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We would like to remind the reader that we again allow for a 5% variation in the charge radii of S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' How- ever, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 11, we present the EFFs only for their cen- tral values for visual clarity, refraining from showing the corresponding band to avoid possible overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' How- ever, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 12, we depict a representative plot with a 5% variation in the charge radius for the lightest scalar me- son, σ, alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Other mesons have similar bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 13, we plot the charge radii, extracted from the EFFs, as a function of the S meson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' In general, the charge radii decrease when the S meson masses increase just as we observed for the PS mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' K * 0 f0 D * 0 D * s0 c0 B * 0 B * s0 B * c0 b0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='7 charge radii [fm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='39 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 13: Charge radii of S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' CONCLUSIONS In this work we present an exhaustive computation of EFFs employing the CI model for twenty ground state PS and S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Note that the CI findings for light mesons and heavy quarkonia are already found in the literature as mentioned before [9, 39, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We include these re- sults for the sake of completeness and as a guide to pin down the best parameters in order to explore heavy-light systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We thus report first results on the latter mesons within this model/formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We expect these new EFFs to be harder than the exact QCD predictions, especially for the PS mesons due to the necessary inclusion of the F-amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We also anticipate the charge radii to be in the ballpark of a (20-25)% error in light of the results where comparison with realistic studies and/or experi- ment has been possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Furthermore, we analyze the sensitivity of the evolu- tion of the EFFs by a change in appropriate parameters to allow for a 5% variation in the charge radii of the corresponding mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The evolution band has been shown explicitly for π, K, σ and ηc alone to avoid over-crowding in other collective plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' However, it is worth mentioning that the corresponding bands in other EFFs are almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Interpolations have also been provided in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' (24, 31) and Tables IV and IX which allow for a convenient algebraic analysis of the behavior of the EFFs in the momentum range that we mentioned 12 TABLE X: The charge radii for S mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' All quantities are reported in fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' u¯d u¯s s¯s c¯u c¯s u¯b s¯b c¯b c¯c b¯b Our Result 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='39 above and for any application the reader may deem useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We plan to recalculate these EFFs for vector and axial vector mesons followed by the same computation within a more realistic algebraic model and in the long run for truncations more akin to full non-perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' It is also straightforward to generalise our analysis to study diquarks EFFs which are crucial in the subsequent computation of baryons EFFs such as the ones reported recently in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' All this is for future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Acknowledgments L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Guti´errez-Guerrero wishes to thank the support from C´atedras CONACyT program of Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Hern´andez-Pinto is supported by CONACyT (Mexico) Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 320856 (Paradigmas y Controver- sias de la Ciencia 2022), Ciencia de Frontera 2021-2042 and Sistema Nacional de Investigadores as well as by PROFAPI 2022 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' PRO A1 024 (Universidad Aut´onoma de Sinaloa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' The work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir is sup- ported in part by the US Department of Energy (DOE) Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' DE-AC05-06OR23177, under which Jef- ferson Science Associates, LLC operates Jefferson Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir also acknowledges Coordinaci´on de la Investi- gaci´on Cient´ıfica of the Universidad Michoacana de San Nicol´as de Hidalgo grant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='10 and the Fulbright-Garc´ıa Robles scholarship for his stay as a visiting scientist at the Thomas Jefferson National Accelerator Facility, Newport News, Virginia, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' We thank Jozef Dudek and Chris- tine Davies for helpful communication on lattice results on EFFs and charge radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Salpeter and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bethe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 84, 1232 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Maris and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Tandy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C 62, 055204 (2000), nucl-th/0005015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Chang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Clo¨et, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Schmidt, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Tandy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 111, 141802 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C77, 025203 (2008), nucl-th/0612069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Shultz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Dudek, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Constantinou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Delmar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Hadjiyiannakou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Koutsou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Lauer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Vaquero (ETM), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D 105, 054502 (2022), 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='08135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Gao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Karthik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Mukherjee, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Petreczky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Syrit- syn, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D 104, 114515 (2021), 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='06047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Davies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Koponen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Lepage, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Lytle, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Zimermmane-Santos (HPQCD), PoS LAT- TICE2018, 298 (2018), 1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='03808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Gutierrez-Guerrero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C 81, 065202 (2010), 1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Xing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Kang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Raya, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Chang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D 106, 054016 (2022), 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='04339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [11] Z.' metadata={'source': 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1 (2009), 0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Wilson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Chang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C85, 025205 (2012), 1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Schmidt, Few Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 55, 1185 (2014), 1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='00046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Raya, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Guti´errez-Guerrero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Chang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Segovia, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' A 57, 266 (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='02306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Segovia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' El-Bennich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rojas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Xu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Aznauryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=', Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' E 22, 1330015 (2013), 1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='4891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Chang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' El-Bennich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Tandy, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 58, 79 (2012), 1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='3366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Gutierrez-Guerrero, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Tandy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C82, 065202 (2010), 1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='0067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Gutierrez-Guerrero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cobos-Mart´ınez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, Few Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 59, 133 (2018), 1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='00383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [40] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Boucaud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Leroy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Yaouanc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Micheli, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Pene, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rodriguez-Quintero, Few Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' 53, 387 (2012), 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Ayala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, D.' metadata={'source': 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Rodriguez-Quintero, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D96, 054026 (2017), 1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='04835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Deur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='08082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rodr´ıguez-Quintero, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Binosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Mezrag, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Papavas- siliou, and C.' metadata={'source': 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Gutierrez-Guerrero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Bashir, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Cloet, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Roberts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' C78, 181 (2018), 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content='06926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' [51] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Binosi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Papavassiliou, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfsC7L/content/2301.11881v1.pdf'} 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Svetogorov, Daniel Loss, and Jelena Klinovaja +Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland +(Dated: January 31, 2023) +We analyze results of a recent experiment [D. Razmadze et al., Phys. Rev. Lett., 125, 116803 +(2020)] on transport through a quantum dot between two full-shell nanowires and show that the +observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic +flux in a full-shell nanowire) rather than the presence of Majorana bound states. Moreover, we +propose that such a setup presents a unique and convenient system to study the competition between +superconductivity and the Kondo effect and has significant advantages in comparison to other known +approaches, as the important parameter is controlled by the magnetic flux through the full-shell +nanowire, which can be significantly varied with small changes of magnetic field, and does not +require additional gates. This competition is of fundamental interest as it results in a quantum +phase transition between an unscreened doublet and a many-body Kondo singlet ground states of +the system. +I. +INTRODUCTION +Semiconducting nanowires with full superconducting +shell were recently introduced as possible realizations +of topological superconductors, which may host Majo- +rana bound states (MBSs) [1]. MBSs in turn have non- +Abelian statistics, which could be exploited to develop +a topologically protected qubit [2, 3]. In nanowires with +a thin shell Little-Parks effect [4] results in modulation +of the superconducting order parameter with the ap- +plied magnetic flux. +In case of a thin nanowire (with +a diameter smaller than the superconducting coherence +length) the Little parks effect is destructive resulting in +a lobe structure of the order parameter as a function +of the flux [5, 6]. The idea of combining effectively one- +dimensional superconductors with a vortex is already fas- +cinating by itself: the vortex can induce a phase wind- +ing of the superconducting order parameter and result in +nontrivial properties, such as the well-known Caroli–de +Gennes–Matricon bound states in 2D case [7]. Further +experiments with full-shell nanowires were performed [8– +10], including those where the observed zero-bias anoma- +lies were shown to have non-topological nature [11]. How- +ever, a recent experiment [12] with a gate-controlled +quantum dot (QD) in a full-shell hybrid interferometer +showed non-trivial features in addition to the zero-bias +peak. Without a magnetic flux (zeroth lobe) through the +shell a change of occupation of the dot (from even to odd) +leads to a change of a sign of the supercurrent through +the dot, which is known as 0−π phase transition [13–16]. +However, with a flux around one magnetic flux quantum +threading the superconducting shell (first lobe) the ef- +fective Josephson junction seems to stay in the 0 phase +even for the odd occupation of the QD, in agreement with +theoretical predictions [17–21] for a Josephson junction +based on QD between two topological superconductors +hosting MBSs. Such peculiar behaviour could be seen as +evidence for the presence of MBSs in the system. Nev- +ertheless, the authors were not completely satisfied with +the interpretation, as they could not explain some of the +observed features, such as an enhancement of the super- +current in the odd state in the first lobe and no signs +of fractional Josephson effect expected in the presence of +MBSs were observed. +In this paper we argue that the results of this ex- +periment can be interpreted as an enhancement of the +Kondo effect [22–26] in the first lobe, including an en- +hancement of the supercurrent. +It was predicted the- +oretically [24, 27–29] and shown experimentally [30–32] +that if the Kondo effect can develop on an odd-occupied +QD between superconducting leads, the ground state is +a many-body Kondo singlet instead of a doublet (un- +screened electron), which restores the 0 phase behaviour +and enhances the critical supercurrent. The phase tran- +sition is determined by the ratio of two energy scales: the +Kondo temperature TK and superconducting gap ∆. In +this paper we show that if the superconducting order pa- +rameter acquires a phase winding around the shell, the +relevant energy scale is not the absolute value of the gap +|∆| but an effective gap which is significantly suppressed +due to destructive interference, which in turn leads to +the enhancement of the Kondo effect. Our interpretation +is further supported by the fact that a bright zero-bias +feature and no π phase is observed at the closing of the +zeroth lobe (around half the flux quantum), where no +topological superconductivity is expected. As a result, +we can claim that due to the detailed experimental data +provided by Razmadze et al. [12] it is possible to establish +a non-trivial effect of the phase winding on the coherent +transport and the ground state properties of a QD be- +tween two full-shell nanowires, which has not been pre- +dicted before. Further experimental studies of the effect +can enrich the understanding of the underlying funda- +mental physics as only elaborate numerical approaches +have been developed to quantitatively capture the QD- +based Josephson junction behaviour in the competition +regime [24, 29, 33–35] (for review see [36]). +arXiv:2301.12442v1 [cond-mat.supr-con] 29 Jan 2023 + +2 +QD +FIG. 1. +Schematic representation of Cooper-pair tunnel- +ing through a QD electron state (green sphere) between two +hollow-cylinder superconductors (blue) representing the shells +and accumulation layers of underlying nanowires. The tunnel- +ing amplitude t(θ) (between the dot and each shell) is angle- +dependent, as the electron wave function on the dot is shifted +from the shell axis due to gating; the superconducting order +parameter on the shell has angle dependence - nonzero phase +winding (n ̸= 0 for large enough magnetic flux Φ > Φ0/2). +II. +MODEL +In this paper we focus on a QD formed in an uncovered +(etched superconductor) and gated region of a full-shell +nanowire. Assuming that electrons in the nanowire form +an accumulation layer at the boundary with the super- +conducting shell [1], we use a hollow-shell approximation +- transport through the system is mostly determined by +Cooper pairs propagation along the shell and tunneling +between the shells through the QD. We assume the shell +to be much thinner than the magnetic penetration length +and the diameter to be smaller than the superconducting +coherence length, which results in no quantization of the +magnetic flux through the shell but reentrant lobe struc- +ture due to the destructive Little-Parks effect: supercon- +ductivity is fully suppressed when the flux around odd +half-integer multiples of a flux quantum is applied [6, 12]. +As a further simplification we model the shell as a hollow +cylinder (in experiments it has rather a hexagonal cross- +section) [1, 37] with radius R, then the total magnetic +flux through the shell is Φ = πR2B. We focus on the +single-level QD limit (large level spacing), the number of +electrons can be considered fixed by the applied gate due +to strong Coulomb interactions on the QD (strong spa- +tial confinement), which is a common experimental situ- +ation. The simplest model to describe such a system is +the superconducting impurity Anderson (SIA) model [38] +modified by the magnetic flux though the shell [37]: +H = HD + +� +s=l/r +(Hs + V s +SD) , +(1) +HD = +� +σ +(ϵd + σVZ) d† +σdσ + Un↑n↓, +(2) +Hs = +� +dxdθ +� +σ +� +ψ† +σθx +� p2 +2ms +− µs +� +ψσθx ++ ∆s(Φ)e−inθ+iφsψ† +σθxψ† +−σθx + h.c. +� +, +(3) +V s +SD = +� +dθ +� +σ +ts(θ)ψ† +σθxdσ/(2π) + h.c. +(4) +Here ϵd is the bare dot energy level, which is controlled +by gate voltage, VZ = µBgB/2 is the Zeeman field (g +is Land´e g-factor and µB is Bohr magneton), σ corre- +sponds to electron spin up/down state (±1 if it is a co- +efficient, ↑ / ↓ if it is an index); d† +σ is the QD electron +creation operator (with spin σ), nσ = d† +σdσ is the occupa- +tion number operator, U is the charging energy (which is +the largest energy scale in the system so that the single- +level approximation is valid); ψσθx are the shell operators +(we omit s = l/r index for the left/right shell), x is the +coordinate along the shells, θ is the angle around the +shell. The superconducting order parameter in the shells +∆l/r(Φ, θ) = +��∆(Φ)l/r +�� e−inθ+iφl/r depends on magnetic +flux Φ as well as angle θ, φl/r is the phase at the same +angle θ = 0 to the left/right of the QD, n denotes the +number of phase windings defined by an integer of the +ratio of magnetic flux to flux quantum (Φ0 = π/e, we +set ℏ = 1 throughout the paper): n = ⌊Φ/Φ0⌉; p2/(2ms) +and µs are the kinetic term and the chemical potential in +the respective shells. The last term describes tunneling +between the left/right shell and the QD, the tunneling +amplitude is given by ts(θ)/(2π) and dependends on the +azimuthal angle θ, as the electron state on the QD is +shifted away from the shell axis due to the dot gating. +We work in the zero-temperature limit (the temperature +in relevant experiments is well below characteristic ener- +gies). The main difference from [37] is the presence of two +superconducting shells instead of one, which adds up and +introduces an additional important parameter, namely +the phase difference φ = φl − φr between the shells (at +the same angle θ). The effective Hamiltonian can be ob- +tained by integrating out the shells and introducing the +dot Green function as G(ω) = [ω − Heff(ω)]−1: +Heff(ω) = Heff,↑(ω) ⊕ Heff,↓(ω) +Heff,σ (ω) = +� +ϵd + σVz +0 +0 +−ϵd + σVz +� ++ ΣU +σ + ΣS, (5) +where the first term corresponds to the bare QD. The last +two terms are self-energies from the Coulomb interaction +U on the dot and from the proximity effect induced by +the superconducting shells, respectively. The proximity + +3 +effect contributes as +ΣS = − +2 ⟨ΓS⟩ +√ +∆2 − ω2 +� +ω +∆eff cos φ +2 +∆eff cos φ +2 +ω +� +. +(6) +Here ⟨ΓS⟩ = πρs +�� π +−π +dθ +2πt (θ) +�2 +stands for the tunneling +energy scale averaged over the angle θ around the shell +(the contribution comes from hopping from QD to shell +and back with amplitude t(θ)/(2π)), ρs is the density of +states at the Fermi energy, φ is the phase difference be- +tween left and right shell (at fixed angle θ). We assume +symmetric tunneling tr(θ) = tl(θ) = t(θ) to the left/right +shell, as corresponding results can be easily generalized +for an asymmetric case [39]. Even more nontrivial contri- +bution comes in the numerator of the off-diagonal terms +(see Appendix A): +∆eff ≈ (1 − δ0n) +����� +2∆ +� π +−π t (θ) eiθn dθ +2π +� π +−π t (θ) dθ +2π +����� + δ0n |∆| . +(7) +One can see that for n ̸= 0 and axially symmetric tun- +neling (t(θ) = const), ∆eff = 0! One should note that +Eq. (7) is approximate, it works well for weak dependence +of tunneling amplitude on θ; for general case see analyses +in Appendix A. +The effect can be interpreted as destructive interfer- +ence, which can be seen from a schematic of a Cooper- +pair tunneling trajectory between two superconducting +hollow cylinders through a QD electronic state, Fig. 1; for +axially symmetric case a Cooper pair has equal probabil- +ity to tunnel between all possible angle positions on su- +perconducting shells. If there is a phase winding around +the shells, each trajectory corresponds to some phase dif- +ference ∆φ ∈ [0, 2πn], then summing over all trajectories +gives exactly zero (which can be written as zero effec- +tive gap ∆eff). +However, the effective QD state (the +bound state wave-function) is rather not centred on the +shell axis, as the QD is gated from one side, which re- +sults in θ-dependent tunneling amplitude t(θ) and, there- +fore, nonzero ∆eff. As was discussed in [37], the angle- +dependence of tunneling is rather weak as the tunneling +is determined by the tails of the wave-function under the +superconducting shell, which is screening the electrical +field form the gate. +The most straightforward conse- +quence of such a destructive interference is the reduction +of the Josephson effect by a factor ∆eff/∆. +However, that is not the whole story. First of all, as +it was already briefly discussed, if a flux around half flux +quantum is applied, superconductivity is completely sup- +pressed due to Little-Parks effect. Second, even an S- +QD-S junction without magnetic flux shows nontrivial +behaviour such as 0 − π phase transition. +This phase +transition was extensively studied theoretically, starting +with the first predictions for transition induced by chang- +ing the occupation of the QD [13–16, 40] and followed by +more sophisticated regimes, when the Kondo effect may +play a significant role [24, 29, 33–35, 41]. These effects +are due to strong Coulomb interactions on the QD repre- +sented by ΣU +σ term in the effective Hamiltonian. In [37] +it was calculated in Hartree-Fock-Bogoliubov approxima- +tion [27, 42] (the lowest U-order expansion) +ΣU +σ ≈ U +� ⟨n−σ⟩ +⟨dσd−σ⟩ +⟨d† +σd† +−σ⟩ +−⟨nσ⟩ +� +, +(8) +which cannot capture the Kondo effect. The latter was +studied with powerful numerical approaches [24, 33–35] +as of now no reliable analytical approach capable of tack- +ling the problem in the most interesting regime of com- +petition between the Kondo effect and superconductivity +has been developed. Fully analytical methods are avail- +able only for special limiting cases such as Hartree–Fock- +Bogoliubov approximation [27, 42] and perturbation in +cotunneling [43, 44] through Yu-Shiba-Rusinov (YSR) +state [45–47] analogs for ∆ ≫ TK or slave-boson mean +field approaches in the opposite limit [48, 49]. Neverthe- +less, it is well established that if the Kondo temperature +TK (characteristic energy scale) is large enough in com- +parison to the superconducting gap ∆, the electron on a +dot can form a Kondo singlet with quasiparticles in the +superconductor (Kondo cloud) and, therefore, the junc- +tion stays in the 0 phase even in the odd sector, the +cotunneling process is enhanced which in turn increases +the supercurrent [25, 29, 35]. The Kondo temperature +depends on tunneling amplitude, Coulomb interaction, +and bare QD level energy [50]: +TK ∼ +√ +UΓ exp +�πϵd +2Γ +� +1 + ϵd +U +�� +, +Γ = 2⟨Γs⟩. +(9) +The exact proportionality factor 0.28 is well defined +only in the middle of the odd occupation region (ϵd = +−U/2) [36, 51]. An important question is how the super- +conducting phase winding affects this competition. +III. +ANALYSES OF THE EXPERIMENT +In this section we analyze the results of an experiment +performed on a S-QD-S junction with a flux through the +full-shell nanowire with the goal to establish topologi- +cal superconductivity [12]. +Several non-trivial features +were reported that could potentially indicate the pres- +ence of Majorana fermions. First, the differential con- +ductance in a voltage-bias configuration was measured, +then the current-phase relation (CPR) was probed in a +SQUID geometry. +The even-occupied regime does not +show anything unexpected: the differential conductance +clearly shows a gap-closing around half flux quantum due +to the destructive Little-Parks effect and a gap-reopening +in the first lobe (Fig. 2c in [12]). Current-bias measure- +ments in the SQUID geometry show a trivial Josephson +effect in both lobes. In the odd-occupied regime a bright +zero-bias peak develops at the closing of the zeroth lobe +(around half flux quantum), which the authors identify as +a Kondo peak. In the destructive regime no peak is vis- +ible (superconductivity is fully suppressed in the shells). + +4 +The zero-bias feature reappears in the first lobe, but less +bright and a bit broadened. In the current-bias measure- +ment a π phase is absent in the odd-occupied regime in +the first lobe (Fig. 4 in [12]), while it is present in the ze- +roth lobe (supercurrent reversal). As it was theoretically +predicted, a full-shell nanowire could potentially acquire +topological properties in the first lobe [1], which could +explain the absence of the π phase and zero-bias peak +by hybridization of a dot electron with MBSs [17–21]. +However, current-bias measurements in SQUID geome- +try show the absence of the π phase in the center of the +odd sector already at the closing of the zeroth lobe (data +in supplemental material of [12], Fig. S8). An enhance- +ment of the supercurrent in the odd state is clearly visible +in comparison to the even state, which is a typical fea- +ture of the Kondo effect in S-QD-S junctions [24, 29, 52]. +And all these features are qualitatively similar to the ones +observed in the first lobe (nicely captured in Figs. 4e-f +in [12]): the CPR of the SQUID is given by a sinusoid, +but in the odd state the critical current is larger (higher +average value), no phase shift is observed. That suggests +that the observed effects are of the same origin. As we +have shown in the previous section the superconducting +phase winding introduces a new important energy scale +- ∆eff, which plays the role of the effective gap for the +QD and which is reduced in comparison to |∆| due to de- +structive interference. In case of nonzero phase winding +n > 0 off-diagonal elements of ΣS [see Eq. (8)] are smaller +by a factor of ∆eff/∆, which suggests that in such a +system the relevant parameter for the quantum phase +transition between the unscreened doublet to the many- +body Kondo singlet ground states is ∆eff/TK. A more +formal way to see that is to perform a renormalization +group (RG) analyses: the RG flow starts at large energy +cutoff, off-diagonal terms of ΣU +σ get renormalized due to +off-diagonal elements of ΣS, see Appendix C. As a result, +we were able to deduce that in the first lobe for odd oc- +cupation of the QD TK > ∆eff and the ground state is a +Kondo singlet, which explains the 0 phase behaviour and +the supercurrent enhancement as well as zero-bias peak +in the differential conductance. Another question arising +is whether the Zeeman effect can play a significant role, +because the Zeeman field can split the Kondo peak if the +g-factor is large enough [53–55]. However, it was shown +that due to spin-orbit interaction the effective g-factor +on a long QD can be renormalized (towards small val- +ues) [56–58]. That significantly complicates theoretical +comparison of Zeeman energy VZ = µBgB and Kondo +temperature TK. In Appendix B we provide some simple +analyses of the experimental data. +Another distinctive feature observed in experiment [12] +is a change of dissipation between zeroth and first lobes: +one can see a strong hysteresis in supercurrent though the +SQUID in the zeroth lobe, which indicates underdamped +junction regime corresponding to low dissipation. +On +contrary, in the first lobe no hysteresis is seen; this ef- +fect is independent of the QD occupation, therefore, it is +not caused by the Kondo effect itself. We suggest that +the higher damping can be attributed to lower effective +gap (and described in terms of subgap states induced +by the vortex [59]). +The different junction dissipation +regime in two lobes also implies a different ratio of critical +and switching current. In the underdamped regime the +switching current (which is measured in the experiment) +can be significantly lower than the actual critical current, +as the macroscopic quantum phase tunneling cannot be +neglected, while in the first lobe strong dissipation (over- +damped regime) should result in the switching current +being in good correspondence with the critical current. +The experimental setup [12] appears to be a very con- +venient and unique device to study competition between +superconductivity and the Kondo effect at relatively low +magnetic fields without additional gates to control the +tunneling amplitude due to destructive Little-Parks ef- +fect. The regime of competition is specifically interesting +to study experimentally as no analytical approach ex- +ists to provide quantitative description of the system in +this regime. The setup allowed the scientists to measure +the CPR and differential conductance in the middle of +the odd occupation sector all the way from the doublet +ground state to the Kondo singlet smoothly varying the +superconducting gap by changing the magnetic flux from +zero to a half flux quantum, which does not even require +going into the first lobe. The well resolved CPR close to +the phase transition (at 40 mT and 45 mT for the first +device; Fig. S8 in [12],) has a drastic difference, which +is an outstanding feature of the change in the character +of the ground state and is in perfect qualitative agree- +ment with theoretical predictions (numerics). We sug- +gest that a measurement of the CPR at different values +of flux with smaller steps around the transition could be +sufficient to establish the transitions between 0, 0′, π′ +and π phases of such a QD-based junction [27, 42] in +the middle of the odd parity sector (before such tran- +sitions have been observed only as a function of gate +voltage [31, 60]). +For this we recommend to fabricate +an asymmetric SQUID with an ancilla junction being in- +dependent of the flux through the shell (i.e. a separate +SIS junction) and having slightly larger critical current +so that the CPR of S-QD-S junction is directly observed +(large difference in critical currents between the junc- +tions forming the SQUID decreases the contrast of the +picture). Further study of the first lobe can provide bet- +ter understanding of the Kondo cloud formation due to +destructive interference. Moreover, it could be interest- +ing to study the effect in shells of larger radius (or thin- +ner shell, so that the superconducting coherence length +is shorter than the shell’s radius [6]), when Little-Parks +effect does not suppress superconducting gap to zero. In +that regime suppression of the gap at half flux quantum +may not be enough to enhance the Kondo effect, how- +ever, the phase winding at higher magnetic fields can still +reduce the effective gap to the values below the Kondo +temperature, which would result in a zero-bias peak only +in the first (or higher) lobe. + +5 +IV. +CONCLUSIONS AND OUTLOOK +We provided a coherent interpretation for the results +observed in an experiment [12] on a transport through +a QD between two full-shell nanowires. +Due to accu- +rate and sufficient data presented, we were able to es- +tablish the effect of superconducting order parameter +phase winding on a ground state of the QD and attribute +it to the Kondo effect. +We showed that the qunatum +phase transition between a doublet and a many-body +Kondo singlet ground state is controlled by a parame- +ter ∆eff/TK ≪ |∆|/TK in case of a superconducting +phase winding. 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B 73, +235337 (2006). +Appendix A: Effective gap +The effective Hamiltonian (5) was derived in [37] +within the SIA model and consists of three parts: bare +QD energy, Coulomb interactions, and proximity effect +from the shell. For a usual S-QD-S junction (no phase +winding) the latter is given by [27, 29] +ΣS = − +� +s +πρSt2 +√ +∆2 − ω2 +� +ω +|∆| eiφs +|∆| e−iφs +ω +� +, +(A1) +where the index s stands for left/right superconductor +(we assume |∆| and t to be the same for the left/right +shells). +As discussed in [37], the approach can be ex- +tended to the hollow-cylinder model, which allows us to +include a magnetic flux in the model. Here we briefly +sketch the main steps of that approach, the difference +with [37] being that here we consider superconducting +shells both to the right and to the left of the QD, there- +fore, we need to take a relative phase into account as +well. One needs to sum over all the possible positions on +the left/right shell, which is done by introducing angle- +dependent tunneling amplitudes t(θ)/(2π) as well as or- +der parameters ∆s = |∆|eiφs−inθ, where n is the num- +ber of the phase windings n = ⌊Φ/Φ0⌉ due to the flux +through the shell, see Fig. 1. As was reported in [37] for +the axially symmetric case t(θ) = const, the off-diagonal +elements for any n ̸= 0 are zero, which can be seen as +destructive interference. Realistically, the wave function +of an electron on the QD cannot be treated as axially +symmetric (with respect to the shell axis) due to gat- +ing from one side and due to inhomogeneities. However, +we can still assume the tunneling to have a relatively +weak angle dependence (as discussed in [37] the tunnel- +ing is determined by the wave-function’s tails underneath +the shell, which are not very sensitive to the gating): +t(θ) = t0 + δt(θ). +It is convenient to decompose this +tunneling amplitude into harmonics t(θ) = � +m tme−imθ, +where tm = +� π +−π t(θ)eimθ dθ +2π. Then, one can get the prox- +imity term in the form of Eq. (A1) by integrating the +shell contribution over the angle θ. The simplest esti- +mation comes from expanding t2(θ) = t2 +0 + 2t0δt(θ) + .... +The off-diagonal term takes the form +ΣS +01 ≈ − +� +s +2πρSt0 |∆| eiφs +√ +∆2 − ω2 +2π +� +0 +t(θ)e−inθdθ +2π +. +(A2) +Let us derive the term in a more accurate fashion. The +thin superconducting shell Hamiltonian can be written +in cylindrical coordinates as [1, 63] +Hs = +� +k2 +s + k2 +r + (kθ + eAτz)2 +2ms +− µs +� +τz+ ++ |∆| (cos [−inθ + iφs] τx + sin [−inθ + iφs] τy) , +(A3) +where ks, kr and kθ are the longitudinal, radial, and tan- +gential components of the momentum operator, ms is the + +7 +effective electron mass, µs is the chemical potential in the +shell; A = +1 +2eR +Φ +Φ0 is the vector potential, τi are Pauli ma- +trices acting in Nambu-Gorkov space. One can introduce +a generalized angular momentum Jz = −i∂θ+ 1 +2nτz+ 1 +2σz, +which is conserved (commutes with Hamiltonian) [1, 59], +then the eigenvalues mJ are good quantum numbers and +can take half-integer values: +� +mJ − 1 +2 − 1 +2n +� +∈ Z. +(A4) +The resulting angular momentum number m is integer +and fixed by mJ and n in each spin and Nambu sec- +tor (note that it is exact only in the absence of radial +spin-orbit interaction [59]). Due to the term 1 +2nτz in the +definition of the generalized angular momentum, for a +fixed spin the Hamiltonian is not block-diagonal in the +(m, −m) Nambu sectors, but in the (m, −m − n) sec- +tors [37]. The retarded Green function for the shell (de- +coupled from the QD) between two positions θ and θ′ is +then given by [37] +gs(ω, θ, θ′) = +� +m +e−im(θ−θ′) +Dm,n +� +� ω + ϵks + [m+n−Φ/(2Φ0)]2 +2msR2 +∆e−iφs+inθ′ +∆eiφs−inθ +� +ω − ϵks − [m+Φ/(2Φ0)]2 +2msR2 +� +e−in(θ−θ′) +� +� , +(A5) +where +Dm,n = ω2 − ∆2 − (ϵks − Lm)2 − (ω − ϵks + Lm) +� +Φn − 2 +� +ΦnLm +� +, +(A6) +with +Lm = +� +m + +Φ +2Φ0 +�2 +2msR2 +and +Φn = +� +n − Φ +Φ0 +�2 +2msR2 +. +(A7) +Then the proximity effect of the shells on the QD can be +described by the self-energy [37] +ΣS(ω) = +� +s +� +dks +� dθ +2π +dθ′ +2π t(θ)gs(ω, θ, θ′)t(θ′). +(A8) +In the middle of the first lobe Φ/Φ0 = n = 1 the result +can be written in the form of Eq. (A1) but with t replaced +by t0 = +� π +−π t(θ) dθ +2π, which is just the tunneling amplitude +averaged over θ, and |∆| in the nnumerator of the off- +diagonal terms replaced by +∆eff = +����� +� +m +t−mtm+n +t2 +0 +∆ +����� . +(A9) +In case of weak modulation compared to the angle- +independent tunneling (t0 ≫ |δt(θ)|) one gets for n ̸= 0 +∆eff ≈ 2 +����� +∆ +� π +−π t (θ) eiθn dθ +2π +� π +−π t (θ) dθ +2π +����� ≪ |∆| , +(A10) +which is the same as Eq. (A2) [if one substitutes ∆eff for +∆ in the numerator of the off-diagonal terms of Eq. (A1)]. +Away from the first lobe center (Φ/Φ0 ̸= 1) the off- +diagonal elements have additional terms in the denomi- +nator: +ΣS +01 = − +� +m +2πρSt−mtm+n |∆| cos φ +2 +� +∆2 − +� +ω + (n/2+m)(n−Φ/Φ0) +2msR2 +�2 . +(A11) +However, these terms do not change the qualitative pic- +ture, therefore, Eq. (A2) gives a reasonable estimation, +which makes ∆eff given by (A10) a relevant energy scale +instead of |∆|. +The spin-orbit interaction was not taken into account +in the simplified hollow-cylinder model, as it can play +a role only in the semiconducting nanowire itself, there- +fore, it should not be crucial for the shell modelling and +can only affect tunneling process to the QD states (intro- +ducing small spin-dependent corrections to the tunneling +amplitudes [64]) and QD parameters. For the latter we +can use empirical effective parameters, i.e. in the main +text we stated that the effective Zeeman splitting is re- +duced due to spin-orbit interactions so that the effective +g factor is small [56–58]. Therefore, we claim that includ- +ing spin-orbit interactions as well as more realistic (i.e. +hexagonal) geometry into consideration should not affect +the result qualitatively. + +8 +Appendix B: Analysis of Kondo conductance peak +Here we provide a simple estimate from below on the +Kondo temperature based on data provided in [12] and +supplemental material therein. From the supplemental +material (Fig. S7) we can see at 45 mT a Kondo enhance- +ment in the odd state (larger supercurrent amplitude in +the odd state in comparison to the even state and no π +phase shift), while at 40 mT we see a π phase behaviour +and similar supercurrent amplitudes in odd and even sec- +tors. +As a result, we can crudely estimate the Kondo +temperature from below as TK ≳ ∆ at 45 mT. NRG cal- +culations predict a transition from a doublet ground state +to a many-body Kondo singlet at TK ≈ 0.3∆ [33, 35, 41], +however, it was defined at zero phase difference, while the +current-phase relation at 45 mT already shows a critical +current enhancement as well as 0 phase behaviour at all +the phases, for nonzero phase the transition occurs at +higher values of TK/∆ [36, 65], which suggests that the +system is already deep in the Kondo regime. The gap is +suppressed at 50 mT, we thus can use a simplified formula +for a thin shell of radius R [66, 67]): +∆(Φ) ≈ ∆0 max +� +1 − ξ2 +R2 +� +n − Φ +Φ0 +�2 +, 0 +� +, +(B1) +where ξ is the superconducting coherence length, n = +⌊Φ/Φ0⌉ the number of phase windings in the shell (n = 0 +for the case here, as the system is still in the zeroth lobe). +Therefore, for an estimate we take 1 − ξ2 +R2 +� +− 50 +120 +�2 = 0, +as the center of the first lobe is at B = 120 mT, which +corresponds exactly to one flux quantum. Then at 45 mT +∆(Φ) ≈ ∆0 +� +1 − 122 +52 +� 45 +120 +�2� += 0.19∆0. +(B2) +The order parameter in the Al shell may be estimated +from below from the differential conductance at zero +magnetic field: ∆0 > 0.1 meV. Then, we can estimate +the Kondo temperature from below: TK > 19 µeV; one +can see that this estimate from below gives a value twice +larger than originally assumed. We stress again that the +estimate is crude and well below the real value, a more +accurate evaluation of the Kondo temperature could be +possible with more data around the transition from the +doublet to the Kondo singlet (40 − 45 mT) or the Kondo +peak analysis in the destructive regime (normal state). +Data on the second device provided in the supplemen- +tal material of Ref. [12] shows similar features suggesting +an enhancement of the Kondo effect. Moreover, in Fig. +S2 the differential conductance peak at low bias voltage +appears to be split in the first lobe, which may be due +to Zeeman splitting of the Kondo peak. Fig. S4 shows +that the critical current in the ancilla junction is larger in +the first lobe than in the QD-based junction (the ampli- +tude of the supercurrent in the SQUID is changing with +the QD occupation), which suggests that the tunneling +amplitude in the second device could be smaller than in +the first one. Therefore, we expect that the Kondo tem- +perature in the second device is lower (or the effective +g-factor can be larger) and the Kondo peak acquires a +visible splitting in the magnetic field (2VZ > TK in the +first lobe), which cannot be explained by MBSs. Further- +more, all three devices show supercurrent enhancement +in the odd state (first lobe) in comparison to even oc- +cupation, which is also a typical feature of the Kondo +effect. In conclusion, the additional data on the second +and third devices in the supplemental material of Ref. [12] +supports our idea of an enhancement of the Kondo effect +in the first lobe. +Appendix C: QD interactions +The self-energy due to interactions can be fully covered +only by elaborate numerical methods, such as NRG [33– +35, 68]. +However, some approaches can give at least +qualitative insights in the system behaviour at differ- +ent values of TK/∆. +For example, in [29] the author +proposed an FRG method [69], which gives within the +lowest-order static approximation flow equations from +high energy cutoff Λ in the Matsubara non-interacting +Green-function G0,Λ(iω) = θ(ω − Λ)G0(iω) in the form +∂ΛΣU +01(Λ) = −U(Λ) +π +ΣU +01(Λ) − ∆eff +2⟨Γs⟩ cos φ +2 +√ +Λ2+|∆|2 +DΛ(iΛ) +, +(C1) +∂ΛU(Λ) = − 2 +π +� U(Λ) +DΛ(iΛ) +�2 +������ +∆eff +2⟨Γs⟩ cos φ +2 +� +Λ2 + |∆|2 − ΣU +01(Λ) +������ +, +(C2) +DΛ(iω) = ω2 +� +�1 + +2⟨Γs⟩ +� +ω2 + |∆|2 +� +� + +������ +∆eff +2⟨Γs⟩ cos φ +2 +� +ω2 + |∆|2 − ΣU +01(Λ) +������ +2 +. +(C3) +The equations are for the middle of the odd sector +(which implies trivial diagonal elements) and symmet- + +9 +ric (left/right) tunneling, ΣU +01(∞) = 0, U(∞) = U. In +case of no phase winding, which was studied in [29], +∆eff = |∆| and ⟨Γs⟩ = Γ/2, while in our case (the first +lobe), the effective gap is reduced, which implies that the +flow is slow and the off-diagonal term ΣU +01(Λ) does not +reach the critical value for the quantum phase transition +into the π phase, which can be seen from the expression +for the supercurrent [29]: +⟨J⟩ = 1 +2π +� � +� +⟨ΓS⟩2∆2 +eff sin φ +DΛ=0(iω)(ω2 + |∆|2) +− 2⟨ΓS⟩∆effΣU +01(0) sin(φ/2) +DΛ=0(iω) +� +ω2 + |∆|2 +� +� dω +2π . +(C4) +If the first term dominates after the renormalization of +ΣU +01, which is the case of low ∆eff, the junction is in +the 0 phase, otherwise in the π phase. One should note +that Eq. (C4) is exact, the 0 or π phase behaviour of +the QD-based junction is determined by the ratio of two +competing terms, however, it is ΣU +01(0) in the second term +which cannot be calculated exactly in the limit of strong +interactions. + diff --git a/ftFMT4oBgHgl3EQf2DEu/content/tmp_files/load_file.txt b/ftFMT4oBgHgl3EQf2DEu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a21616eb6b402eb5b0fa4448d98ce1925d13d701 --- /dev/null +++ b/ftFMT4oBgHgl3EQf2DEu/content/tmp_files/load_file.txt @@ -0,0 +1,813 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf,len=812 +page_content='Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Aleksandr E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Svetogorov, Daniel Loss, and Jelena Klinovaja Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Dated: January 31, 2023) We analyze results of a recent experiment [D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Razmadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=', 125, 116803 (2020)] on transport through a quantum dot between two full-shell nanowires and show that the observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic flux in a full-shell nanowire) rather than the presence of Majorana bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Moreover, we propose that such a setup presents a unique and convenient system to study the competition between superconductivity and the Kondo effect and has significant advantages in comparison to other known approaches, as the important parameter is controlled by the magnetic flux through the full-shell nanowire, which can be significantly varied with small changes of magnetic field, and does not require additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' This competition is of fundamental interest as it results in a quantum phase transition between an unscreened doublet and a many-body Kondo singlet ground states of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' INTRODUCTION Semiconducting nanowires with full superconducting shell were recently introduced as possible realizations of topological superconductors, which may host Majo- rana bound states (MBSs) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' MBSs in turn have non- Abelian statistics, which could be exploited to develop a topologically protected qubit [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In nanowires with a thin shell Little-Parks effect [4] results in modulation of the superconducting order parameter with the ap- plied magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In case of a thin nanowire (with a diameter smaller than the superconducting coherence length) the Little parks effect is destructive resulting in a lobe structure of the order parameter as a function of the flux [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The idea of combining effectively one- dimensional superconductors with a vortex is already fas- cinating by itself: the vortex can induce a phase wind- ing of the superconducting order parameter and result in nontrivial properties, such as the well-known Caroli–de Gennes–Matricon bound states in 2D case [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Further experiments with full-shell nanowires were performed [8– 10], including those where the observed zero-bias anoma- lies were shown to have non-topological nature [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' How- ever, a recent experiment [12] with a gate-controlled quantum dot (QD) in a full-shell hybrid interferometer showed non-trivial features in addition to the zero-bias peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Without a magnetic flux (zeroth lobe) through the shell a change of occupation of the dot (from even to odd) leads to a change of a sign of the supercurrent through the dot, which is known as 0−π phase transition [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, with a flux around one magnetic flux quantum threading the superconducting shell (first lobe) the ef- fective Josephson junction seems to stay in the 0 phase even for the odd occupation of the QD, in agreement with theoretical predictions [17–21] for a Josephson junction based on QD between two topological superconductors hosting MBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Such peculiar behaviour could be seen as evidence for the presence of MBSs in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Nev- ertheless, the authors were not completely satisfied with the interpretation, as they could not explain some of the observed features, such as an enhancement of the super- current in the odd state in the first lobe and no signs of fractional Josephson effect expected in the presence of MBSs were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In this paper we argue that the results of this ex- periment can be interpreted as an enhancement of the Kondo effect [22–26] in the first lobe, including an en- hancement of the supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' It was predicted the- oretically [24, 27–29] and shown experimentally [30–32] that if the Kondo effect can develop on an odd-occupied QD between superconducting leads, the ground state is a many-body Kondo singlet instead of a doublet (un- screened electron), which restores the 0 phase behaviour and enhances the critical supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The phase tran- sition is determined by the ratio of two energy scales: the Kondo temperature TK and superconducting gap ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In this paper we show that if the superconducting order pa- rameter acquires a phase winding around the shell, the relevant energy scale is not the absolute value of the gap |∆| but an effective gap which is significantly suppressed due to destructive interference, which in turn leads to the enhancement of the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Our interpretation is further supported by the fact that a bright zero-bias feature and no π phase is observed at the closing of the zeroth lobe (around half the flux quantum), where no topological superconductivity is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As a result, we can claim that due to the detailed experimental data provided by Razmadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [12] it is possible to establish a non-trivial effect of the phase winding on the coherent transport and the ground state properties of a QD be- tween two full-shell nanowires, which has not been pre- dicted before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Further experimental studies of the effect can enrich the understanding of the underlying funda- mental physics as only elaborate numerical approaches have been developed to quantitatively capture the QD- based Josephson junction behaviour in the competition regime [24, 29, 33–35] (for review see [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='12442v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='supr-con] 29 Jan 2023 2 QD FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Schematic representation of Cooper-pair tunnel- ing through a QD electron state (green sphere) between two hollow-cylinder superconductors (blue) representing the shells and accumulation layers of underlying nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The tunnel- ing amplitude t(θ) (between the dot and each shell) is angle- dependent, as the electron wave function on the dot is shifted from the shell axis due to gating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' the superconducting order parameter on the shell has angle dependence - nonzero phase winding (n ̸= 0 for large enough magnetic flux Φ > Φ0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' MODEL In this paper we focus on a QD formed in an uncovered (etched superconductor) and gated region of a full-shell nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Assuming that electrons in the nanowire form an accumulation layer at the boundary with the super- conducting shell [1], we use a hollow-shell approximation transport through the system is mostly determined by Cooper pairs propagation along the shell and tunneling between the shells through the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We assume the shell to be much thinner than the magnetic penetration length and the diameter to be smaller than the superconducting coherence length, which results in no quantization of the magnetic flux through the shell but reentrant lobe struc- ture due to the destructive Little-Parks effect: supercon- ductivity is fully suppressed when the flux around odd half-integer multiples of a flux quantum is applied [6, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As a further simplification we model the shell as a hollow cylinder (in experiments it has rather a hexagonal cross- section) [1, 37] with radius R, then the total magnetic flux through the shell is Φ = πR2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We focus on the single-level QD limit (large level spacing), the number of electrons can be considered fixed by the applied gate due to strong Coulomb interactions on the QD (strong spa- tial confinement), which is a common experimental situ- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The simplest model to describe such a system is the superconducting impurity Anderson (SIA) model [38] modified by the magnetic flux though the shell [37]: H = HD + � s=l/r (Hs + V s SD) , (1) HD = � σ (ϵd + σVZ) d† σdσ + Un↑n↓, (2) Hs = � dxdθ � σ � ψ† σθx � p2 2ms − µs � ψσθx + ∆s(Φ)e−inθ+iφsψ† σθxψ† −σθx + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' � , (3) V s SD = � dθ � σ ts(θ)ψ† σθxdσ/(2π) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (4) Here ϵd is the bare dot energy level, which is controlled by gate voltage, VZ = µBgB/2 is the Zeeman field (g is Land´e g-factor and µB is Bohr magneton), σ corre- sponds to electron spin up/down state (±1 if it is a co- efficient, ↑ / ↓ if it is an index);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' d† σ is the QD electron creation operator (with spin σ), nσ = d† σdσ is the occupa- tion number operator, U is the charging energy (which is the largest energy scale in the system so that the single- level approximation is valid);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' ψσθx are the shell operators (we omit s = l/r index for the left/right shell), x is the coordinate along the shells, θ is the angle around the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The superconducting order parameter in the shells ∆l/r(Φ, θ) = ��∆(Φ)l/r �� e−inθ+iφl/r depends on magnetic flux Φ as well as angle θ, φl/r is the phase at the same angle θ = 0 to the left/right of the QD, n denotes the number of phase windings defined by an integer of the ratio of magnetic flux to flux quantum (Φ0 = π/e, we set ℏ = 1 throughout the paper): n = ⌊Φ/Φ0⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' p2/(2ms) and µs are the kinetic term and the chemical potential in the respective shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The last term describes tunneling between the left/right shell and the QD, the tunneling amplitude is given by ts(θ)/(2π) and dependends on the azimuthal angle θ, as the electron state on the QD is shifted away from the shell axis due to the dot gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We work in the zero-temperature limit (the temperature in relevant experiments is well below characteristic ener- gies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The main difference from [37] is the presence of two superconducting shells instead of one, which adds up and introduces an additional important parameter, namely the phase difference φ = φl − φr between the shells (at the same angle θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The effective Hamiltonian can be ob- tained by integrating out the shells and introducing the dot Green function as G(ω) = [ω − Heff(ω)]−1: Heff(ω) = Heff,↑(ω) ⊕ Heff,↓(ω) Heff,σ (ω) = � ϵd + σVz 0 0 −ϵd + σVz � + ΣU σ + ΣS, (5) where the first term corresponds to the bare QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The last two terms are self-energies from the Coulomb interaction U on the dot and from the proximity effect induced by the superconducting shells, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The proximity 3 effect contributes as ΣS = − 2 ⟨ΓS⟩ √ ∆2 − ω2 � ω ∆eff cos φ 2 ∆eff cos φ 2 ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (6) Here ⟨ΓS⟩ = πρs �� π −π dθ 2πt (θ) �2 stands for the tunneling energy scale averaged over the angle θ around the shell (the contribution comes from hopping from QD to shell and back with amplitude t(θ)/(2π)), ρs is the density of states at the Fermi energy, φ is the phase difference be- tween left and right shell (at fixed angle θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We assume symmetric tunneling tr(θ) = tl(θ) = t(θ) to the left/right shell, as corresponding results can be easily generalized for an asymmetric case [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Even more nontrivial contri- bution comes in the numerator of the off-diagonal terms (see Appendix A): ∆eff ≈ (1 − δ0n) ����� 2∆ � π −π t (θ) eiθn dθ 2π � π −π t (θ) dθ 2π ����� + δ0n |∆| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (7) One can see that for n ̸= 0 and axially symmetric tun- neling (t(θ) = const), ∆eff = 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' One should note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (7) is approximate, it works well for weak dependence of tunneling amplitude on θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' for general case see analyses in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The effect can be interpreted as destructive interfer- ence, which can be seen from a schematic of a Cooper- pair tunneling trajectory between two superconducting hollow cylinders through a QD electronic state, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' for axially symmetric case a Cooper pair has equal probabil- ity to tunnel between all possible angle positions on su- perconducting shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' If there is a phase winding around the shells, each trajectory corresponds to some phase dif- ference ∆φ ∈ [0, 2πn], then summing over all trajectories gives exactly zero (which can be written as zero effec- tive gap ∆eff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, the effective QD state (the bound state wave-function) is rather not centred on the shell axis, as the QD is gated from one side, which re- sults in θ-dependent tunneling amplitude t(θ) and, there- fore, nonzero ∆eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As was discussed in [37], the angle- dependence of tunneling is rather weak as the tunneling is determined by the tails of the wave-function under the superconducting shell, which is screening the electrical field form the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The most straightforward conse- quence of such a destructive interference is the reduction of the Josephson effect by a factor ∆eff/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, that is not the whole story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' First of all, as it was already briefly discussed, if a flux around half flux quantum is applied, superconductivity is completely sup- pressed due to Little-Parks effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Second, even an S- QD-S junction without magnetic flux shows nontrivial behaviour such as 0 − π phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' This phase transition was extensively studied theoretically, starting with the first predictions for transition induced by chang- ing the occupation of the QD [13–16, 40] and followed by more sophisticated regimes, when the Kondo effect may play a significant role [24, 29, 33–35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' These effects are due to strong Coulomb interactions on the QD repre- sented by ΣU σ term in the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In [37] it was calculated in Hartree-Fock-Bogoliubov approxima- tion [27, 42] (the lowest U-order expansion) ΣU σ ≈ U � ⟨n−σ⟩ ⟨dσd−σ⟩ ⟨d† σd† −σ⟩ −⟨nσ⟩ � , (8) which cannot capture the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The latter was studied with powerful numerical approaches [24, 33–35] as of now no reliable analytical approach capable of tack- ling the problem in the most interesting regime of com- petition between the Kondo effect and superconductivity has been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Fully analytical methods are avail- able only for special limiting cases such as Hartree–Fock- Bogoliubov approximation [27, 42] and perturbation in cotunneling [43, 44] through Yu-Shiba-Rusinov (YSR) state [45–47] analogs for ∆ ≫ TK or slave-boson mean field approaches in the opposite limit [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Neverthe- less, it is well established that if the Kondo temperature TK (characteristic energy scale) is large enough in com- parison to the superconducting gap ∆, the electron on a dot can form a Kondo singlet with quasiparticles in the superconductor (Kondo cloud) and, therefore, the junc- tion stays in the 0 phase even in the odd sector, the cotunneling process is enhanced which in turn increases the supercurrent [25, 29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The Kondo temperature depends on tunneling amplitude, Coulomb interaction, and bare QD level energy [50]: TK ∼ √ UΓ exp �πϵd 2Γ � 1 + ϵd U �� , Γ = 2⟨Γs⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (9) The exact proportionality factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='28 is well defined only in the middle of the odd occupation region (ϵd = −U/2) [36, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' An important question is how the super- conducting phase winding affects this competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' ANALYSES OF THE EXPERIMENT In this section we analyze the results of an experiment performed on a S-QD-S junction with a flux through the full-shell nanowire with the goal to establish topologi- cal superconductivity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Several non-trivial features were reported that could potentially indicate the pres- ence of Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' First, the differential con- ductance in a voltage-bias configuration was measured, then the current-phase relation (CPR) was probed in a SQUID geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The even-occupied regime does not show anything unexpected: the differential conductance clearly shows a gap-closing around half flux quantum due to the destructive Little-Parks effect and a gap-reopening in the first lobe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 2c in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Current-bias measure- ments in the SQUID geometry show a trivial Josephson effect in both lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In the odd-occupied regime a bright zero-bias peak develops at the closing of the zeroth lobe (around half flux quantum), which the authors identify as a Kondo peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In the destructive regime no peak is vis- ible (superconductivity is fully suppressed in the shells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 4 The zero-bias feature reappears in the first lobe, but less bright and a bit broadened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In the current-bias measure- ment a π phase is absent in the odd-occupied regime in the first lobe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 4 in [12]), while it is present in the ze- roth lobe (supercurrent reversal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As it was theoretically predicted, a full-shell nanowire could potentially acquire topological properties in the first lobe [1], which could explain the absence of the π phase and zero-bias peak by hybridization of a dot electron with MBSs [17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, current-bias measurements in SQUID geome- try show the absence of the π phase in the center of the odd sector already at the closing of the zeroth lobe (data in supplemental material of [12], Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' An enhance- ment of the supercurrent in the odd state is clearly visible in comparison to the even state, which is a typical fea- ture of the Kondo effect in S-QD-S junctions [24, 29, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' And all these features are qualitatively similar to the ones observed in the first lobe (nicely captured in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 4e-f in [12]): the CPR of the SQUID is given by a sinusoid, but in the odd state the critical current is larger (higher average value), no phase shift is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' That suggests that the observed effects are of the same origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As we have shown in the previous section the superconducting phase winding introduces a new important energy scale ∆eff, which plays the role of the effective gap for the QD and which is reduced in comparison to |∆| due to de- structive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In case of nonzero phase winding n > 0 off-diagonal elements of ΣS [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (8)] are smaller by a factor of ∆eff/∆, which suggests that in such a system the relevant parameter for the quantum phase transition between the unscreened doublet to the many- body Kondo singlet ground states is ∆eff/TK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' A more formal way to see that is to perform a renormalization group (RG) analyses: the RG flow starts at large energy cutoff, off-diagonal terms of ΣU σ get renormalized due to off-diagonal elements of ΣS, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As a result, we were able to deduce that in the first lobe for odd oc- cupation of the QD TK > ∆eff and the ground state is a Kondo singlet, which explains the 0 phase behaviour and the supercurrent enhancement as well as zero-bias peak in the differential conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Another question arising is whether the Zeeman effect can play a significant role, because the Zeeman field can split the Kondo peak if the g-factor is large enough [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, it was shown that due to spin-orbit interaction the effective g-factor on a long QD can be renormalized (towards small val- ues) [56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' That significantly complicates theoretical comparison of Zeeman energy VZ = µBgB and Kondo temperature TK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In Appendix B we provide some simple analyses of the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Another distinctive feature observed in experiment [12] is a change of dissipation between zeroth and first lobes: one can see a strong hysteresis in supercurrent though the SQUID in the zeroth lobe, which indicates underdamped junction regime corresponding to low dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' On contrary, in the first lobe no hysteresis is seen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' this ef- fect is independent of the QD occupation, therefore, it is not caused by the Kondo effect itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We suggest that the higher damping can be attributed to lower effective gap (and described in terms of subgap states induced by the vortex [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The different junction dissipation regime in two lobes also implies a different ratio of critical and switching current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In the underdamped regime the switching current (which is measured in the experiment) can be significantly lower than the actual critical current, as the macroscopic quantum phase tunneling cannot be neglected, while in the first lobe strong dissipation (over- damped regime) should result in the switching current being in good correspondence with the critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The experimental setup [12] appears to be a very con- venient and unique device to study competition between superconductivity and the Kondo effect at relatively low magnetic fields without additional gates to control the tunneling amplitude due to destructive Little-Parks ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The regime of competition is specifically interesting to study experimentally as no analytical approach ex- ists to provide quantitative description of the system in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The setup allowed the scientists to measure the CPR and differential conductance in the middle of the odd occupation sector all the way from the doublet ground state to the Kondo singlet smoothly varying the superconducting gap by changing the magnetic flux from zero to a half flux quantum, which does not even require going into the first lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The well resolved CPR close to the phase transition (at 40 mT and 45 mT for the first device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' S8 in [12],) has a drastic difference, which is an outstanding feature of the change in the character of the ground state and is in perfect qualitative agree- ment with theoretical predictions (numerics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We sug- gest that a measurement of the CPR at different values of flux with smaller steps around the transition could be sufficient to establish the transitions between 0, 0′, π′ and π phases of such a QD-based junction [27, 42] in the middle of the odd parity sector (before such tran- sitions have been observed only as a function of gate voltage [31, 60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' For this we recommend to fabricate an asymmetric SQUID with an ancilla junction being in- dependent of the flux through the shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' a separate SIS junction) and having slightly larger critical current so that the CPR of S-QD-S junction is directly observed (large difference in critical currents between the junc- tions forming the SQUID decreases the contrast of the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Further study of the first lobe can provide bet- ter understanding of the Kondo cloud formation due to destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Moreover, it could be interest- ing to study the effect in shells of larger radius (or thin- ner shell, so that the superconducting coherence length is shorter than the shell’s radius [6]), when Little-Parks effect does not suppress superconducting gap to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In that regime suppression of the gap at half flux quantum may not be enough to enhance the Kondo effect, how- ever, the phase winding at higher magnetic fields can still reduce the effective gap to the values below the Kondo temperature, which would result in a zero-bias peak only in the first (or higher) lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK We provided a coherent interpretation for the results observed in an experiment [12] on a transport through a QD between two full-shell nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Due to accu- rate and sufficient data presented, we were able to es- tablish the effect of superconducting order parameter phase winding on a ground state of the QD and attribute it to the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We showed that the qunatum phase transition between a doublet and a many-body Kondo singlet ground state is controlled by a parame- ter ∆eff/TK ≪ |∆|/TK in case of a superconducting phase winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We discussed the consequences of this transitions and suggested experiments to test the exist- ing theoretical results on the regime of competition of the Kondo effect and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Finally, theoret- ical analyses of the results suggest that the conductance enhancement due to the Kondo effect in a vortex may as well explain zero-bias anomalies observed in different systems, such as the vortexes localized at magnetic im- purities of some superconductors [61, 62], which requires further studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We thank Wolfgang Belzig, Mikhail Pletyukhov, Charles Marcus and Saulius Vaitiek˙enas for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' This project received funding from the European Union’s Horizon 2020 re- search and innovation program (ERC Starting Grant, grant agreement No 757725).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Vaitiek˙enas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Winkler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Petersson, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Marcus, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 125, 156804 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Aligia, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' von Oppen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 119, 046801 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} 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Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' B 101, 014512 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Kouwenhoven and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Glazman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' World 14, 33 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [23] M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Mart´ın-Rodero and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Levy Yeyati, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' : Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 24, 385303 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [69] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Karrasch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Enss, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Meden, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' B 73, 235337 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Appendix A: Effective gap The effective Hamiltonian (5) was derived in [37] within the SIA model and consists of three parts: bare QD energy, Coulomb interactions, and proximity effect from the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' For a usual S-QD-S junction (no phase winding) the latter is given by [27, 29] ΣS = − � s πρSt2 √ ∆2 − ω2 � ω |∆| eiφs |∆| e−iφs ω � , (A1) where the index s stands for left/right superconductor (we assume |∆| and t to be the same for the left/right shells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As discussed in [37], the approach can be ex- tended to the hollow-cylinder model, which allows us to include a magnetic flux in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Here we briefly sketch the main steps of that approach, the difference with [37] being that here we consider superconducting shells both to the right and to the left of the QD, there- fore, we need to take a relative phase into account as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' One needs to sum over all the possible positions on the left/right shell, which is done by introducing angle- dependent tunneling amplitudes t(θ)/(2π) as well as or- der parameters ∆s = |∆|eiφs−inθ, where n is the num- ber of the phase windings n = ⌊Φ/Φ0⌉ due to the flux through the shell, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As was reported in [37] for the axially symmetric case t(θ) = const, the off-diagonal elements for any n ̸= 0 are zero, which can be seen as destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Realistically, the wave function of an electron on the QD cannot be treated as axially symmetric (with respect to the shell axis) due to gat- ing from one side and due to inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, we can still assume the tunneling to have a relatively weak angle dependence (as discussed in [37] the tunnel- ing is determined by the wave-function’s tails underneath the shell, which are not very sensitive to the gating): t(θ) = t0 + δt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' It is convenient to decompose this tunneling amplitude into harmonics t(θ) = � m tme−imθ, where tm = � π −π t(θ)eimθ dθ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Then, one can get the prox- imity term in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A1) by integrating the shell contribution over the angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The simplest esti- mation comes from expanding t2(θ) = t2 0 + 2t0δt(θ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='. The off-diagonal term takes the form ΣS 01 ≈ − � s 2πρSt0 |∆| eiφs √ ∆2 − ω2 2π � 0 t(θ)e−inθdθ 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A2) Let us derive the term in a more accurate fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The thin superconducting shell Hamiltonian can be written in cylindrical coordinates as [1, 63] Hs = � k2 s + k2 r + (kθ + eAτz)2 2ms − µs � τz+ + |∆| (cos [−inθ + iφs] τx + sin [−inθ + iφs] τy) , (A3) where ks, kr and kθ are the longitudinal, radial, and tan- gential components of the momentum operator, ms is the 7 effective electron mass, µs is the chemical potential in the shell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' A = 1 2eR Φ Φ0 is the vector potential, τi are Pauli ma- trices acting in Nambu-Gorkov space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' One can introduce a generalized angular momentum Jz = −i∂θ+ 1 2nτz+ 1 2σz, which is conserved (commutes with Hamiltonian) [1, 59], then the eigenvalues mJ are good quantum numbers and can take half-integer values: � mJ − 1 2 − 1 2n � ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A4) The resulting angular momentum number m is integer and fixed by mJ and n in each spin and Nambu sec- tor (note that it is exact only in the absence of radial spin-orbit interaction [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Due to the term 1 2nτz in the definition of the generalized angular momentum, for a fixed spin the Hamiltonian is not block-diagonal in the (m, −m) Nambu sectors, but in the (m, −m − n) sec- tors [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The retarded Green function for the shell (de- coupled from the QD) between two positions θ and θ′ is then given by [37] gs(ω, θ, θ′) = � m e−im(θ−θ′) Dm,n � � ω + ϵks + [m+n−Φ/(2Φ0)]2 2msR2 ∆e−iφs+inθ′ ∆eiφs−inθ � ω − ϵks − [m+Φ/(2Φ0)]2 2msR2 � e−in(θ−θ′) � � , (A5) where Dm,n = ω2 − ∆2 − (ϵks − Lm)2 − (ω − ϵks + Lm) � Φn − 2 � ΦnLm � , (A6) with Lm = � m + Φ 2Φ0 �2 2msR2 and Φn = � n − Φ Φ0 �2 2msR2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A7) Then the proximity effect of the shells on the QD can be described by the self-energy [37] ΣS(ω) = � s � dks � dθ 2π dθ′ 2π t(θ)gs(ω, θ, θ′)t(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A8) In the middle of the first lobe Φ/Φ0 = n = 1 the result can be written in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A1) but with t replaced by t0 = � π −π t(θ) dθ 2π, which is just the tunneling amplitude averaged over θ, and |∆| in the nnumerator of the off- diagonal terms replaced by ∆eff = ����� � m t−mtm+n t2 0 ∆ ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A9) In case of weak modulation compared to the angle- independent tunneling (t0 ≫ |δt(θ)|) one gets for n ̸= 0 ∆eff ≈ 2 ����� ∆ � π −π t (θ) eiθn dθ 2π � π −π t (θ) dθ 2π ����� ≪ |∆| , (A10) which is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A2) [if one substitutes ∆eff for ∆ in the numerator of the off-diagonal terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Away from the first lobe center (Φ/Φ0 ̸= 1) the off- diagonal elements have additional terms in the denomi- nator: ΣS 01 = − � m 2πρSt−mtm+n |∆| cos φ 2 � ∆2 − � ω + (n/2+m)(n−Φ/Φ0) 2msR2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A11) However, these terms do not change the qualitative pic- ture, therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (A2) gives a reasonable estimation, which makes ∆eff given by (A10) a relevant energy scale instead of |∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The spin-orbit interaction was not taken into account in the simplified hollow-cylinder model, as it can play a role only in the semiconducting nanowire itself, there- fore, it should not be crucial for the shell modelling and can only affect tunneling process to the QD states (intro- ducing small spin-dependent corrections to the tunneling amplitudes [64]) and QD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' For the latter we can use empirical effective parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' in the main text we stated that the effective Zeeman splitting is re- duced due to spin-orbit interactions so that the effective g factor is small [56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Therefore, we claim that includ- ing spin-orbit interactions as well as more realistic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' hexagonal) geometry into consideration should not affect the result qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' 8 Appendix B: Analysis of Kondo conductance peak Here we provide a simple estimate from below on the Kondo temperature based on data provided in [12] and supplemental material therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' From the supplemental material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' S7) we can see at 45 mT a Kondo enhance- ment in the odd state (larger supercurrent amplitude in the odd state in comparison to the even state and no π phase shift), while at 40 mT we see a π phase behaviour and similar supercurrent amplitudes in odd and even sec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' As a result, we can crudely estimate the Kondo temperature from below as TK ≳ ∆ at 45 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' NRG cal- culations predict a transition from a doublet ground state to a many-body Kondo singlet at TK ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='3∆ [33, 35, 41], however, it was defined at zero phase difference, while the current-phase relation at 45 mT already shows a critical current enhancement as well as 0 phase behaviour at all the phases, for nonzero phase the transition occurs at higher values of TK/∆ [36, 65], which suggests that the system is already deep in the Kondo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' The gap is suppressed at 50 mT, we thus can use a simplified formula for a thin shell of radius R [66, 67]): ∆(Φ) ≈ ∆0 max � 1 − ξ2 R2 � n − Φ Φ0 �2 , 0 � , (B1) where ξ is the superconducting coherence length, n = ⌊Φ/Φ0⌉ the number of phase windings in the shell (n = 0 for the case here, as the system is still in the zeroth lobe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Therefore, for an estimate we take 1 − ξ2 R2 � − 50 120 �2 = 0, as the center of the first lobe is at B = 120 mT, which corresponds exactly to one flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Then at 45 mT ∆(Φ) ≈ ∆0 � 1 − 122 52 � 45 120 �2� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='19∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (B2) The order parameter in the Al shell may be estimated from below from the differential conductance at zero magnetic field: ∆0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='1 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Then, we can estimate the Kondo temperature from below: TK > 19 µeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' one can see that this estimate from below gives a value twice larger than originally assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' We stress again that the estimate is crude and well below the real value, a more accurate evaluation of the Kondo temperature could be possible with more data around the transition from the doublet to the Kondo singlet (40 − 45 mT) or the Kondo peak analysis in the destructive regime (normal state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Data on the second device provided in the supplemen- tal material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [12] shows similar features suggesting an enhancement of the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' S2 the differential conductance peak at low bias voltage appears to be split in the first lobe, which may be due to Zeeman splitting of the Kondo peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' S4 shows that the critical current in the ancilla junction is larger in the first lobe than in the QD-based junction (the ampli- tude of the supercurrent in the SQUID is changing with the QD occupation), which suggests that the tunneling amplitude in the second device could be smaller than in the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Therefore, we expect that the Kondo tem- perature in the second device is lower (or the effective g-factor can be larger) and the Kondo peak acquires a visible splitting in the magnetic field (2VZ > TK in the first lobe), which cannot be explained by MBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Further- more, all three devices show supercurrent enhancement in the odd state (first lobe) in comparison to even oc- cupation, which is also a typical feature of the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In conclusion, the additional data on the second and third devices in the supplemental material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' [12] supports our idea of an enhancement of the Kondo effect in the first lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' Appendix C: QD interactions The self-energy due to interactions can be fully covered only by elaborate numerical methods, such as NRG [33– 35, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' However, some approaches can give at least qualitative insights in the system behaviour at differ- ent values of TK/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' in [29] the author proposed an FRG method [69],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' which gives within the lowest-order static approximation flow equations from high energy cutoff Λ in the Matsubara non-interacting Green-function G0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content='Λ(iω) = θ(ω − Λ)G0(iω) in the form ∂ΛΣU 01(Λ) = −U(Λ) π ΣU 01(Λ) − ∆eff 2⟨Γs⟩ cos φ 2 √ Λ2+|∆|2 DΛ(iΛ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (C1) ∂ΛU(Λ) = − 2 π � U(Λ) DΛ(iΛ) �2 ������ ∆eff 2⟨Γs⟩ cos φ 2 � Λ2 + |∆|2 − ΣU 01(Λ) ������ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (C2) DΛ(iω) = ω2 � �1 + 2⟨Γs⟩ � ω2 + |∆|2 � � + ������ ∆eff 2⟨Γs⟩ cos φ 2 � ω2 + |∆|2 − ΣU 01(Λ) ������ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (C3) The equations are for the middle of the odd sector (which implies trivial diagonal elements) and symmet- 9 ric (left/right) tunneling, ΣU 01(∞) = 0, U(∞) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' In case of no phase winding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' which was studied in [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' ∆eff = |∆| and ⟨Γs⟩ = Γ/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' while in our case (the first lobe),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' the effective gap is reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' which implies that the flow is slow and the off-diagonal term ΣU 01(Λ) does not reach the critical value for the quantum phase transition into the π phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' which can be seen from the expression for the supercurrent [29]: ⟨J⟩ = 1 2π � � � ⟨ΓS⟩2∆2 eff sin φ DΛ=0(iω)(ω2 + |∆|2) − 2⟨ΓS⟩∆effΣU 01(0) sin(φ/2) DΛ=0(iω) � ω2 + |∆|2 � � dω 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (C4) If the first term dominates after the renormalization of ΣU 01, which is the case of low ∆eff, the junction is in the 0 phase, otherwise in the π phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' One should note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} +page_content=' (C4) is exact, the 0 or π phase behaviour of the QD-based junction is determined by the ratio of two competing terms, however, it is ΣU 01(0) in the second term which cannot be calculated exactly in the limit of strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFMT4oBgHgl3EQf2DEu/content/2301.12442v1.pdf'} diff --git a/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf b/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..25363c07663910dbf3590218943f790a79483731 --- /dev/null +++ b/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf @@ -0,0 +1,3 @@ +version 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a/idE0T4oBgHgl3EQfYACE/content/tmp_files/2301.02301v1.pdf.txt b/idE0T4oBgHgl3EQfYACE/content/tmp_files/2301.02301v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..88d17f76a5a541e57b5a4026ac85ac6bdc7061dc --- /dev/null +++ b/idE0T4oBgHgl3EQfYACE/content/tmp_files/2301.02301v1.pdf.txt @@ -0,0 +1,893 @@ +LINEAR RESPONSE DUE TO SINGULARITIES +WAEL BAHSOUN AND STEFANO GALATOLO +Abstract. It is well known that a family of tent-like maps with bounded +derivatives has no linear response for typical deterministic perturbations chang- +ing the value of the turning point. In this note we prove a rather unexpected +result: if we consider a tent-like family with a cusp at the turning point, we re- +cover the linear response. More precisely, let Tε be a family of such cusp maps +generated by changing the value of the turning point of T0 by a deterministic +perturbation and let hε be the corresponding invariant density. We prove that +ε �→ hε is differentiable in L1 and provide a formula for its derivative. +1. Introduction +Let M be a compact manifold with a reference volume and T : M → M be a map, +whose iterates determine the dynamics. A T-invariant measure µ; i.e. T∗µ = µ, is +said to be physical if there is a positive volume set B such that for any continuous +observable f : M → R +lim +n→∞ +1 +n +n−1 +� +i=0 +f(T i(x)) = +� +M +f dµ +for all x ∈ B. An important question is to study how such measures vary, in an +appropriate topology, under suitable perturbations of T. Namely, consider a family +{(M, Tε)}ε∈V , where V is a small neighbourhood of 0, T0 := T and Tε → T0, as +ε → 0, in some suitable topology. For many chaotic systems it turns out that Tε +admits a unique physical measure µε. +The system (M, T0, µ0) is called statistically stable if the map ε �−→ µε is +continuous, in an appropriate topology, at ε = 0. +Quantitative statistical sta- +bility is provided by quantitative estimates on its modulus of continuity (see e.g. +[1, 26, 20, 21, 32]). In the case where ε �−→ µε is differentiable in some sense, the +system is also said to admit linear response; i.e., the ‘derivative’ ˙µ0 represents the +first order term of the response of the system to the perturbation +µδ = µ0 + ˙µ0ε + o(ε) +where the error is understood in an appropriate topology. +Linear response results in the context of deterministic dynamics was first ob- +tained in the case of uniformly hyperbolic systems [31] (see also [25]). Nowadays +linear response results are known for systems outside the uniformly hyperbolic set- +ting (see [13, 15, 17, 22, 26], the survey article [8] and references therein.) For +systems with discontinuities or critical points the situation is quite complicated +even in the presence of uniform expansion. Indeed, for suitable small perturbations +Date: January 9, 2023. +2020 Mathematics Subject Classification. Primary 37A05, 37E05. +Key words and phrases. Linear response, Lorenz like maps. +1 +arXiv:2301.02301v1 [math.DS] 5 Jan 2023 + +2 +WAEL BAHSOUN AND STEFANO GALATOLO +of piecewise expanding maps there are several examples that lack statistical sta- +bility or linear response (see [26, 30, 9] and [3] for recent results in this direction). +In the literature there are some results indicating that perturbations which are not +changing the topological class1 or tangent to the topological class, linear response is +likely to occur, while for perturbations which are transversal to the topological class +there is no linear response (see [13] for the case of piecewise expanding unimodal +maps, [10] and [14] for smooth unimodal maps and [22] for results along this line +in the case of rotations). +In this paper we study perturbations of one dimensional tent-like piecewise ex- +panding maps with unbounded derivatives at the turning point (a cusp singular- +ity). We prove linear response for a large class of deterministic perturbations also +changing the turning points and hence the topological class. The existence of such +a linear response is due to the singularity. +Indeed, unlike the usual tent maps +with bounded derivative, which do not admit linear response for the same kind of +perturbations([8, 30]), linear response in the singular case is attained due to fact +that the cusp has a ‘regularizing effect’ at the level of the action of the associated +transfer operators. Cusp like singularities and unbounded derivatives appear in +several important systems including Lorenz-type maps [5, 6] and billiard maps (see +e.g. [12]). Unlike the case of Anosov flows [16] where linear and higher order re- +sponse is proved, in the case of the classical Lorenz flow only statistical stability is +known [2, 5, 6]. By highlighting the role of singularities in studying linear response, +we hope that the results of the present paper contribute to the understanding of +linear response for Lorenz-like flows and billiard maps. +The paper is organised as follows. In Section 2 we introduce the class of systems +we consider and state the main result, Theorem 1, of the paper. In Section 3 we +prove Theorem 1 in a series of lemmas. +Acknowledgements. The research of W. Bahsoun is supported by EPSRC grant +EP/V053493/1. W. Bahsoun would like to thank the University of Pisa for its +hospitality during his visit to S. Galatolo. The authors thank M. Ruziboev and D. +Smania for fruitful discussions. +2. Family of maps and statement of the main result +Let C ≥ 0, c ∈ (0, 1), β ∈ (−1, − 3 +4), δ > 0. For ε ∈ (−δ, δ) consider a family of +maps Tε : [0, 1] → [0, 1] satisfying the following assumptions: +(A1) T0,ε := Tε|[0,c) and T1,ε := Tε|(c,1] are one to one. +(A2) Tε(0) = 0, Tε(1) = 0, Tε(c) = aε ∈ [0, 1]. +(A3) Tε|[0,1]\{c} ∈ C3. +(A4) infx∈[0,1]\{c} |T ′ +ε(x)| ≥ θ > 1. +(A5) Tε is transitive. +(A6) limx→c± T ′ +ε(x) = ±∞ and limx→c± |T ′ +ε(x)| +|x−c|β = C1. +(A7) limx→c± +|T ′′ +ε (x)| +|x−c|β−1 = C2, limx→c± +|T ′′′ +ε (x)| +|x−c|β−2 = C3. +(A8) supε∈(−δ,δ),i∈{0,1,2} supx | T (i+1) +ε +(x) +(x−c)β−i | < ∞. +1A perturbation does not change the topological class if it changes a system to a system which +is topologically conjugated to it + +LINEAR RESPONSE DUE TO SINGULARITIES +3 +(A9) For2 f ∈ L1 set +( 1 +|T ′ε| · f) ◦ T −1 +0,ε (x) · 1T0,ε[0,c)(x) + ( 1 +|T ′ε| · f) ◦ T −1 +1,ε (x) · 1T1,ε(c,1](x) +:= ψ1,ε(x) + ψ2,ε(x). +For3 f ∈ W 1,1, ε �→ ψj,ε, j = 1, 2, is continuous at ε = 0. Moreover, for +f ∈ C1 the partial derivatives ∂εψj,ε, ∂xψj,ε, ∂x∂εψj,ε, ∂ε∂xψj,ε, j = 1, 2, +exist and are jointly continuous4. +Figure 1 below presents an example of a map from the above family. +Figure 1. An example of Tε with aε = 0.8 and c = 1 +2. +Let m denote Lebesgue measure on [0, 1]. The transfer operator associated with +Tε, denoted by LTε, is defined by duality as follows: for f ∈ L1, g ∈ L∞ +� 1 +0 +LTεf · gdm = +� 1 +0 +f · g ◦ Tεdm. +Although it is known that LTε admits a spectral gap when acting on the space of +functions of generalized bounded variations [27], we study the action of LTε on finer +Banach spaces. Namely, the Sobolev spaces W i,1, i = 1, 2. In particular, we show +that LTε admits a spectral gap on W i,1, i = 1, 2. This will allow us to conclude +that Tε admits an invariant density which is regular enough, in x. The latter is +essential to derive the linear response formula that we are after in this work. The +following result is the main result of the paper. +Theorem 1. Tε admits a unique invariant density hε ∈ W 2,1; moreover, ε �→ hε +is differentiable in L1. In particular, +hε = h0 + ε(I − LT0)−1(q) + o(ε), +2Note that the definition of ψi depends on f. +3We use the notation W i,1, i = 1, 2 to denote the usual Sobolev spaces equipped with the +norms ∥f∥W i,1 = �i +k=0 ∥f(k)∥1. +4This implies +(2.1) +∂ε(ψ(i) +j,ε) = (∂εψj,ε)(i) +i = 0, 1; j = 0, 1. + +1.0 +y +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +X4 +WAEL BAHSOUN AND STEFANO GALATOLO +here +(2.2) +q(x) = +� LT0[A0h′ + B0h](x) +for x ∈ [0, a0) +0 +for x ∈ [a0, 1] , +with +Aε = − +�∂εTε +T ′ε +� +, +Bε = +�∂εTε · T ′′ +ε +T ′2 +ε +− ∂εT ′ +ε +T ′ε +� +and the o is in the L1-topology. +3. Proof of Theorem 1 +In this section we prove Theorem 1 in two steps. First, in subsection 3.1 we show +that LTε admits a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1. This +implies, in particular, that Tε has a unique invariant density hε ∈ W 2,1. Then in +subsection 3.2 we show that ε �→ hε is differentiable in L1 and obtain a formula for +the derivative. +3.1. Uniform spectral gap on W 1,1 and on W 2,1. In this section we prove +that the transfer operators associated with our class of systems admit a uniform +spectral gap when acting on suitable Sobolev spaces (Lemma 3). We obtain this as +a consequence of classical results which are recalled below. +3.1.1. Classical results. We first recall Lemma 2.2 of [11] which is a modification +of Theorem XIV.3 of [24]. +Lemma 1. Let (B, ∥ ∥s) be a Banach space, let ∥ ∥w be a continuous semi-norm +on B and Q a bounded linear operator on B, such that for any sequence xn with +∥xn∥s ≤ 1 it contains a Cauchy subsequence for ∥ ∥w. +Assume there is λ ≥ 0 and C > 0 such that, for any f ∈ B +∥Qf∥s ≤ λ∥f∥s + C∥f∥w. +Then the essential spectral radius of Q is bounded by λ. +We also recall the main result of [28], stating it in a simplified form suitable for +our purposes. +Lemma 2. Let (B, ∥ ∥s) and ∥ ∥w as above, and Pε : B → B with ε ≥ 0 be a +family of bounded linear operators. Assume: there are C1 > 0 and M ≥ 1 such that +(3.1) +∥P n +ε ∥w ≤ C1M n; +there are C2, C3 > 0 and 0 ≤ λ < 1 such that, for any f ∈ B, ε ≥ 0, n ∈ R +(3.2) +∥P n +ε f∥s ≤ C2λn∥f∥s + C3M n∥f∥w; +1 is not in the residual spectrum of Pε for any ε ≥ 0; +(3.3) +∥P0 − Pε∥s→w ≤ τ(ε) +where τ(ε) → 0 monotonically and upper semicontinuously. Then there are ϵ0, a, b > +0 such that for all 0 ≤ ε ≤ ϵ0 and f ∈ B +(3.4) +∥(Id − Pε)−1f∥s ≤ a∥f∥s + b∥f∥w +and +(3.5) +lim +ε→0 ∥(Id − P0)−1 − (Id − Pε)−1∥s→w = 0. + +LINEAR RESPONSE DUE TO SINGULARITIES +5 +3.1.2. Uniform spectral gap for the associated transfer operators. We now state the +main result of this section. +Lemma 3. For any ϵ ∈ (−δ, δ) LTε admits a unique invariant density hε ∈ W 2,1. +Furthermore, LTε has a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1. +In particular, ∃ C > 0 such that for all ϵ ∈ (−δ, δ) +(3.6) +∥(Id − LTε)−1∥W 1,1→W 1,1 ≤ C. +To prove Lemma 3 we apply Lemma 1 and Lemma 2. We first start with an +auxiliary lemma that verifies condition 3.3 in Lemma 2. +Lemma 4. Let f ∈ W 1,1. Then +(3.7) +lim +ϵ→0 ∥(LT0 − LTε)f∥L2 = 0. +Proof of Lemma 4. By (3.8) we have +∥(LT0 − LTε)f∥L2 ≤ ∥ψ1,ε − ψ1,0∥L2 + ∥ψ2,ε − ψ2,0∥L2. +Since f ∈ W 1,1, the proof follows from assumption (A9). +□ +Proof of Lemma 3. We are going to prove that LTε admits a uniform Lasota-Yorke +inequality when acting on W i,1, i = 1, 2, with L2 being the weak space. +Consider the pointwise representation of the transfer operator associated with +Tε +(3.8) +(LTεf)(x) = +� � +y∈T −1 +ε +(x) +1 +|T ′ε(y)|f(y) if x ∈ [0, aϵ] +0 if x ∈ (aϵ, 1] +and, for f ∈ W 1,1, its derivative with respect to x: +(3.9) +(LTεf) +′(x) = +� � +y∈T −1 +ε +(x) +f ′(y) +|T ′ε(y)|T ′ε(y) − +T ′′ +ε (y) +(T ′ε(y))3 f(y) if x ∈ [0, aϵ] +0 if x ∈ (aϵ, 1] +Notice that LTεf ∈ W 1,1. Indeed since f is bounded, by (3.8) limx→aϵ LTεf = 0. +LTεf is then continuous, with derivative almost everywhere. Furthermore, we note +that in (3.9) when x → aϵ then y → c and5 +T ′′ +ε (y) +(T ′ +ε)3(y) ∼ |y−c|β−1 +|y−c|3β += |y − c|−2β−1 . +Therefore, +T ′′ +ε (y) +(T ′ +ε)3(y) → 0 as y → c, this shows that LTεf ∈ W 1,1. +Now, observe that +(3.10) +(LTεf) +′ = LTε( 1 +T ′ε +f ′) − LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +f). +By (3.10) we get +∥(LTεf) +′∥1 +≤ +∥LTε( 1 +T ′ε +f ′)∥1 + ∥LTε( T ′′ +ε +(T +′ +ε)2 f)∥1 +(3.11) +≤ +∥ 1 +T ′ε +f ′∥1 + ∥ T ′′ +ε +(T +′ +ε)2 f∥1 +(3.12) +≤ +∥ 1 +T ′ε +∥∞∥f ′∥1 + ∥ T ′′ +ε +(T +′ +ε)2 f∥1 +(3.13) +≤ +λ∥f ′∥1 + ∥ T ′′ +ε +(T +′ +ε)2 ∥2∥f∥2 +(3.14) +5Here the asymptotic equivalence f ∼ g stands for limx→c +|f(x)| +|g(x)| = C with 0 < C < +∞. + +6 +WAEL BAHSOUN AND STEFANO GALATOLO +where for each ε, λ := ∥ 1 +T ′ε ∥∞ < 1 and ∥ T ′′ +ε +(T ′ +ε)2 ∥2 < +∞. Note that furthermore +by assumptions (A4) and (A8) the quantities ∥ 1 +T ′ε ∥∞ and ∥ T ′′ +ε +(T ′ +ε)2 ∥2 can be bounded +uniformly for ε ∈ (−δ, δ). +We now show that the transfer operators are continuous in the weak norm. By +(3.8) we have +(3.15) +∥LTε∥L2→L2 ≤ 2. +Indeed we have +∥(LTεf)∥L2 ≤ ∥ψ1,ε∥L2 + ∥ψ2,ε∥L2 +while +� +[0,1] +(ψ1,ε)2dm += +� +[0,1] +[( 1 +|T ′ε| · f) ◦ T −1 +0,ε (x) · 1T0,ε[0,c)(x)]2dx += +� +[0,aε] +1 +|T ′ε(T −1 +0,ε (x))| +1 +|T ′ε(T −1 +0,ε (x))| · [f(T −1 +0,ε (x))]2dx +≤ +sup +x [ +1 +|T ′(x)|] +� +[0,aε] +1 +|T ′ε(T −1 +0,ε (x))| · [f(T −1 +0,ε (x))]2dx += +sup +x [ +1 +|T ′(x)|]∥LTε(f 2 · 1[0,c))∥L1 ≤ sup +x [ +1 +|T ′(x)|]∥f 2 · 1[0,c)∥L1 +≤ +sup +x [ +1 +|T ′(x)|][∥f∥L2]2. +A similar estimate holds for ∥ψ2,ε∥2 +L2. +Using the Lasota Yorke inequalities (3.11)-(3.14) and the fact that W 1,1 is com- +pactly embedded in L2 by the Rellich-Kondrakov theorem, applying Lemma 1, we +get that the essential spectral radius ρess ≤ λ < 1. By this, any element of the +spectrum with modulus strictly bigger than λ is an isolated eigenvalue and we can +prove that spectral radius of LTε is 1 (in fact it cannot be smaller than 1 because +the transfer operator is Markov; consequently, iterating the uniform density does +not converge to 0, and for similar reasons there cannot be an eigenvalue of modulus +greater than 1, since the corresponding eigenfunction cannot be expanded by iter- +ating the transfer operator which is a weak contraction in the L1 norm). Since the +map is transitive there are no eigenvalues on the unit circle other than the simple +eigenvalue 1. This implies that Tε admits a unique invariant density hε ∈ W 1,1. +Differentiating (3.10), we get +(LTεf)′′ = (LTε( 1 +T ′ε +f ′) − LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +f))′ += (LTε( 1 +T ′ε +f ′))′ − (LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +f))′ += LTε( 1 +T ′ε +( 1 +T ′ε +f ′)′) − LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +1 +T ′ε +f ′) +− [LTε( 1 +T ′ε +( +T ′′ +ε +|T +′ +ε|T +′ +ε +f)′) − LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +T ′′ +ε +|T +′ +ε|T +′ +ε +f)] +(3.16) + +LINEAR RESPONSE DUE TO SINGULARITIES +7 +where +LTε( 1 +T ′ε +( 1 +T ′ε +f ′)′) − LTε( T ′′ +ε +|T ′ε|T ′ε +1 +T ′ε +f ′) = LTε( 1 +T ′ε +( −T ′′ +ε +(T +′ +ε)2 f ′ + 1 +T ′ε +f ′′)) − LTε( +T ′′ +ε +|T +′ +ε|T +′ +ε +1 +T ′ε +f ′) += LTε(( 1 +T ′ε +)2f ′′)) − LTε([ T ′′ +ε +(T +′ +ε)2 + +T ′′ +ε +|T +′ +ε|T +′ +ε +] 1 +T ′ε +f ′) +(3.17) +and +LTε( 1 +T ′ε +( +T ′′ +ε +|T +′ +ε|T +′ +ε +f)′) = LTε( 1 +T ′ε +[( +T ′′ +ε +|T +′ +ε|T +′ +ε +)′f + ( +T ′′ +ε +|T +′ +ε|T +′ +ε +)f ′]) += LTε( 1 +T ′ε +(T ′′′ +ε |T +′ +ε|T +′ +ε − 2(|T ′ +ε|T ′′ +ε )T ′′ +ε +(T +′ +ε)4 +)f + 1 +T ′ε +( +T ′′ +ε +|T +′ +ε|T +′ +ε +)f ′)). +(3.18) +Since β ∈ (−1, − 3 +4), we have +• ( 1 +T ′ε )2 ∈ L∞; +• g1,a := +T ′′ +ε +|T ′ +ε|T ′ +ε +1 +T ′ε ∼ |y − c|−2β−1; i.e, g1,a ∈ L∞; +• g1,b := +T ′′ +ε +(T ′ +ε)2 +1 +T ′ε ∼ |y − c|−2β−1; i.e, g1,b ∈ L∞; +• g2 := +1 +T ′ε ( T ′′′ +ε |T +′ +ε|T +′ +ε +(T ′ +ε)4 +) ∼ |y − c|−2β−2, ; i.e., g2 ∈ L2; +• g3 := +1 +T ′ε ( 2(|T ′ +ε|T ′′ +ε )T ′′ +ε +(T ′ +ε)4 +) ∼ |y − c|−2−2β; i.e., g3 ∈ L2; +• g4 := +T ′′ +ε +|T ′ +ε|T ′ +ε +T ′′ +ε +|T ′ +ε|T ′ +ε ∼ |y − c|−2−2β; i.e., g4 ∈ L2. +Consequently, for f ∈ W 2,1 then LTεf ∈ W 2,1. Moreover, using the notation +above, (3.16), (3.17) and (3.18), we have +(LTεf)′′ +≤ +LTε(( 1 +T ′ε +)2f ′′)) − LTε([g1,a + g1,b]f ′) − LTε((g2 − g3)f) − LTε(g1,af ′) + LTε(g4f) +≤ +LTε(( 1 +T ′ε +)2f ′′)) − LTε([2g1,a + g1,b]f ′) − LTε((g2 − g3 + g4)f), +and +∥(LTεf)′′∥1 ≤ ∥LTε(( 1 +T ′ε +)2f ′′))∥1 − ∥LTε([2g1,a + g1,b]f ′)∥1 − ∥LTε((g2 − g3 + g4)f)∥1 +≤ λ2∥f ′′∥1 − ∥[2g1,a + g1,b]∥∞∥f ′∥1 − ∥g2 − g3 + g4∥2∥f∥2. +(3.19) +Thus, ∃ M ≥ 0 such that +(3.20) +∥LTεf∥W 2,1 ≤ λ2∥f∥W 2,1 + M∥f∥W 1,1. +By (3.11)-(3.14) LTε is bounded when acting on W 1,1. Moreover, W 2,1 is compactly +embedded in W 1,1 +and thus Lemma 1 also implies the essential spectral radius +of LTε when acting on W 2,1 is smaller than λ. By the transitivity and the same +reasoning as before we get that the spectral radius is 1 and that the unique invariant +probability density is in W 2,1. +To prove (3.6) we apply Lemma 2 to LTε when acting on V0: the space of zero +average functions in W 1,1, V0 = {f ∈ W 1,1| +� +fdm = 0}, considering ∥ ∥L2 as +the weak norm. First notice that 1 is not in the spectrum of LT0 : V0 → V0 and +by Lemma 1 the essential spectral radius of LTε is bounded by λ. The operators + +8 +WAEL BAHSOUN AND STEFANO GALATOLO +LTε indeed satisfy a uniform Lasota-Yorke Inequality (3.11)-(3.14) iterating this +inequality and using (3.15) we get +(3.21) +∥Ln +Tεf∥W 1,1 ≤ λn∥f∥W 1,1 + M2n∥f∥w +verifying assumption (3.2). +The assumption (3.3) is verified in Lemma 4. +The +application of Lemma 2 gives then directly (3.6). +□ +3.2. Linear response derivation. +Lemma 5. ε �→ hε is differentiable in L1. In particular, +hε = h + ε(I − LT0)−1(q) + o(ε), +Where +(3.22) +q(x) = +� +LT0[A0h′ + B0h](x) +for x ∈ [0, a0) +0 +for x ∈ [a0, 1] . +with +Aε = − +�∂εTε +T ′ε +� +, +Bε = +�∂εTε · T ′′ +ε +T ′2 +ε +− ∂εT ′ +ε +T ′ε +� +and the o is in the L1-topology. +Proof. Consider +(3.23) +( 1 +|T ′ε| ·h)◦T −1 +0,ε (x)·1T0,ε[0,c)(x)+( 1 +|T ′ε| ·h)◦T −1 +1,ε (x)·1T1,ε(c,1](x) = ψ1,ε(x)+ψ2,ε(x). +Notice that the functions ψj,ε, j = 1, 2 are in W 2,1. Moreover, we have +(3.24) +∂ε(ψ(i) +j,ε) = (∂εψj,ε)(i) +i = 0, 1; j = 1, 2, +and these are continuous functions on [0, 1] × (−δ, δ). Further, we introduce the +following notation: +Hε := LTε − LT0; +Gε := (I − LTε)−1. +Since LTε has a spectral gap on W 1,1 it eventually contracts exponentially on the +subset of zero average functions V0 and the following relation is well defined: +(3.25) +hε = GεHεh + h. +Let ν ∈ V , V a compact subset of (−δ, δ) and ε be small. We have +(3.26) ∥ψj,ε+ν−ψj,ν−ε(∂δψj,δ|δ=ν)∥W 1,1 = +1 +� +i=0 +∥ψ(i) +j,ε+ν−ψ(i) +j,ν−ε(∂δψj,δ|δ=ν)(i)∥L1. +For each x, by the mean value theorem, there exists ηi +x,j,ε such that ψ(i) +j,ε+ν(x) − +ψ(i) +j,ν(x) = ε∂δψ(i) +j,δ|δ=ηi +x,j,ε, with |ηi +x,j,ε − ν| < ε. Therefore, +1 +� +i=0 +∥ψ(i) +j,ε+ν−ψ(i) +j,ν−ε(∂δψj,δ|δ=ν)(i)∥L1 ≤ |ε| +1 +� +i=0 +∥∂δψ(i) +j,δ|δ=ηi +x,j,ε−∂δψ(i) +j,δ|j,δ=ν∥L1 = o(ε). +We conclude by (3.26) and the commutation relation (3.24) that +(3.27) +Hεh = εq + o(ε), +for some q ∈ W 1,1, with the error o(ε) understood in W 1,1. + +LINEAR RESPONSE DUE TO SINGULARITIES +9 +To obtain a formula for q, for x ∈ [0, aε), set gj,ε := T −1 +j,ε (x) and consider +∂ε(h ◦ gj,εg′ +j,ε) = ∂ε(h ◦ gj,ε)g′ +j,ε + h ◦ gj,ε∂εg′ +j,ε += h′ ◦ gω,ε∂εgω,εg′ +ω,ε + h ◦ gj,ε∂εg′ +j,ε. +To prove the statement for x ∈ [0, aε), we start from the relation Tε ◦ gj,ε(x) = x +and differentiate it with respect to ε and get T ′ +ε ◦ gj,ε∂εgj,ε + ∂εTε ◦ gj,ε = 0. This +gives ∂εgj,ε = Aε ◦ gj,ε. This also implies that ∂εg′ +j,ε = A′ +ε ◦ gj,εg′ +j,ε = Bε ◦ gj,εg′ +j,ε. +This provides the formula for q in (3.22). +To continue, recall that LTε admits a uniform, in ε, spectral gap on W 1,1. There- +fore, Gε is uniformly bounded in L(W 1,1 +0 +W 1,1) and we have +(3.28) +GεHεh = εGεq + o(ε), +where the above error is understood in W 1,1. Now, observe that +Gεq − G0q = (I − LTε)−1(Lε − LT0)(I − LT0)−1q, +and set φ = (I − LT0)−1q ∈ W 1,1. By Lemma 4 +(3.29) +∥(LTε − LT0)φ∥1 → 0 +as ε → 0, and by the uniform boundedness of (I − LTε)−1 as operators on W 1,1, +we get +(3.30) +lim +ε→0 ∥Gε(q) − G0(q)∥L1 = 0. +Using (3.27), (3.28) and (3.30) together with (3.25) we obtain in L1 +hε = h + εG0(q) + o(ε), +which proves differentiability of hε and completes the proof of the theorem. +□ +References +[1] J. F. Alves. Strong statistical stability of non-uniformly expanding maps. Nonlinearity, 17, +(2004), no. 4, 1193–1215. +[2] J.F. Alves, M. Soufi, M. 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An applica- +tion to maps with indifferent fixed points Chaos, Solitons & Fractals 103, 596-601 (2017) +https://doi.org/10.1016/j.chaos.2017.07.005 +[22] S. Galatolo, A. Sorrentino, Quantitative statistical stability and linear response for irrational +rotations and diffeomorphisms of the circle. Discrete Contin. Dyn. Syst. 42(2), (2022), 815- +839. +[23] H. Hennion. Sur un th´eor`eme spectral et son application aux noyaux Lipchitziens. Proc. +Amer. Math. Soc. 118 (1993), 627–634. +[24] H. Hennion and L. Herve, Limit Theorems for Markov Chains and Stochastic Properties +of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, Vol. 1766 +(Springer-Verlag, 2001). +[25] A. Katok, G. Knieper, M. Pollicott, H. Weiss, Differentiability and analyticity of topological +entropy for Anosov and geodesic flows. Invent. Math. 98 (1989), no. 3, 581–597. +[26] G. Keller, Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94 (1982), +no. 4, 313–333. +[27] G. 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Viana Stochastic Dynamics of Deterministic Systems IMPA, 1997 +Department of Mathematical Sciences, Loughborough University, Loughborough, +Leicestershire, LE11 3TU, UK +Email address: W.Bahsoun@lboro.ac.uk +Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 1,Pisa - Italy +Email address: galatolo@dm.unipi.it + diff --git a/idE0T4oBgHgl3EQfYACE/content/tmp_files/load_file.txt b/idE0T4oBgHgl3EQfYACE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fb7315c01ee5734a7c68c5a646e361cced9f648 --- /dev/null +++ b/idE0T4oBgHgl3EQfYACE/content/tmp_files/load_file.txt @@ -0,0 +1,504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf,len=503 +page_content='LINEAR RESPONSE DUE TO SINGULARITIES WAEL BAHSOUN AND STEFANO GALATOLO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations chang- ing the value of the turning point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In this note we prove a rather unexpected result: if we consider a tent-like family with a cusp at the turning point, we re- cover the linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' More precisely, let Tε be a family of such cusp maps generated by changing the value of the turning point of T0 by a deterministic perturbation and let hε be the corresponding invariant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We prove that ε �→ hε is differentiable in L1 and provide a formula for its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Introduction Let M be a compact manifold with a reference volume and T : M → M be a map, whose iterates determine the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' A T-invariant measure µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' T∗µ = µ, is said to be physical if there is a positive volume set B such that for any continuous observable f : M → R lim n→∞ 1 n n−1 � i=0 f(T i(x)) = � M f dµ for all x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' An important question is to study how such measures vary, in an appropriate topology, under suitable perturbations of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Namely, consider a family {(M, Tε)}ε∈V , where V is a small neighbourhood of 0, T0 := T and Tε → T0, as ε → 0, in some suitable topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' For many chaotic systems it turns out that Tε admits a unique physical measure µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The system (M, T0, µ0) is called statistically stable if the map ε �−→ µε is continuous, in an appropriate topology, at ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Quantitative statistical sta- bility is provided by quantitative estimates on its modulus of continuity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' [1, 26, 20, 21, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In the case where ε �−→ µε is differentiable in some sense, the system is also said to admit linear response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=', the ‘derivative’ ˙µ0 represents the first order term of the response of the system to the perturbation µδ = µ0 + ˙µ0ε + o(ε) where the error is understood in an appropriate topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Linear response results in the context of deterministic dynamics was first ob- tained in the case of uniformly hyperbolic systems [31] (see also [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Nowadays linear response results are known for systems outside the uniformly hyperbolic set- ting (see [13, 15, 17, 22, 26], the survey article [8] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=') For systems with discontinuities or critical points the situation is quite complicated even in the presence of uniform expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Indeed, for suitable small perturbations Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Primary 37A05, 37E05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Linear response, Lorenz like maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='02301v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='DS] 5 Jan 2023 2 WAEL BAHSOUN AND STEFANO GALATOLO of piecewise expanding maps there are several examples that lack statistical sta- bility or linear response (see [26, 30, 9] and [3] for recent results in this direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In the literature there are some results indicating that perturbations which are not changing the topological class1 or tangent to the topological class, linear response is likely to occur, while for perturbations which are transversal to the topological class there is no linear response (see [13] for the case of piecewise expanding unimodal maps, [10] and [14] for smooth unimodal maps and [22] for results along this line in the case of rotations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In this paper we study perturbations of one dimensional tent-like piecewise ex- panding maps with unbounded derivatives at the turning point (a cusp singular- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We prove linear response for a large class of deterministic perturbations also changing the turning points and hence the topological class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The existence of such a linear response is due to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Indeed, unlike the usual tent maps with bounded derivative, which do not admit linear response for the same kind of perturbations([8, 30]), linear response in the singular case is attained due to fact that the cusp has a ‘regularizing effect’ at the level of the action of the associated transfer operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Cusp like singularities and unbounded derivatives appear in several important systems including Lorenz-type maps [5, 6] and billiard maps (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Unlike the case of Anosov flows [16] where linear and higher order re- sponse is proved, in the case of the classical Lorenz flow only statistical stability is known [2, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By highlighting the role of singularities in studying linear response, we hope that the results of the present paper contribute to the understanding of linear response for Lorenz-like flows and billiard maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In Section 2 we introduce the class of systems we consider and state the main result, Theorem 1, of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In Section 3 we prove Theorem 1 in a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The research of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Bahsoun is supported by EPSRC grant EP/V053493/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Bahsoun would like to thank the University of Pisa for its hospitality during his visit to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Galatolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The authors thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Ruziboev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Smania for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Family of maps and statement of the main result Let C ≥ 0, c ∈ (0, 1), β ∈ (−1, − 3 4), δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' For ε ∈ (−δ, δ) consider a family of maps Tε : [0, 1] → [0, 1] satisfying the following assumptions: (A1) T0,ε := Tε|[0,c) and T1,ε := Tε|(c,1] are one to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A2) Tε(0) = 0, Tε(1) = 0, Tε(c) = aε ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A3) Tε|[0,1]\\{c} ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A4) infx∈[0,1]\\{c} |T ′ ε(x)| ≥ θ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A5) Tε is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A6) limx→c± T ′ ε(x) = ±∞ and limx→c± |T ′ ε(x)| |x−c|β = C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A7) limx→c± |T ′′ ε (x)| |x−c|β−1 = C2, limx→c± |T ′′′ ε (x)| |x−c|β−2 = C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (A8) supε∈(−δ,δ),i∈{0,1,2} supx | T (i+1) ε (x) (x−c)β−i | < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 1A perturbation does not change the topological class if it changes a system to a system which is topologically conjugated to it LINEAR RESPONSE DUE TO SINGULARITIES 3 (A9) For2 f ∈ L1 set ( 1 |T ′ε| · f) ◦ T −1 0,ε (x) · 1T0,ε[0,c)(x) + ( 1 |T ′ε| · f) ◦ T −1 1,ε (x) · 1T1,ε(c,1](x) := ψ1,ε(x) + ψ2,ε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' For3 f ∈ W 1,1, ε �→ ψj,ε, j = 1, 2, is continuous at ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Moreover, for f ∈ C1 the partial derivatives ∂εψj,ε, ∂xψj,ε, ∂x∂εψj,ε, ∂ε∂xψj,ε, j = 1, 2, exist and are jointly continuous4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Figure 1 below presents an example of a map from the above family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' An example of Tε with aε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8 and c = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Let m denote Lebesgue measure on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The transfer operator associated with Tε, denoted by LTε, is defined by duality as follows: for f ∈ L1, g ∈ L∞ � 1 0 LTεf · gdm = � 1 0 f · g ◦ Tεdm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Although it is known that LTε admits a spectral gap when acting on the space of functions of generalized bounded variations [27], we study the action of LTε on finer Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Namely, the Sobolev spaces W i,1, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In particular, we show that LTε admits a spectral gap on W i,1, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This will allow us to conclude that Tε admits an invariant density which is regular enough, in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The latter is essential to derive the linear response formula that we are after in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The following result is the main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Tε admits a unique invariant density hε ∈ W 2,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' moreover, ε �→ hε is differentiable in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In particular, hε = h0 + ε(I − LT0)−1(q) + o(ε), 2Note that the definition of ψi depends on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 3We use the notation W i,1, i = 1, 2 to denote the usual Sobolev spaces equipped with the norms ∥f∥W i,1 = �i k=0 ∥f(k)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 4This implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1) ∂ε(ψ(i) j,ε) = (∂εψj,ε)(i) i = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='0 X4 WAEL BAHSOUN AND STEFANO GALATOLO here (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2) q(x) = � LT0[A0h′ + B0h](x) for x ∈ [0, a0) 0 for x ∈ [a0, 1] , with Aε = − �∂εTε T ′ε � , Bε = �∂εTε · T ′′ ε T ′2 ε − ∂εT ′ ε T ′ε � and the o is in the L1-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Proof of Theorem 1 In this section we prove Theorem 1 in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' First, in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1 we show that LTε admits a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This implies, in particular, that Tε has a unique invariant density hε ∈ W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Then in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2 we show that ε �→ hε is differentiable in L1 and obtain a formula for the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Uniform spectral gap on W 1,1 and on W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In this section we prove that the transfer operators associated with our class of systems admit a uniform spectral gap when acting on suitable Sobolev spaces (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We obtain this as a consequence of classical results which are recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Classical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We first recall Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2 of [11] which is a modification of Theorem XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='3 of [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Let (B, ∥ ∥s) be a Banach space, let ∥ ∥w be a continuous semi-norm on B and Q a bounded linear operator on B, such that for any sequence xn with ∥xn∥s ≤ 1 it contains a Cauchy subsequence for ∥ ∥w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Assume there is λ ≥ 0 and C > 0 such that, for any f ∈ B ∥Qf∥s ≤ λ∥f∥s + C∥f∥w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Then the essential spectral radius of Q is bounded by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We also recall the main result of [28], stating it in a simplified form suitable for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Let (B, ∥ ∥s) and ∥ ∥w as above, and Pε : B → B with ε ≥ 0 be a family of bounded linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Assume: there are C1 > 0 and M ≥ 1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1) ∥P n ε ∥w ≤ C1M n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' there are C2, C3 > 0 and 0 ≤ λ < 1 such that, for any f ∈ B, ε ≥ 0, n ∈ R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2) ∥P n ε f∥s ≤ C2λn∥f∥s + C3M n∥f∥w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 1 is not in the residual spectrum of Pε for any ε ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='3) ∥P0 − Pε∥s→w ≤ τ(ε) where τ(ε) → 0 monotonically and upper semicontinuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Then there are ϵ0, a, b > 0 such that for all 0 ≤ ε ≤ ϵ0 and f ∈ B (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='4) ∥(Id − Pε)−1f∥s ≤ a∥f∥s + b∥f∥w and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='5) lim ε→0 ∥(Id − P0)−1 − (Id − Pε)−1∥s→w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' LINEAR RESPONSE DUE TO SINGULARITIES 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Uniform spectral gap for the associated transfer operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We now state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' For any ϵ ∈ (−δ, δ) LTε admits a unique invariant density hε ∈ W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Furthermore, LTε has a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In particular, ∃ C > 0 such that for all ϵ ∈ (−δ, δ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='6) ∥(Id − LTε)−1∥W 1,1→W 1,1 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' To prove Lemma 3 we apply Lemma 1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We first start with an auxiliary lemma that verifies condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='3 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Let f ∈ W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='7) lim ϵ→0 ∥(LT0 − LTε)f∥L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8) we have ∥(LT0 − LTε)f∥L2 ≤ ∥ψ1,ε − ψ1,0∥L2 + ∥ψ2,ε − ψ2,0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Since f ∈ W 1,1, the proof follows from assumption (A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' □ Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We are going to prove that LTε admits a uniform Lasota-Yorke inequality when acting on W i,1, i = 1, 2, with L2 being the weak space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Consider the pointwise representation of the transfer operator associated with Tε (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8) (LTεf)(x) = � � y∈T −1 ε (x) 1 |T ′ε(y)|f(y) if x ∈ [0, aϵ] 0 if x ∈ (aϵ, 1] and, for f ∈ W 1,1, its derivative with respect to x: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='9) (LTεf) ′(x) = � � y∈T −1 ε (x) f ′(y) |T ′ε(y)|T ′ε(y) − T ′′ ε (y) (T ′ε(y))3 f(y) if x ∈ [0, aϵ] 0 if x ∈ (aϵ, 1] Notice that LTεf ∈ W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Indeed since f is bounded, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8) limx→aϵ LTεf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' LTεf is then continuous, with derivative almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Furthermore, we note that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='9) when x → aϵ then y → c and5 T ′′ ε (y) (T ′ ε)3(y) ∼ |y−c|β−1 |y−c|3β = |y − c|−2β−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Therefore, T ′′ ε (y) (T ′ ε)3(y) → 0 as y → c, this shows that LTεf ∈ W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Now, observe that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='10) (LTεf) ′ = LTε( 1 T ′ε f ′) − LTε( T ′′ ε |T ′ ε|T ′ ε f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='10) we get ∥(LTεf) ′∥1 ≤ ∥LTε( 1 T ′ε f ′)∥1 + ∥LTε( T ′′ ε (T ′ ε)2 f)∥1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='11) ≤ ∥ 1 T ′ε f ′∥1 + ∥ T ′′ ε (T ′ ε)2 f∥1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='12) ≤ ∥ 1 T ′ε ∥∞∥f ′∥1 + ∥ T ′′ ε (T ′ ε)2 f∥1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='13) ≤ λ∥f ′∥1 + ∥ T ′′ ε (T ′ ε)2 ∥2∥f∥2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='14) 5Here the asymptotic equivalence f ∼ g stands for limx→c |f(x)| |g(x)| = C with 0 < C < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' 6 WAEL BAHSOUN AND STEFANO GALATOLO where for each ε, λ := ∥ 1 T ′ε ∥∞ < 1 and ∥ T ′′ ε (T ′ ε)2 ∥2 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Note that furthermore by assumptions (A4) and (A8) the quantities ∥ 1 T ′ε ∥∞ and ∥ T ′′ ε (T ′ ε)2 ∥2 can be bounded uniformly for ε ∈ (−δ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We now show that the transfer operators are continuous in the weak norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='8) we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='15) ∥LTε∥L2→L2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Indeed we have ∥(LTεf)∥L2 ≤ ∥ψ1,ε∥L2 + ∥ψ2,ε∥L2 while � [0,1] (ψ1,ε)2dm = � [0,1] [( 1 |T ′ε| · f) ◦ T −1 0,ε (x) · 1T0,ε[0,c)(x)]2dx = � [0,aε] 1 |T ′ε(T −1 0,ε (x))| 1 |T ′ε(T −1 0,ε (x))| · [f(T −1 0,ε (x))]2dx ≤ sup x [ 1 |T ′(x)|] � [0,aε] 1 |T ′ε(T −1 0,ε (x))| · [f(T −1 0,ε (x))]2dx = sup x [ 1 |T ′(x)|]∥LTε(f 2 · 1[0,c))∥L1 ≤ sup x [ 1 |T ′(x)|]∥f 2 · 1[0,c)∥L1 ≤ sup x [ 1 |T ′(x)|][∥f∥L2]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' A similar estimate holds for ∥ψ2,ε∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Using the Lasota Yorke inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='14) and the fact that W 1,1 is com- pactly embedded in L2 by the Rellich-Kondrakov theorem, applying Lemma 1, we get that the essential spectral radius ρess ≤ λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By this, any element of the spectrum with modulus strictly bigger than λ is an isolated eigenvalue and we can prove that spectral radius of LTε is 1 (in fact it cannot be smaller than 1 because the transfer operator is Markov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' consequently, iterating the uniform density does not converge to 0, and for similar reasons there cannot be an eigenvalue of modulus greater than 1, since the corresponding eigenfunction cannot be expanded by iter- ating the transfer operator which is a weak contraction in the L1 norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Since the map is transitive there are no eigenvalues on the unit circle other than the simple eigenvalue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This implies that Tε admits a unique invariant density hε ∈ W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='10), we get (LTεf)′′ = (LTε( 1 T ′ε f ′) − LTε( T ′′ ε |T ′ ε|T ′ ε f))′ = (LTε( 1 T ′ε f ′))′ − (LTε( T ′′ ε |T ′ ε|T ′ ε f))′ = LTε( 1 T ′ε ( 1 T ′ε f ′)′) − LTε( T ′′ ε |T ′ ε|T ′ ε 1 T ′ε f ′) − [LTε( 1 T ′ε ( T ′′ ε |T ′ ε|T ′ ε f)′) − LTε( T ′′ ε |T ′ ε|T ′ ε T ′′ ε |T ′ ε|T ′ ε f)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='16) LINEAR RESPONSE DUE TO SINGULARITIES 7 where LTε( 1 T ′ε ( 1 T ′ε f ′)′) − LTε( T ′′ ε |T ′ε|T ′ε 1 T ′ε f ′) = LTε( 1 T ′ε ( −T ′′ ε (T ′ ε)2 f ′ + 1 T ′ε f ′′)) − LTε( T ′′ ε |T ′ ε|T ′ ε 1 T ′ε f ′) = LTε(( 1 T ′ε )2f ′′)) − LTε([ T ′′ ε (T ′ ε)2 + T ′′ ε |T ′ ε|T ′ ε ] 1 T ′ε f ′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='17) and LTε( 1 T ′ε ( T ′′ ε |T ′ ε|T ′ ε f)′) = LTε( 1 T ′ε [( T ′′ ε |T ′ ε|T ′ ε )′f + ( T ′′ ε |T ′ ε|T ′ ε )f ′]) = LTε( 1 T ′ε (T ′′′ ε |T ′ ε|T ′ ε − 2(|T ′ ε|T ′′ ε )T ′′ ε (T ′ ε)4 )f + 1 T ′ε ( T ′′ ε |T ′ ε|T ′ ε )f ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='18) Since β ∈ (−1, − 3 4), we have ( 1 T ′ε )2 ∈ L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' g1,a := T ′′ ε |T ′ ε|T ′ ε 1 T ′ε ∼ |y − c|−2β−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e, g1,a ∈ L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' g1,b := T ′′ ε (T ′ ε)2 1 T ′ε ∼ |y − c|−2β−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e, g1,b ∈ L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' g2 := 1 T ′ε ( T ′′′ ε |T ′ ε|T ′ ε (T ′ ε)4 ) ∼ |y − c|−2β−2, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=', g2 ∈ L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' g3 := 1 T ′ε ( 2(|T ′ ε|T ′′ ε )T ′′ ε (T ′ ε)4 ) ∼ |y − c|−2−2β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=', g3 ∈ L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' g4 := T ′′ ε |T ′ ε|T ′ ε T ′′ ε |T ′ ε|T ′ ε ∼ |y − c|−2−2β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=', g4 ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Consequently, for f ∈ W 2,1 then LTεf ∈ W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Moreover, using the notation above, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='18), we have (LTεf)′′ ≤ LTε(( 1 T ′ε )2f ′′)) − LTε([g1,a + g1,b]f ′) − LTε((g2 − g3)f) − LTε(g1,af ′) + LTε(g4f) ≤ LTε(( 1 T ′ε )2f ′′)) − LTε([2g1,a + g1,b]f ′) − LTε((g2 − g3 + g4)f), and ∥(LTεf)′′∥1 ≤ ∥LTε(( 1 T ′ε )2f ′′))∥1 − ∥LTε([2g1,a + g1,b]f ′)∥1 − ∥LTε((g2 − g3 + g4)f)∥1 ≤ λ2∥f ′′∥1 − ∥[2g1,a + g1,b]∥∞∥f ′∥1 − ∥g2 − g3 + g4∥2∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='19) Thus, ∃ M ≥ 0 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='20) ∥LTεf∥W 2,1 ≤ λ2∥f∥W 2,1 + M∥f∥W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='14) LTε is bounded when acting on W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Moreover, W 2,1 is compactly embedded in W 1,1 and thus Lemma 1 also implies the essential spectral radius of LTε when acting on W 2,1 is smaller than λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By the transitivity and the same reasoning as before we get that the spectral radius is 1 and that the unique invariant probability density is in W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='6) we apply Lemma 2 to LTε when acting on V0: the space of zero average functions in W 1,1, V0 = {f ∈ W 1,1| � fdm = 0}, considering ∥ ∥L2 as the weak norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' First notice that 1 is not in the spectrum of LT0 : V0 → V0 and by Lemma 1 the essential spectral radius of LTε is bounded by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The operators 8 WAEL BAHSOUN AND STEFANO GALATOLO LTε indeed satisfy a uniform Lasota-Yorke Inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='14) iterating this inequality and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='15) we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='21) ∥Ln Tεf∥W 1,1 ≤ λn∥f∥W 1,1 + M2n∥f∥w verifying assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='3) is verified in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' The application of Lemma 2 gives then directly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Linear response derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' ε �→ hε is differentiable in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' In particular, hε = h + ε(I − LT0)−1(q) + o(ε), Where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='22) q(x) = � LT0[A0h′ + B0h](x) for x ∈ [0, a0) 0 for x ∈ [a0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' with Aε = − �∂εTε T ′ε � , Bε = �∂εTε · T ′′ ε T ′2 ε − ∂εT ′ ε T ′ε � and the o is in the L1-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='23) ( 1 |T ′ε| ·h)◦T −1 0,ε (x)·1T0,ε[0,c)(x)+( 1 |T ′ε| ·h)◦T −1 1,ε (x)·1T1,ε(c,1](x) = ψ1,ε(x)+ψ2,ε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Notice that the functions ψj,ε, j = 1, 2 are in W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Moreover, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='24) ∂ε(ψ(i) j,ε) = (∂εψj,ε)(i) i = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' j = 1, 2, and these are continuous functions on [0, 1] × (−δ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Further, we introduce the following notation: Hε := LTε − LT0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Gε := (I − LTε)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Since LTε has a spectral gap on W 1,1 it eventually contracts exponentially on the subset of zero average functions V0 and the following relation is well defined: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='25) hε = GεHεh + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Let ν ∈ V , V a compact subset of (−δ, δ) and ε be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='26) ∥ψj,ε+ν−ψj,ν−ε(∂δψj,δ|δ=ν)∥W 1,1 = 1 � i=0 ∥ψ(i) j,ε+ν−ψ(i) j,ν−ε(∂δψj,δ|δ=ν)(i)∥L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' For each x, by the mean value theorem, there exists ηi x,j,ε such that ψ(i) j,ε+ν(x) − ψ(i) j,ν(x) = ε∂δψ(i) j,δ|δ=ηi x,j,ε, with |ηi x,j,ε − ν| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Therefore, 1 � i=0 ∥ψ(i) j,ε+ν−ψ(i) j,ν−ε(∂δψj,δ|δ=ν)(i)∥L1 ≤ |ε| 1 � i=0 ∥∂δψ(i) j,δ|δ=ηi x,j,ε−∂δψ(i) j,δ|j,δ=ν∥L1 = o(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' We conclude by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='26) and the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='24) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='27) Hεh = εq + o(ε), for some q ∈ W 1,1, with the error o(ε) understood in W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' LINEAR RESPONSE DUE TO SINGULARITIES 9 To obtain a formula for q, for x ∈ [0, aε), set gj,ε := T −1 j,ε (x) and consider ∂ε(h ◦ gj,εg′ j,ε) = ∂ε(h ◦ gj,ε)g′ j,ε + h ◦ gj,ε∂εg′ j,ε = h′ ◦ gω,ε∂εgω,εg′ ω,ε + h ◦ gj,ε∂εg′ j,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' To prove the statement for x ∈ [0, aε), we start from the relation Tε ◦ gj,ε(x) = x and differentiate it with respect to ε and get T ′ ε ◦ gj,ε∂εgj,ε + ∂εTε ◦ gj,ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This gives ∂εgj,ε = Aε ◦ gj,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This also implies that ∂εg′ j,ε = A′ ε ◦ gj,εg′ j,ε = Bε ◦ gj,εg′ j,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' This provides the formula for q in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' To continue, recall that LTε admits a uniform, in ε, spectral gap on W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' There- fore, Gε is uniformly bounded in L(W 1,1 0 W 1,1) and we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='28) GεHεh = εGεq + o(ε), where the above error is understood in W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Now, observe that Gεq − G0q = (I − LTε)−1(Lε − LT0)(I − LT0)−1q, and set φ = (I − LT0)−1q ∈ W 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' By Lemma 4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='29) ∥(LTε − LT0)φ∥1 → 0 as ε → 0, and by the uniform boundedness of (I − LTε)−1 as operators on W 1,1, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='30) lim ε→0 ∥Gε(q) − G0(q)∥L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='27), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='30) together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='25) we obtain in L1 hε = h + εG0(q) + o(ε), which proves differentiability of hε and completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Alves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' ICM Seoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='In: Proceedings, Volume III, 525–545 [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Baladi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' On the susceptibility function of piecewise expanding interval maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content=' Math.' metadata={'source': 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+page_content='unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQfYACE/content/2301.02301v1.pdf'} diff --git a/idE_T4oBgHgl3EQf4Rxl/content/2301.08351v1.pdf b/idE_T4oBgHgl3EQf4Rxl/content/2301.08351v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1dddf6bafa7a4c275135b06cbcd329aea3a7ef1d --- /dev/null +++ b/idE_T4oBgHgl3EQf4Rxl/content/2301.08351v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:36cd207f28eb57b0dcbb9aa50b71cbc251778a66f550d46b9b2e48625ed5859a +size 22946571 diff --git a/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf b/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b3af6756b03d03dabdf2efbbf0dba6eb8b4e5676 --- /dev/null +++ b/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf @@ 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RAOUAFI +,1 G. STENBORG +,1 D. B. SEATON +,2 H. WANG +,3, 4, 5 J. WANG +,3, 4, 5 C. E. DEFOREST +,2 +S. D. BALE +,6, 7 J. F. DRAKE +,8 V. M. URITSKY +,9, 10 J. T. KARPEN +,10 C. R. DEVORE +,10 A. C. STERLING +,11 +T. S. HORBURY +,12 L. K. HARRA +,13, 14 S. BOUROUAINE +,1 J. C. KASPER +,15 P. KUMAR +,16, 10 T. D. PHAN +,7 AND +M. VELLI +17 +1The Johns Hopkins Applied Physics Laboratory, Laurel, MD 20723, USA +2Southwest Research Institute, Boulder, CO 80302, USA +3Institute for Space Weather Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA +4Big Bear Solar Observatory, New Jersey Institute of Technology, Big Bear City, CA 92314, USA +5Center for Solar-Terrestrial Research, New Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA +6Physics Department, University of California, Berkeley, CA 94720, USA +7Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA +8Department of Physics, University of Maryland, College Park, MD 20742, USA +9Catholic University of America, 620 Michigan Avenue NE, Washington, DC 20061, USA +10Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +11NASA/Marshall Space Flight Center, Huntsville, AL 35812, USA +12The Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK +13PMOD/WRC, Dorfstrasse33 7260 Davos Dorf, Switzerland +14ETH-Zurich, H¨onggerberg campus, HIT building, Z¨urich, Switzerland +15BWX Technologies, Inc., Washington DC 20002, USA +16Department of Physics, American University, Washington, DC 20016, USA +17Earth Planetary and Space Sciences, UCLA, CA 90095, USA +ABSTRACT +We present EUV solar observations showing evidence for omnipresent jetting activity driven by small-scale +magnetic reconnection at the base of the solar corona. We argue that the physical mechanism that heats and +drives the solar wind at its source is ubiquitous magnetic reconnection in the form of small-scale jetting activity +(i.e., a.k.a. jetlets). This jetting activity, like the solar wind and the heating of the coronal plasma, are ubiqui- +tous regardless of the solar cycle phase. Each event arises from small-scale reconnection of opposite polarity +magnetic fields producing a short-lived jet of hot plasma and Alfv´en waves into the corona. The discrete na- +ture of these jetlet events leads to intermittent outflows from the corona, which homogenize as they propagate +away from the Sun and form the solar wind. This discovery establishes the importance of small-scale magnetic +reconnection in solar and stellar atmospheres in understanding ubiquitous phenomena such as coronal heating +and solar wind acceleration. Based on previous analyses linking the switchbacks to the magnetic network, we +also argue that these new observations might provide the link between the magnetic activity at the base of the +corona and the switchback solar wind phenomenon. These new observations need to be put in the bigger picture +of the role of magnetic reconnection and the diverse form of jetting in the solar atmosphere. +Keywords: Magnetic reconnection — Sun: solar wind — Sun: magnetic fields — Sun: corona — Sun: UV +radiation — Plasmas — Waves — Stars: winds, outflows — Methods: observational — Techniques: +image processing +1. INTRODUCTION +Solar and stellar winds are ubiquitous flows of charged par- +ticles (i.e., electrons, protons, and heavier ions) permeating +the astral spheres (Neugebauer & Snyder 1962). Through +these winds, stars lose angular momentum, slow down their +rotation as they age, shape planetary systems, and affect the +composition and the physical and chemical evolution of plan- +etary atmospheres and, consequently, the habitability of these +planets (L¨uftinger et al. 2014; Gallet et al. 2017). How the +solar wind is generated at the source, heated, and accelerated, +and what determines its variability, are long-standing funda- +mental questions. +The genesis of the hot and highly dynamic plasma in the +corona and the solar wind is among astrophysics’ most chal- +lenging and long-standing questions. +The solar wind has +three main regimes: fast, slow, and transient. The fast solar +arXiv:2301.00903v1 [astro-ph.SR] 3 Jan 2023 + +ID2 +RAOUAFI ET AL. 2023 +wind, with speeds typically over 500 km/s, originates from +the interior of coronal holes (i.e., open magnetic-field re- +gions). The source of the slow wind is highly debated (Abbo +et al. 2016). It apparently arises from the interfaces between +closed-field regions, such as active regions and quiet Sun, +and the edges of open-field coronal holes (D’Amicis & Bruno +2015). The fast solar wind is less dense and hotter than the +slow wind, and has photospheric composition, whereas the +slow wind has coronal composition. At 1 AU, the fast wind +is mainly Alfv´enic , whereas most slow wind is more vari- +able and non-Alfv´enic (Grappin et al. 1991; Bruno & Car- +bone 2005; Bale et al. 2019; Kasper et al. 2019; Bourouaine +et al. 2022), although uncommon streams of Alfv´enic slow +wind have been reported (D’Amicis & Bruno 2015; Marsch +et al. 1981; D’Amicis et al. 2019; Perrone et al. 2020). Two +major theories have been proposed to explain the solar wind’s +genesis via heating and acceleration: magnetic reconnection +(Parker 1988; Axford & McKenzie 1992; Fisk 2003) and +magnetohydrodynamic (MHD) wave turbulence (Belcher & +Davis 1971). +Transients, such as coronal mass ejections +(CMEs), are considered a third solar-wind regime that drives +space weather and correlates with the sunspot cycle (Raouafi +et al. 2021). +Jetting in the solar atmosphere manifests in different forms +(e.g., spicules [Beckers 1968, 1972; Sterling 2000; de Pon- +tieu et al. 2007], jets [Shibata et al. 1992; Raouafi et al. +2016], and surges [Canfield et al. 1996]). There is grow- +ing evidence that this jetting plays a key role in supplying +the corona and the solar wind with mass and momentum, and +may provide enough energy to power the solar wind (De Pon- +tieu et al. 2007; McIntosh et al. 2011; Tian et al. 2014). Coro- +nal jet signatures have been traced out to several Mm in X- +ray/extreme ultraviolet (EUV) observations, up to several so- +lar radii in white-light images (Wang et al. 1998), and beyond +1 AU in in situ measurements (Wang et al. 2006; Nitta et al. +2008; Neugebauer 2012). At lower altitudes, de Pontieu et al. +(2007) identified a similar phenomenon in the chromosphere, +the Type II spicules, which are typically observed in the chro- +mospheric Ca II 854.2 nm and Hα lines (Rouppe van der +Voort et al. 2009), and are heated as they propagate upward. +However, there is no evidence for Type II spicules reaching +coronal temperatures and altitudes as coronal plumes and jets +do. Observations from the IRIS (De Pontieu et al. 2014) and +SDO missions suggest that the spicular cool plasma falls back +to the solar surface (Samanta et al. 2015). Whether spicules +contribute to the solar wind and how much is not well known. +A recent study by Sow Mondal et al. (2022) suggests that sig- +nificantly more spicules than observed were needed to drive +the solar wind. +Close to the Sun, Parker Solar Probe (PSP) measurements +reveal a highly structured solar wind dominated by high- +amplitude Alfv´en waves. The magnetic field is often ob- +served to rotate over 90◦ forming reversals or switchbacks +(Bale et al. 2019; Kasper et al. 2019), which were observed +before by Ulysses (Balogh et al. 1999), WIND (Gosling et al. +2011), and Helios (Horbury et al. 2018). They were, how- +ever, scarce at large heliodistances. These switchbacks also +occur in patches separated by quiet periods where the field is +nearly radial. Several reports discuss the potential origins of +these structures, which can be put in two categories: coronal +origin (Fisk & Kasper 2020; Sterling & Moore 2020; Drake +et al. 2021) and in situ solar-wind origin (Squire et al. 2020; +Ruffolo et al. 2020; Shoda et al. 2021; Mallet et al. 2021; He +et al. 2021; Schwadron & McComas 2021). Bale et al. (2021) +and Fargette et al. (2021) found that the scale size of switch- +back patches correlates with the scale size of supergranules +on the solar surface, favoring a coronal origin for the switch- +backs. However, how the switchbacks form in the corona and +the driving physical mechanism near the solar surface remain +unclear. This topic is hotly debated (Sterling & Moore 2020; +Drake et al. 2021; Squire et al. 2020; Shoda et al. 2021; Bale +et al. 2021), partly because of the lack of clear observational +evidence of the processes responsible for heating and driving +the solar wind near the base of the solar atmosphere. +Small-scale jetting, or jetlets, was discovered in coronal +plumes in equatorial coronal holes by Raouafi & Stenborg +(2014). Coronal plumes are bright structures extending from +the magnetic network into high coronal altitudes (Wang et al. +1997). They are particularly prominent in images of total +solar eclipses, and were historically known as coronal rays +(van de Hulst 1950). Plumes are also observed to extend to +solar wind altitudes (i.e., ∼ 45 R⊙; DeForest et al. 2001). +They are brighter but cooler than surrounding interplumes +regions observed as darker (i.e., lower density) lanes in EUV +and white-light images of the solar corona. For further de- +tails on coronal plumes and jets, see the reviews by Wilhelm +et al. (2011) and Raouafi et al. (2016). Raouafi & Stenborg +(2014) showed that the small-scale and high-frequency jet- +ting (i.e., jetlets) at the base of coronal plumes is driven by +interchange magnetic reconnection, and that it sustains them +for long periods of time (see also Panesar et al. 2018, 2019; +Uritsky et al. 2021; Kumar et al. 2022). +Here we show evidence that ubiquitous jetting at tiny +scales (a few hundred km) driven by interchange magnetic +reconnection near the base of the corona could be the origin +of the heating and acceleration of the solar wind. We inter- +pret the magnetic field switchbacks (Bale et al. 2019; Kasper +et al. 2019) as tracers of this small-scale explosive magnetic +activity. +2. UBIQUITOUS JETTING ACTIVITY AT THE BASE +OF THE SOLAR CORONA +The high-resolution, +high-cadence observations from +space missions such as SOHO (Domingo et al. 1995), Hinode +(Kosugi et al. 2007), STEREO (Kaiser et al. 2008), SDO (Pes- +nell et al. 2012), and SolO (M¨uller et al. 2020) show tremen- +dous diversity of multi-scale explosive activity ranging from +enormous flares (Shibata & Magara 2011) and CMEs (Chen +2011) down to bright-point eruptions (Madjarska 2019) and +coronal jets (Shibata et al. 1992; Raouafi et al. 2016). So- +lar observations suggest that magnetic reconnection plays a +predominant role in the evolution of these structures by en- +abling the impulsive conversion of stored magnetic energy to +plasma kinetic and thermal energy and to nonthermal parti- + +MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND +3 +2021-04-28 03:48:09 +(a) +Figure 1. (a) Composite of SDO/AIA and GOES-R/SUVI 171 ˚A images showing the small-scale activity at the base of the solar corona and its +extension to higher altitudes (see movies in the Supplemental material). The maximum extent of the jetlets in the AIA field of view is limited by +the instrument sensitivity. Estimates of their occurrence rate and size are also limited by the temporal and spatial resolution of the instrument. +The SUVI image maps the structures observed at the coronal base into the solar wind. The accompanying movies illustrate the highly dynamic +and continuous nature of this phenomenon. (b) AIA image (171 ˚A) showing the jetlet structures as elongated features above the solar polar +limb. Examples of jetlet events are indicated by the arrows. +cles. In contrast to sunspots and active regions that nearly +disappear at the minimum of the solar cycle, small-scale +activity (e.g., jets, bright points, etc.) +is omnipresent re- +gardless of the solar-cycle phase (see, e.g., McIntosh et al. +2014; Madjarska 2019). In fact, jetting resulting from mag- +netic reconnection is evidently a fundamental process on the +Sun. Reconnection-driven jets are not restricted to the open +magnetic-field regions (i.e., coronal holes) but also occur in +closed structures, heating the plasma to high temperatures. A +particular example of this activity is the tiny jets (i.e., jetlets) +observed at the base of plumes within equatorial coronal +holes (Raouafi & Stenborg 2014). Previous analyses were, +however, confined to particular coronal structures, namely +plumes in equatorial coronal holes (Uritsky et al. 2021; Ku- +mar et al. 2022) or singular jetlets (Panesar et al. 2018, 2019). +Jetlets are minuscule reconnection events between open and +closed magnetic flux resulting in collimated plasma ejections +into the solar corona (Raouafi & Stenborg 2014; Kumar et al. +2022; Panesar et al. 2018, 2019). (Raouafi & Stenborg 2014) +found that jetlets are the primary driver of coronal plumes +sustaining them for days and weeks. (Kumar et al. 2022) also +found quasiperiodic energy releases (equivalent to nanoflare +energies, i.e., 1024 ergs) and associated jetlets at the base of +plumes that could contribute significant mass flux to the solar +wind. +The mechanism producing jetlets within plumes also oc- +curs elsewhere on the solar disk. A careful analysis of the +SDO/AIA and GOES-R/SUVI observations reveals that this +phenomenon is much more pervasive than merely coronal +plumes. Figure 1a and Figure 1b are a composite of AIA +and SUVI wide-field images (Seaton et al. 2021) and an AIA +zoomed view of the southern polar region, respectively. The +raw images show hazy structures extending to high coronal +altitudes. The processed images using the multi-resolution +image-processing technique reveal tiny bursts of hot plasma +permeating nearly all coronal structures. +The lifetime of +these events ranges from tens of seconds to several minutes. +The ubiquity and the highly dynamic nature of this activity in + +(b)4 +RAOUAFI ET AL. 2023 +the off-limb corona are striking (see the movies provided in +the Supplemental material for details). Based on the analysis +of long series of continuous SDO/AIA images, we find that +this off-limb small-scale activity persists over time, indicat- +ing that it extends over the whole solar surface (i.e., coronal +holes, the quiet Sun, and active regions; see the movies in the +supplemental material). +3. MAGNETIC RECONNECTION DRIVING THE +SMALL-SCALE JETTING +Figure 2. Co-temporal magnetograms from the SDO/HMI (a) and +BBSO/GST-NIRIS (b) instruments with respective spatial resolu- +tions of 1′′ and 0′′.2. The magnitude NIRIS magnetograms is scaled +to the HMI unit scale. The displayed magnetic fields saturate at +±200 G. The corresponding EUV images are in the 193 ˚A (c) and +211 ˚A (d) channels of SDO/AIA, respectively. Red (blue) contours +represent positive (negative) network fields and magnetic elements. +Panel (a) shows that only strong-field regions are resolved at low +resolution, and most of the solar disk area seems to be void of any +significant flux. The magnetogram images change dramatically with +increasing spatial resolution and instrument sensitivity (panel (b)). +In particular, apparent void regions and unipolar patches show sig- +nificantly more mixed polarities, a favored landscape for magnetic +reconnection. +Figure 2a,b displays magnetograms from the SDO Helio- +seismic and Magnetic Imager (HMI; Scherrer et al. 2012) and +the 1.6 m Goode Solar Telescope (GST) at the Big Bear So- +lar Observatory (BBSO). The HMI and GST magnetograms +have a spatial resolution of 1′′ and 0′′.2, respectively. Only +relatively strong field regions can be observed at the low reso- +lution and sensitivity of HMI, with hints of a diffuse opposite +polarity. This clear view is primarily due to two instrumental +factors: the polarimetric sensitivity and the low resolution +that leads to the Zeeman-cancellation of opposite-polarity +fields within the resolution element. The magnetic-field land- +scape changes dramatically by improving the instrumental +sensitivity and increasing the spatial resolution. Most unipo- +lar flux concentrations observed with low-resolution instru- +ments become fragmented at high resolution, as can be seen +clearly in the GST sub-arcsecond magnetograms. A multi- +tude of multi-scale magnetic elements of highly mixed po- +larity are present throughout the instrument’s field of view. +Magnetic reconnection between background network mag- +netic fields (at the supergranule boundaries, the base of the +open fields in coronal holes) and opposite-polarity intranet- +work fields are likely the cause of small-scale jetting. This +comports with the finding that switchbacks have a modula- +tion scale of supergranules (Bale et al. 2021; Fargette et al. +2021). An animated sequence of GST magnetograms is pro- +vided in the Supplemental material. +In the GST magnetograms1, a significant number of mag- +netic bipoles appeared in the regions devoid of magnetic flux +at coarser resolution. These small-scale, highly mixed po- +larity fields are a rich medium for magnetic reconnection. +During about 90 minutes of continuous observations, 1434 +cancellation events were identified in the quiet Sun/coronal- +hole boundary region in the GST 70′′ × 70′′ field of view. +Most notably, the distribution of these sites seems to be uni- +form as there is no appreciable difference between the quiet +Sun and the coronal hole (Figure 2c,d), an indicator of the +universality of small-scale reconnection in the lower solar +atmosphere. The magnetic flux-cancellation rate in the ob- +servation is 1–2 × 1018 Mx Mm−2 hr−1. 88 cancellations +were associated with Hα spicules, of which 61 were rooted +in network field concentrations presumably open to the so- +lar wind. Among them, 7 produced detectable EUV jetlets +above the spicules. Assuming that these occurrences are typ- +ical of those over the whole Sun, scaling the observed fre- +quencies yields about 600 flux-cancellation events s−1 gen- +erating about 35 Hα spicules s−1 and 3 EUV jetlets s−1. We +expect that such cancellation/reconnection sites would pro- +duce additional eruptive events below the smallest currently +observable scales. +Magnetic reconnection also generates Alfv´enic pertur- +bations (waves, fronts, and shocks). +The simultaneous +generation of the radial flows associated with jetlets and +Alfv´enic perturbations is a natural consequence of reconnec- +tion (Karpen et al. 2017; Uritsky et al. 2017; Roberts et al. +2018). +Cirtain et al. (2007) showed evidence for Alfv´en +waves in solar X-ray jet. These Alfv´enic waves are crucial +for heating and accelerating the solar wind plasma, and for +generating turbulent flows at higher coronal altitudes (Chan- +dran et al. 2011). +3.1. Coronal Jetting Rate, Mass and Energy Fluxes +1 For more details on how the BBSO/GST magnetograms were produced +and the magnetic fine structures were identified, see Wang et al. (2022) and +Appendix A. + +2018-07-29T16:44:08.000UT +200 +2018-07-29T16:45:21UT +a +(b) +150 +-100" +-100 +(Solar-Y) +100 +Latitude(Solar +110 +Latitude +50 +-120" + [G] +Helioprojective L +-120 +Helioprojective +50 +130 +-100 +-140" +140 +150 + (a) SDO/HMI Magnetogram +(b) GST/NIRIS Magnetogram +-200 +150- +640" +-620" +600" +-580" +640 +630 +620 +-610 +-600 +590 +Helioprojective Longitude (Solar-X) +Helioprojective Longitude (Solar X) +2018-07-29T16:45:16.843UT +2018-07-29T16:45:09.626UT +(c) +(d) +D +150 +-100" +-100" +Latitude (Solar-Y) +100 +(Solar-Y) +50 +Latitude +-120" +[G] +-120" +Helioprojective +Helioprojective +los +b +50 +Q +-100 +140" +-140" +-150 +(c) SDO/AIA 193 +(d) SDO/AIA 211 +-640" +-620" +-600" +580" +-640" +-620" +-600" +-580" +Helioprojective Longitude (Solar-X) +Helioprojective Longitude (Solar-X)MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND +5 +Figure 3. (a) Polar projection of AIA 171 ˚A images of the north- +ern solar polar region showing coronal bright (dark) structures (e.g., +plumes, interplume regions, etc.). (b) Time-distance diagram along +the virtual slit marked by the dark lines in (a). +Jetlets emanat- +ing from small-scale magnetic reconnection persist at all times and +dominate the activity at the base of the solar corona. (c) Diagram +showing the coronal structures crossing the orange virtual slit in (a) +as a function of time at an altitude of 40 Mm above the solar polar +limb. The jetlets are the bright structures. +Figure 3a shows an SDO/AIA 171 ˚A polar projection of +the northern polar region (±25◦). The two black lines mark +the area used to build the time-distance diagram in Figure 3b. +The jetlet events along the virtual slit show as streaks whose +slopes yield an average speed of ∼ 150 km s−1. Figure 3c +shows the signal in the EUV image crossing a circular slit +at an altitude of 40 Mm above the solar polar limb (i.e., or- +ange line in panel (a)) as a function of time. The jetlets are +the bright structures with typical lifetimes of several minutes. +To estimate the jetlets crossing the artificial slit (orange line +in Figure 3a), we apply a Fourier analysis that gives us the +jetting rate. We then inverted this rate into the total number +of jetlets over the several-hour period we considered for this +analysis. Our analysis of these two diagrams provides an oc- +currence rate of ∼ 2500 jetlets per day over a position-angle +interval of 50◦ above the northern polar cap. This jetlet rate +is orders of magnitude higher than the reported ∼ 160 X- +ray jets per day per solar hemisphere (Savcheva et al. 2007; +Paraschiv et al. 2015), even though our analysis underesti- +mates the jetlet occurrence rate by about 30%. The jetlet +detection is also limited by the instrument sensitivity and the +spatial and temporal resolution of the data. The X-ray jets +are, in contrast, significantly larger and longer-lived, and in- +dividually more energetic. +The jetlets typically have a width of 2′′ − 3′′ (Kumar et al. +2022), a speed of 150 km s−1, and a lifetime of 5 − 10 min- +utes. Assuming a coronal density of 5×108 cm−3 at the base +of these events, the particle ejection rate into the corona re- +sulting from each small-scale jetlet is about 7.5×1015 cm−2 +s−1. +This amounts to about 3 × 1032 protons s−1 and +1×1035 protons total over the lifetime of the jetlet, assuming +that all of the ejected plasma escapes (hence this value should +be regarded as an upper limit). To account for the entire solar +wind loss of 6 × 1035 protons s−1 (von Steiger et al. 2000; +Wang 2016, 2020) requires roughly 2 × 103 jetlets to be ac- +tive at any instant and 6 jetlets s−1 to be initiated over the full +Sun. This last number is comparable to the rate extrapolated +from the BBSO/GST measurements discussed above. +The kinetic energy injected into the corona by each jetlet +is 1.2 × 106 erg cm−2 s−1 or 5 × 1022 erg s−1, assuming +the same jetlet width as quoted above. If 2 × 103 jetlets are +active at any instant, the total jetlet kinetic-energy injection +rate is 1 × 1026 erg s−1. By comparison, the overall solar +kinetic-energy loss rate (assuming an asymptotic flow speed +of 500 km s−1) is about 1×1027 erg s−1. Clearly the injected +jetlet plasma must be accelerated further by the coronal ther- +mal pressure plus wave pressure to reach the asymptotic wind +speed. +The BBSO/GST analysis shows that the magnetic flux +density of the reconnecting bipoles at the photosphere is +∼ 190 G. At coronal altitudes, we estimate the strength of the +reconnecting field to lie in the range 5 − 10 G. The magnetic +energy released to the plasma during the reconnection pro- +cess is assumed to be partitioned between plasma bulk flow +and heating. (see the Supplemental material for the detailed +calculations.) +The total solar jetlet-generation rate of ∼ 6 jetlets s−1 (i.e., +5 × 105 jetlets d−1) greatly exceeds the estimated ∼ 0.03 +jetlets s−1 (i.e., 2.5 × 103 jetlets d−1) estimated from the +limb observations with SDO/AIA. However, the latter en- +compassed just 1/2π of the solar circumference, and it was +restricted to jetlets within a narrow, undetermined angle δ +from the limb onto and behind the disk. The full-Sun and +limb-detected results are consistent for δ ≈ 0.03 rad, equiva- +lent to a linear distance d ≈ 20 Mm from the limb. +All of these rates fall in the ranges required to drive the +solar-wind plasma at the base of the corona in the quiet Sun +and coronal holes. Thus, our analysis supports our contention +that the ubiquitous, small-scale jetting activity (jetlets) driven +by magnetic reconnection can account for essentially all of +the mass and energy lost by the Sun to the solar wind. +A critical aspect of measuring the reconnection-driven jet- +ting at the base of the corona is the dependence on the res- +olution of the magnetic-field data and the EUV images. We + +(a) +100 +Height (Mm) +80 +60 +40 +20 +25 +20 +15 +10 +5 +0 +Angle from North (Deg.) +50 +(b) +40 +Height (Mm) +30 +20 +10 +0 +5 +10 +15 +Time (Hours UT)6 +RAOUAFI ET AL. 2023 +expect that magnetic field data with significantly higher res- +olution and polarization accuracy, for instance from the 4-m +Daniel K. Inouye Solar Telescope (DKIST), would provide a +substantially higher incidence of reconnection events, result- +ing in a considerably higher jetting rate. This will, of course, +affect the estimated heating and acceleration of the coronal +solar-wind plasma. +4. PARKER SOLAR PROBE OBSERVATIONS AND +THEIR CONNECTION TO THE CORONA +Using Ulysses’ fast solar wind measurements above the so- +lar poles (> 1 AU), Neugebauer et al. (1995) showed that the +so-called micro-streams, where the solar wind speed deviates +by > 20 km s−1 from the average, are of solar origin. His- +torically, micro-streams were thought to be related to coronal +plumes, although this relationship cannot fully explain their +properties. +Neugebauer (2012) argued that micro-streams +are related to episodic rather than quasi-stationary sources. +Based on the work by Raouafi et al. (2008), which found +a causal relationship between jets and plumes, (Neugebauer +2012) confirmed that the micro-streams are of solar origin, +and their properties can be explained if the fast ones result +from jetting activity at the base of the corona. +0 +20 +40 +60 +80 +Radius [RS] +0 +200 +400 +600 +800 +vR [km/s] +Figure 4. The vertical bars show the spread of solar wind velocities +measured by PSP as a function of helio-distance during the first ten +encounters. The red curve, which bounds the measurements on the +lower end, seems to indicate that the base solar wind behaves like +the Parker model (Parker 1958). Above that boundary, plasma jets +dominate the solar wind, which may suggest tracers in the solar +wind of the coronal jetting activity. +Closer to the Sun, PSP observed predominantly Alfv´enic +solar wind (both fast and slow) during its perihelion encoun- +ters (< 0.25 AU). Streams of non-Alfv´enic flows have been +reported, but they represent only a very small fraction of the +observations. The prevalence of Alfv´enic flows in the inner +solar wind is an important clue about the nature of the solar +wind as it emerges from its source(s). During the first per- +ihelion encounter , a small equatorial hole was identified as +the source of the observed slow wind (Bale et al. 2019). For +the other encounters, the models show that the spacecraft is +frequently connected magnetically to the edges of the polar +coronal holes and their equatorial extensions. At the base of +the solar atmosphere, only remote-sensing data are available +to assess the physical processes that might occur at the origin +of the solar wind flow. Current data quality is much higher +than in previous decades, so with PSP flying so close to the +Sun, linking the in situ measurements to remote-sensing ob- +servations at much lower altitudes in the solar atmosphere is +an exciting possibility. +The data also show that the solar wind speed, as measured +by PSP, is dominated by radial-velocity jets (Kasper et al. +2019) superimposed on the background Parker-like wind (see +Figure 4). The vertical bars represent the spread of the speeds +of these plasma jets. The red curve marks the low bound +of these speeds, remarkably resembling a Parker-like solar +wind. The structuring of the inner solar wind and the dom- +inance of in situ plasma jetting may also indicate the sig- +nature of the small-scale magnetic reconnection and jetting +at the base of the solar corona. Magnetic reconnection pro- +duces Alfv´enic waves that eventually make their way to high +altitudes and whose dissipation heats and accelerates the so- +lar wind plasma (see, e.g., De Pontieu et al. 2007; McIntosh +et al. 2011). +Switchbacks are short magnetic field rotations that are +ubiquitously observed in the solar wind. They are consis- +tent with local folds in the magnetic field rather than changes +in the magnetic connectivity to solar source regions (Bale +et al. 2019; Kasper et al. 2019). The large number of the +omnipresent eruptive jetting events observed at the base of +the corona is a credible explanation of the source of the +magnetic-field switchbacks observed by PSP. The jetlets are +the direct product of ubiquitous magnetic reconnection at +small spatial and temporal scales. The EUV images from +SDO and GOES-R/SUVI and the magnetic-field data from +the BBSO/GST provide clear evidence for the preponder- +ance of small-scale reconnection at these sites. The EUV +images, particularly in the solar polar regions, show a semi- +regular spacing of brighter and darker coronal structures. The +brighter areas exhibit a much higher jetlet occurrence than +the darker ones. Hence, the switchback patchiness could be +explained by the varying magnetic connection of the space- +craft to sites with different susceptibilities to reconnection +events. The quiet periods (dark in EUV) would correspond +to locations with lower event rates, while the times of strong +connectivity to regions with higher event rates characterized +by more jetlets/switchbacks. +Different models have been suggested to explain the for- +mation of switchbacks: (1) interchange reconnection (e.g., +Fisk & Kasper 2020; Sterling & Moore 2020; He et al. +2021; Drake et al. 2021; Agapitov et al. 2022); (2) steep- +ening of Alfv´en waves and/or Alfv´enic turbulence (Squire +et al. 2020; Mallet et al. 2021; Shoda et al. 2021); (3) +due to roll up from nonlinear Kelvin-Helmholtz instabili- + +MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND +7 +ties (Ruffolo et al. 2020); and (4) through magnetic field +lines that stretch between sources of slower and faster wind +(Schwadron & McComas 2021). We postulate that the om- +nipresent magnetic reconnection and the resulting jetting in +the corona satisfy most if not the proposed switchback mod- +els. Magnetic reconnection produces impulsive plasma jets +and Alfv´en waves, which are the principal inputs for most +switchback models (e.g., Alfv´en waves, shear flows, etc.). +5. CONCLUSIONS. HOW IS THE SOLAR WIND BORN? +The coronal holes where the fast solar wind originates are +regions of open magnetic fields. (Hassler et al. 1999) used +coronal observations in the Ne7+ 770 ˚A spectral line to find +evidence for strong outflows coinciding with the boundaries +of the chromospheric network. Although they did not discuss +the physical mechanism generating these outflows, they sug- +gested that the wind is rooted in the boundaries of this net- +work. (Tu et al. 2005) suggested that these areas are open +to the corona and could be the source of the wind. +The +present data show the predominance of intermittently driven +hot plasma outflows at small scales. These jetlets are om- +nipresent, much like the solar wind, regardless of the phase of +the sunspot cycle. Evidence for reconnection in the low solar +atmosphere is present across the entire solar disk, particularly +at the boundaries of the chromospheric network (i.e., super- +granules). Although these areas are typically dominated by +unipolar fields, high-resolution magnetic-field measurements +show the presence of minority-polarity intrusions (i.e., the +salt and pepper fields) actively moving amongst and cancel- +ing with the dominant polarity field to drive the jetlets. +We believe that magnetic activity at small scales plays the +dominant role in shaping the solar atmosphere, heating the +corona, and driving the solar wind. We believe the jetlets +analyzed here are part of a whole spectrum that extends to +much smaller scales. With higher spatial and temporal reso- +lution and greater instrumental sensitivity, therefore, we ex- +pect to detect more frequent signatures of magnetic recon- +nection at finer scales. For instance, DKIST will provide +observations with spatial resolution three times better than +BBSO/GST. With these data, we expect to identify signif- +icantly more fine-scale magnetic reconnection sites provid- +ing hot and impulsive plasma jets to the corona and the solar +wind, with significant implications for coronal heating and +solar-wind acceleration. +Our proposed scenario applies most obviously to the fast +solar wind. This originates in coronal hole regions, where the +magnetic field is open. Therefore, jets on coronal hole open +fields have a direct route to the heliosphere, and therefore +can explain the fast solar wind in a straightforward manner. +In contrast, the origin of the slow solar wind is not yet clear, +but there is evidence that it originates in closed-field regions +and/or at the boundaries between open- and closed-field re- +gions. Because jets/jetlets occur in the close-field regions +also, we expect that they are source of not only the fast wind, +but also the slow wind too. Further analyses are, however, +required to confirm this. We hope to clarify these points with +future PSP observations. +One crucial aspect of PSP measurement close to the Sun +is that almost all the observed solar wind is highly Alfv´enic. +The observed Alfv´enicity of the wind seems independent of +the wind regime. It might indicate a common physical pro- +cess at the origin of the solar wind and that the difference +between the slow and fast wind might result from evolution +at higher altitudes. Wang & Sheeley (1990) suggested that +super-radial expansion of the coronal magnetic field can gen- +erate a slow solar wind such as at the boundaries of coronal +holes (see also Panasenco et al. 2019). Future PSP measure- +ments, during the upcoming closest perihelia together with +Solar Orbiter and DKIST observations, hold promise for con- +firming the links between small-scale magnetic activity and +the solar wind, hopefully by inferring direct connections be- +tween small-scale reconnection or other magnetic events and +small-scale structures in the solar wind. +We are grateful to Dr. Valentin Martinez Pillet for the con- +structive comments and suggestions, which helped improve +the quality of the paper. +Parker Solar Probe was designed, built, and is now oper- +ated by the Johns Hopkins Applied Physics Laboratory as +part of NASA’s Living with a Star (LWS) program (contract +NNN06AA01C). Support from the LWS management and +technical team has played a critical role in the success of the +Parker Solar Probe mission. +SDO is the first mission to be launched for NASA’s Living +With a Star (LWS) Program. The SDO/AIA and SDO/HMI +data are provided by the Joint Science Operations Center +(JSOC) Science Data Processing (SDP). +Solar UltraViolet Imager (SUVI) product development, +analysis, calibration, validation, and data stewardship by +CIRES-affiliated authors within National Centers for Envi- +ronmental Information (NCEI) was supported by National +Oceanic and Atmospheric Administration cooperative agree- +ment no. NA17OAR4320101. +We gratefully acknowledge the use of data from the Goode +Solar Telescope (GST) of the Big Bear Solar Observatory +(BBSO). BBSO operation is supported by US NSF AGS- +1821294 grant and New Jersey Institute of Technology. GST +operation is partly supported by the Korea Astronomy and +Space Science Institute and the Seoul National University. +APPENDIX + +8 +RAOUAFI ET AL. 2023 +A. BBSO/GST MAGNETIC FIELD DATA. +Taking advantage of high-order correction by the adaptive optics system with 308 sub-apertures (Cao et al. 2010) and the solar +speckle interferometric data-reconstruction technique (W¨oger et al. 2008), the observation during ∼16:34 – 18:38 UT achieved +diffraction-limited resolution under a favorable seeing condition. There is a ∼ 20-minute observation gap between 18:07 – 18:27 +UT due to bad seeing. Spectroscopic polarization measurements of Fe I 1.56 µm were taken by NIRIS with a 0′′.24 resolution +and a 42 s cadence. +Due to weak polarization signal in the quiet-Sun regions, line-of-sight (LOS) magnetograms are reduced by summing Stokes- +V profiles from GST observations to enhance the SNR. The magnetic field strength is scaled the contemporal HMI magnetic- +field measurements. The small-scale magnetic elements are tracked with Southwest Automatic Magnetic Identification Suite +(SWAMIS; DeForest et al. 2007) based on similarity heuristics across a time series of magnetograms, by which the magnetic +cancellation events are detected and their corresponding magnetic fluxes are calculated. +B. ENERGY AND PARTICLE FLUXES FROM MAGNETIC RECONNECTION +This section estimates the magnetic energy flux resulting from the small-scale reconnection episodes. This energy flux is +transferred to the plasma in the form of bulk flows and heating. We start from the observed particle and kinetic energy ejection +rates into the corona and wind, using the following average jetlet properties: +• Transverse scale (i.e., width): LJ ≈ 3′′ ≈ 2000 km +• Speed: VJ ≈ 150 km s−1 +• Lifetime: τ ≈ 5 min = 300 s +• Density: n ≈ 5 × 108 cm−3 +The jetlet speed is determined from the time-distance diagram. It is the projected speed on the plane of the sky, which should +be considered as a lower limit on the real jetlet speed. For the electron density, we used a typical plume density at the base of +the corona. The jetlets may be denser, perhaps by as much as a factor of 4 (Sterling & Moore 2020), but we employ the ambient +coronal density to be conservative. There are variations by at least a factor of 2 in the width (LJ) and lifetime (τ) of the jetlets +that decrease or increase our estimated jetlet contributions to the wind and, therefore, the number of jetlets required to drive the +entire solar-wind flux. It is not possible to be precise about these contributions beyond a factor of about 2 in either direction. We +have endeavored here to demonstrate that conservatively estimated jetlet contributions to the wind are comparable to the total +estimated mass and energy fluxes from the Sun. The resulting values are given in the main text of the paper. +The reconnecting magnetic field strength in the corona is much smaller than the measured average photospheric flux density, +Bph ≈ 100 G in quiet Sun and slightly higher in the coronal hole boundary region. Because we do not have direct coronal +magnetic-field measurements, we must infer the strength of the reconnecting field, BR. We do this by requiring the magnetic +energy released by the reconnection to be sufficient to power the jetlet outflow, plus an assumed equivalent amount of plasma +heating. For simplicity, we ignore the unknown but plausible contribution of released magnetic energy to accelerated nonthermal +particles, which is a very important and well-known consequence of reconnection in large CMEs and flares. +The magnetic reconnection inflow speed, VR, is assumed to be a fraction 0.1 of the Alfv´en speed, VA = BR/√4πρ, associated +with the reconnecting field strength, BR. This dimensionless reconnection rate (i.e., the inflow Alfv´en Mach number) is well +established from numerical MHD simulations of fast reconnection, including those specifically of reconnection-driven coronal +jets (Karpen et al. 2017). The reconnection occurs over a transverse scale LR that defines the width of the reconnection region in +the corona, and which may be smaller than the width LJ of the jetlet. For further detail on the theory of magnetic reconnection, +see Lin & Lee (1993). +The kinetic energy flux density and total release rate into the bulk outflow are +wKE = 1 +2ρV 2 +J VJ, +(B1) +WKE = wKEL2 +J +(B2) +whence wKE ≈ 1.2 × 106 erg cm−2 s−1 and WKE ≈ 5 × 1022 erg s−1. Assuming that half of the magnetic energy is +transferred to the plasma in the form of heating, the total magnetic energy release rate during the reconnection, WME, satisfies +WME = 2 WKE ≈ 1 × 1023 erg s−1. We have +wME = 1 +4π B2 +R VR = 0.1 +4π B2 +R +BR +√4πρ, +(B3) +WME = wME L2 +R, +(B4) +wME = WME/L2 +R = 2 WKE/L2 +R = 2 wKE L2 +J/L2 +R, +(B5) + +MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND +9 +which can be solved for the reconnection field strength, +B3 +R = 80 π3/2 √ρ wME +(B6) += 160 π3/2 L2 +J +L2 +R +wKE. +(B7) +Substituting values from above, we obtain BR ≈ 5 (LJ/LR)2/3 G. The implied field strength depends, as is to be expected, +on the ratio of characteristic scales in the reconnection region and in the resultant jetlet. The minimum strength is about 5 G, 5% +of the average photospheric flux density of ∼ 100 G, obtained for LR = LJ. This is a reasonable value for the coronal magnetic +field. The implied field strength doubles to about 10 G if we assume that LR = LJ/3. +C. SUPPLEMENTARY MATERIAL +The supplementary material package contains five video files and can be downloaded HERE +Movie M1: Composite of SDO/AIA and GOES-R/SUVI 171 ˚A image sequences showing the small-scale activity at the base +of the solar corona and its extension to higher altitudes. The movie begins on April 28th, 2021 at 00:00:09 and ends the same day +09:57:09. Its real-time duration is 8 seconds. +Movie M2: SDO/AIA 171 ˚A image sequence showing the small-scale activity at the base of the solar corona. The movie starts +on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09. Its real-time duration is 8 seconds. +Movie M3: Zoom on the SDO/AIA 171 ˚A image sequence of the northern solar polar region where small-scale activity is +clearly visible. The movie starts on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09. Its real-time duration is 8 +seconds. +Movie M4: Zoom on the SDO/AIA 171 ˚A image sequence of the southern solar polar region where small-scale activity is +clearly visible. The movie starts on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09. Its real-time duration is 8 +seconds. +Movie M5: Animation of high-resolution magnetograms from the BBSO/GST-NIRIS instrument showing the highly-dynamic +magnetic fields at small-scales. The number of the cancelling bipoles is much greater than that at 1′′ resolution. The movie begins +on June 29th, 2018 at 16:32:18 UT and ends the same day 18:38:36 UT. Its real-time duration is 14 seconds. +REFERENCES +Abbo, L., et al. 2016, SSRv, 201, 55 +Agapitov, O. V., et al. 2022, ApJ, 925, 213 +Axford, W. I., & McKenzie, J. F. 1992, in Solar Wind Seven +Colloquium, ed. E. Marsch & R. Schwenn, 1 +Bale, S. D., et al. 2019, Nature, 576, 237 +Bale, S. D., et al. 2021, ApJ, 923, 174 +Balogh, A., Forsyth, R. J., Lucek, E. A., Horbury, T. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Huntsville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' AL 35812,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' USA 12The Blackett Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' SW7 2AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' UK 13PMOD/WRC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Dorfstrasse33 7260 Davos Dorf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Switzerland 14ETH-Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' H¨onggerberg campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' HIT building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Switzerland 15BWX Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', Washington DC 20002, USA 16Department of Physics, American University, Washington, DC 20016, USA 17Earth Planetary and Space Sciences, UCLA, CA 90095, USA ABSTRACT We present EUV solar observations showing evidence for omnipresent jetting activity driven by small-scale magnetic reconnection at the base of the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We argue that the physical mechanism that heats and drives the solar wind at its source is ubiquitous magnetic reconnection in the form of small-scale jetting activity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' jetlets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This jetting activity, like the solar wind and the heating of the coronal plasma, are ubiqui- tous regardless of the solar cycle phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Each event arises from small-scale reconnection of opposite polarity magnetic fields producing a short-lived jet of hot plasma and Alfv´en waves into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The discrete na- ture of these jetlet events leads to intermittent outflows from the corona, which homogenize as they propagate away from the Sun and form the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This discovery establishes the importance of small-scale magnetic reconnection in solar and stellar atmospheres in understanding ubiquitous phenomena such as coronal heating and solar wind acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Based on previous analyses linking the switchbacks to the magnetic network, we also argue that these new observations might provide the link between the magnetic activity at the base of the corona and the switchback solar wind phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' These new observations need to be put in the bigger picture of the role of magnetic reconnection and the diverse form of jetting in the solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Keywords: Magnetic reconnection — Sun: solar wind — Sun: magnetic fields — Sun: corona — Sun: UV radiation — Plasmas — Waves — Stars: winds, outflows — Methods: observational — Techniques: image processing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' INTRODUCTION Solar and stellar winds are ubiquitous flows of charged par- ticles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', electrons, protons, and heavier ions) permeating the astral spheres (Neugebauer & Snyder 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Through these winds, stars lose angular momentum, slow down their rotation as they age, shape planetary systems, and affect the composition and the physical and chemical evolution of plan- etary atmospheres and, consequently, the habitability of these planets (L¨uftinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Gallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' How the solar wind is generated at the source, heated, and accelerated, and what determines its variability, are long-standing funda- mental questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The genesis of the hot and highly dynamic plasma in the corona and the solar wind is among astrophysics’ most chal- lenging and long-standing questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The solar wind has three main regimes: fast, slow, and transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The fast solar arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='00903v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='SR] 3 Jan 2023 ID2 RAOUAFI ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2023 wind, with speeds typically over 500 km/s, originates from the interior of coronal holes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', open magnetic-field re- gions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The source of the slow wind is highly debated (Abbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' It apparently arises from the interfaces between closed-field regions, such as active regions and quiet Sun, and the edges of open-field coronal holes (D’Amicis & Bruno 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The fast solar wind is less dense and hotter than the slow wind, and has photospheric composition, whereas the slow wind has coronal composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' At 1 AU, the fast wind is mainly Alfv´enic , whereas most slow wind is more vari- able and non-Alfv´enic (Grappin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Bruno & Car- bone 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Bourouaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022), although uncommon streams of Alfv´enic slow wind have been reported (D’Amicis & Bruno 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Marsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' D’Amicis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Perrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Two major theories have been proposed to explain the solar wind’s genesis via heating and acceleration: magnetic reconnection (Parker 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Axford & McKenzie 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Fisk 2003) and magnetohydrodynamic (MHD) wave turbulence (Belcher & Davis 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Transients, such as coronal mass ejections (CMEs), are considered a third solar-wind regime that drives space weather and correlates with the sunspot cycle (Raouafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Jetting in the solar atmosphere manifests in different forms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', spicules [Beckers 1968, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Sterling 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' de Pon- tieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007], jets [Shibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Raouafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2016], and surges [Canfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1996]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' There is grow- ing evidence that this jetting plays a key role in supplying the corona and the solar wind with mass and momentum, and may provide enough energy to power the solar wind (De Pon- tieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Coro- nal jet signatures have been traced out to several Mm in X- ray/extreme ultraviolet (EUV) observations, up to several so- lar radii in white-light images (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1998), and beyond 1 AU in in situ measurements (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Nitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Neugebauer 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' At lower altitudes, de Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2007) identified a similar phenomenon in the chromosphere, the Type II spicules, which are typically observed in the chro- mospheric Ca II 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='2 nm and Hα lines (Rouppe van der Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2009), and are heated as they propagate upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' However, there is no evidence for Type II spicules reaching coronal temperatures and altitudes as coronal plumes and jets do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Observations from the IRIS (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2014) and SDO missions suggest that the spicular cool plasma falls back to the solar surface (Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Whether spicules contribute to the solar wind and how much is not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A recent study by Sow Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2022) suggests that sig- nificantly more spicules than observed were needed to drive the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Close to the Sun, Parker Solar Probe (PSP) measurements reveal a highly structured solar wind dominated by high- amplitude Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic field is often ob- served to rotate over 90◦ forming reversals or switchbacks (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019), which were observed before by Ulysses (Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1999), WIND (Gosling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2011), and Helios (Horbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' They were, how- ever, scarce at large heliodistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' These switchbacks also occur in patches separated by quiet periods where the field is nearly radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Several reports discuss the potential origins of these structures, which can be put in two categories: coronal origin (Fisk & Kasper 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Sterling & Moore 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021) and in situ solar-wind origin (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Ruffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Shoda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Mallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Schwadron & McComas 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2021) and Fargette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2021) found that the scale size of switch- back patches correlates with the scale size of supergranules on the solar surface, favoring a coronal origin for the switch- backs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' However, how the switchbacks form in the corona and the driving physical mechanism near the solar surface remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This topic is hotly debated (Sterling & Moore 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Shoda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021), partly because of the lack of clear observational evidence of the processes responsible for heating and driving the solar wind near the base of the solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Small-scale jetting, or jetlets, was discovered in coronal plumes in equatorial coronal holes by Raouafi & Stenborg (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Coronal plumes are bright structures extending from the magnetic network into high coronal altitudes (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' They are particularly prominent in images of total solar eclipses, and were historically known as coronal rays (van de Hulst 1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Plumes are also observed to extend to solar wind altitudes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', ∼ 45 R⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' DeForest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' They are brighter but cooler than surrounding interplumes regions observed as darker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', lower density) lanes in EUV and white-light images of the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For further de- tails on coronal plumes and jets, see the reviews by Wilhelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2011) and Raouafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Raouafi & Stenborg (2014) showed that the small-scale and high-frequency jet- ting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', jetlets) at the base of coronal plumes is driven by interchange magnetic reconnection, and that it sustains them for long periods of time (see also Panesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Uritsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Here we show evidence that ubiquitous jetting at tiny scales (a few hundred km) driven by interchange magnetic reconnection near the base of the corona could be the origin of the heating and acceleration of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We inter- pret the magnetic field switchbacks (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019) as tracers of this small-scale explosive magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' UBIQUITOUS JETTING ACTIVITY AT THE BASE OF THE SOLAR CORONA The high-resolution, high-cadence observations from space missions such as SOHO (Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1995), Hinode (Kosugi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007), STEREO (Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2008), SDO (Pes- nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2012), and SolO (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020) show tremen- dous diversity of multi-scale explosive activity ranging from enormous flares (Shibata & Magara 2011) and CMEs (Chen 2011) down to bright-point eruptions (Madjarska 2019) and coronal jets (Shibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Raouafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' So- lar observations suggest that magnetic reconnection plays a predominant role in the evolution of these structures by en- abling the impulsive conversion of stored magnetic energy to plasma kinetic and thermal energy and to nonthermal parti- MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND 3 2021-04-28 03:48:09 (a) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (a) Composite of SDO/AIA and GOES-R/SUVI 171 ˚A images showing the small-scale activity at the base of the solar corona and its extension to higher altitudes (see movies in the Supplemental material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The maximum extent of the jetlets in the AIA field of view is limited by the instrument sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Estimates of their occurrence rate and size are also limited by the temporal and spatial resolution of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The SUVI image maps the structures observed at the coronal base into the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The accompanying movies illustrate the highly dynamic and continuous nature of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (b) AIA image (171 ˚A) showing the jetlet structures as elongated features above the solar polar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Examples of jetlet events are indicated by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' In contrast to sunspots and active regions that nearly disappear at the minimum of the solar cycle, small-scale activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', jets, bright points, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=') is omnipresent re- gardless of the solar-cycle phase (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Madjarska 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' In fact, jetting resulting from mag- netic reconnection is evidently a fundamental process on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Reconnection-driven jets are not restricted to the open magnetic-field regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', coronal holes) but also occur in closed structures, heating the plasma to high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A particular example of this activity is the tiny jets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', jetlets) observed at the base of plumes within equatorial coronal holes (Raouafi & Stenborg 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Previous analyses were, however, confined to particular coronal structures, namely plumes in equatorial coronal holes (Uritsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Ku- mar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022) or singular jetlets (Panesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Jetlets are minuscule reconnection events between open and closed magnetic flux resulting in collimated plasma ejections into the solar corona (Raouafi & Stenborg 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Panesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (Raouafi & Stenborg 2014) found that jetlets are the primary driver of coronal plumes sustaining them for days and weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022) also found quasiperiodic energy releases (equivalent to nanoflare energies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', 1024 ergs) and associated jetlets at the base of plumes that could contribute significant mass flux to the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The mechanism producing jetlets within plumes also oc- curs elsewhere on the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A careful analysis of the SDO/AIA and GOES-R/SUVI observations reveals that this phenomenon is much more pervasive than merely coronal plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Figure 1a and Figure 1b are a composite of AIA and SUVI wide-field images (Seaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021) and an AIA zoomed view of the southern polar region, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The raw images show hazy structures extending to high coronal altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The processed images using the multi-resolution image-processing technique reveal tiny bursts of hot plasma permeating nearly all coronal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The lifetime of these events ranges from tens of seconds to several minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The ubiquity and the highly dynamic nature of this activity in (b)4 RAOUAFI ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2023 the off-limb corona are striking (see the movies provided in the Supplemental material for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Based on the analysis of long series of continuous SDO/AIA images, we find that this off-limb small-scale activity persists over time, indicat- ing that it extends over the whole solar surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', coronal holes, the quiet Sun, and active regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' see the movies in the supplemental material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' MAGNETIC RECONNECTION DRIVING THE SMALL-SCALE JETTING Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Co-temporal magnetograms from the SDO/HMI (a) and BBSO/GST-NIRIS (b) instruments with respective spatial resolu- tions of 1′′ and 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnitude NIRIS magnetograms is scaled to the HMI unit scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The displayed magnetic fields saturate at ±200 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The corresponding EUV images are in the 193 ˚A (c) and 211 ˚A (d) channels of SDO/AIA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Red (blue) contours represent positive (negative) network fields and magnetic elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Panel (a) shows that only strong-field regions are resolved at low resolution, and most of the solar disk area seems to be void of any significant flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetogram images change dramatically with increasing spatial resolution and instrument sensitivity (panel (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' In particular, apparent void regions and unipolar patches show sig- nificantly more mixed polarities, a favored landscape for magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Figure 2a,b displays magnetograms from the SDO Helio- seismic and Magnetic Imager (HMI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Scherrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2012) and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='6 m Goode Solar Telescope (GST) at the Big Bear So- lar Observatory (BBSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The HMI and GST magnetograms have a spatial resolution of 1′′ and 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Only relatively strong field regions can be observed at the low reso- lution and sensitivity of HMI, with hints of a diffuse opposite polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This clear view is primarily due to two instrumental factors: the polarimetric sensitivity and the low resolution that leads to the Zeeman-cancellation of opposite-polarity fields within the resolution element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic-field land- scape changes dramatically by improving the instrumental sensitivity and increasing the spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Most unipo- lar flux concentrations observed with low-resolution instru- ments become fragmented at high resolution, as can be seen clearly in the GST sub-arcsecond magnetograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A multi- tude of multi-scale magnetic elements of highly mixed po- larity are present throughout the instrument’s field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Magnetic reconnection between background network mag- netic fields (at the supergranule boundaries, the base of the open fields in coronal holes) and opposite-polarity intranet- work fields are likely the cause of small-scale jetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This comports with the finding that switchbacks have a modula- tion scale of supergranules (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Fargette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' An animated sequence of GST magnetograms is pro- vided in the Supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' In the GST magnetograms1, a significant number of mag- netic bipoles appeared in the regions devoid of magnetic flux at coarser resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' These small-scale, highly mixed po- larity fields are a rich medium for magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' During about 90 minutes of continuous observations, 1434 cancellation events were identified in the quiet Sun/coronal- hole boundary region in the GST 70′′ × 70′′ field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Most notably, the distribution of these sites seems to be uni- form as there is no appreciable difference between the quiet Sun and the coronal hole (Figure 2c,d), an indicator of the universality of small-scale reconnection in the lower solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic flux-cancellation rate in the ob- servation is 1–2 × 1018 Mx Mm−2 hr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 88 cancellations were associated with Hα spicules, of which 61 were rooted in network field concentrations presumably open to the so- lar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Among them, 7 produced detectable EUV jetlets above the spicules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Assuming that these occurrences are typ- ical of those over the whole Sun, scaling the observed fre- quencies yields about 600 flux-cancellation events s−1 gen- erating about 35 Hα spicules s−1 and 3 EUV jetlets s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We expect that such cancellation/reconnection sites would pro- duce additional eruptive events below the smallest currently observable scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Magnetic reconnection also generates Alfv´enic pertur- bations (waves, fronts, and shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The simultaneous generation of the radial flows associated with jetlets and Alfv´enic perturbations is a natural consequence of reconnec- tion (Karpen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Uritsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Cirtain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2007) showed evidence for Alfv´en waves in solar X-ray jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' These Alfv´enic waves are crucial for heating and accelerating the solar wind plasma, and for generating turbulent flows at higher coronal altitudes (Chan- dran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Coronal Jetting Rate, Mass and Energy Fluxes 1 For more details on how the BBSO/GST magnetograms were produced and the magnetic fine structures were identified, see Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2022) and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018-07-29T16:44:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='000UT 200 2018-07-29T16:45:21UT a (b) 150 100" 100 (Solar-Y) 100 Latitude(Solar 110 Latitude 50 120" [G] Helioprojective L 120 Helioprojective 50 130 100 140" 140 150 (a) SDO/HMI Magnetogram (b) GST/NIRIS Magnetogram 200 150- 640" 620" 600" 580" 640 630 620 610 600 590 Helioprojective Longitude (Solar-X) Helioprojective Longitude (Solar X) 2018-07-29T16:45:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='843UT 2018-07-29T16:45:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='626UT (c) (d) D 150 100" 100" Latitude (Solar-Y) 100 (Solar-Y) 50 Latitude 120" [G] 120" Helioprojective Helioprojective los b 50 Q 100 140" 140" 150 (c) SDO/AIA 193 (d) SDO/AIA 211 640" 620" 600" 580" 640" 620" 600" 580" Helioprojective Longitude (Solar-X) Helioprojective Longitude (Solar-X)MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (a) Polar projection of AIA 171 ˚A images of the north- ern solar polar region showing coronal bright (dark) structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', plumes, interplume regions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (b) Time-distance diagram along the virtual slit marked by the dark lines in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Jetlets emanat- ing from small-scale magnetic reconnection persist at all times and dominate the activity at the base of the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (c) Diagram showing the coronal structures crossing the orange virtual slit in (a) as a function of time at an altitude of 40 Mm above the solar polar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlets are the bright structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Figure 3a shows an SDO/AIA 171 ˚A polar projection of the northern polar region (±25◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The two black lines mark the area used to build the time-distance diagram in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlet events along the virtual slit show as streaks whose slopes yield an average speed of ∼ 150 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Figure 3c shows the signal in the EUV image crossing a circular slit at an altitude of 40 Mm above the solar polar limb (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', or- ange line in panel (a)) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlets are the bright structures with typical lifetimes of several minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' To estimate the jetlets crossing the artificial slit (orange line in Figure 3a), we apply a Fourier analysis that gives us the jetting rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We then inverted this rate into the total number of jetlets over the several-hour period we considered for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Our analysis of these two diagrams provides an oc- currence rate of ∼ 2500 jetlets per day over a position-angle interval of 50◦ above the northern polar cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This jetlet rate is orders of magnitude higher than the reported ∼ 160 X- ray jets per day per solar hemisphere (Savcheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Paraschiv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2015), even though our analysis underesti- mates the jetlet occurrence rate by about 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlet detection is also limited by the instrument sensitivity and the spatial and temporal resolution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The X-ray jets are, in contrast, significantly larger and longer-lived, and in- dividually more energetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlets typically have a width of 2′′ − 3′′ (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022), a speed of 150 km s−1, and a lifetime of 5 − 10 min- utes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Assuming a coronal density of 5×108 cm−3 at the base of these events, the particle ejection rate into the corona re- sulting from each small-scale jetlet is about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='5×1015 cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This amounts to about 3 × 1032 protons s−1 and 1×1035 protons total over the lifetime of the jetlet, assuming that all of the ejected plasma escapes (hence this value should be regarded as an upper limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' To account for the entire solar wind loss of 6 × 1035 protons s−1 (von Steiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Wang 2016, 2020) requires roughly 2 × 103 jetlets to be ac- tive at any instant and 6 jetlets s−1 to be initiated over the full Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This last number is comparable to the rate extrapolated from the BBSO/GST measurements discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The kinetic energy injected into the corona by each jetlet is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='2 × 106 erg cm−2 s−1 or 5 × 1022 erg s−1, assuming the same jetlet width as quoted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' If 2 × 103 jetlets are active at any instant, the total jetlet kinetic-energy injection rate is 1 × 1026 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' By comparison, the overall solar kinetic-energy loss rate (assuming an asymptotic flow speed of 500 km s−1) is about 1×1027 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Clearly the injected jetlet plasma must be accelerated further by the coronal ther- mal pressure plus wave pressure to reach the asymptotic wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The BBSO/GST analysis shows that the magnetic flux density of the reconnecting bipoles at the photosphere is ∼ 190 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' At coronal altitudes, we estimate the strength of the reconnecting field to lie in the range 5 − 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic energy released to the plasma during the reconnection pro- cess is assumed to be partitioned between plasma bulk flow and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (see the Supplemental material for the detailed calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=') The total solar jetlet-generation rate of ∼ 6 jetlets s−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', 5 × 105 jetlets d−1) greatly exceeds the estimated ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='03 jetlets s−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='5 × 103 jetlets d−1) estimated from the limb observations with SDO/AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' However, the latter en- compassed just 1/2π of the solar circumference, and it was restricted to jetlets within a narrow, undetermined angle δ from the limb onto and behind the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The full-Sun and limb-detected results are consistent for δ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='03 rad, equiva- lent to a linear distance d ≈ 20 Mm from the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' All of these rates fall in the ranges required to drive the solar-wind plasma at the base of the corona in the quiet Sun and coronal holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Thus, our analysis supports our contention that the ubiquitous, small-scale jetting activity (jetlets) driven by magnetic reconnection can account for essentially all of the mass and energy lost by the Sun to the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A critical aspect of measuring the reconnection-driven jet- ting at the base of the corona is the dependence on the res- olution of the magnetic-field data and the EUV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We (a) 100 Height (Mm) 80 60 40 20 25 20 15 10 5 0 Angle from North (Deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=') 50 (b) 40 Height (Mm) 30 20 10 0 5 10 15 Time (Hours UT)6 RAOUAFI ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2023 expect that magnetic field data with significantly higher res- olution and polarization accuracy, for instance from the 4-m Daniel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Inouye Solar Telescope (DKIST), would provide a substantially higher incidence of reconnection events, result- ing in a considerably higher jetting rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This will, of course, affect the estimated heating and acceleration of the coronal solar-wind plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' PARKER SOLAR PROBE OBSERVATIONS AND THEIR CONNECTION TO THE CORONA Using Ulysses’ fast solar wind measurements above the so- lar poles (> 1 AU), Neugebauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (1995) showed that the so-called micro-streams, where the solar wind speed deviates by > 20 km s−1 from the average, are of solar origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' His- torically, micro-streams were thought to be related to coronal plumes, although this relationship cannot fully explain their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Neugebauer (2012) argued that micro-streams are related to episodic rather than quasi-stationary sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Based on the work by Raouafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2008), which found a causal relationship between jets and plumes, (Neugebauer 2012) confirmed that the micro-streams are of solar origin, and their properties can be explained if the fast ones result from jetting activity at the base of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 0 20 40 60 80 Radius [RS] 0 200 400 600 800 vR [km/s] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The vertical bars show the spread of solar wind velocities measured by PSP as a function of helio-distance during the first ten encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The red curve, which bounds the measurements on the lower end, seems to indicate that the base solar wind behaves like the Parker model (Parker 1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Above that boundary, plasma jets dominate the solar wind, which may suggest tracers in the solar wind of the coronal jetting activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Closer to the Sun, PSP observed predominantly Alfv´enic solar wind (both fast and slow) during its perihelion encoun- ters (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='25 AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Streams of non-Alfv´enic flows have been reported, but they represent only a very small fraction of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The prevalence of Alfv´enic flows in the inner solar wind is an important clue about the nature of the solar wind as it emerges from its source(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' During the first per- ihelion encounter , a small equatorial hole was identified as the source of the observed slow wind (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For the other encounters, the models show that the spacecraft is frequently connected magnetically to the edges of the polar coronal holes and their equatorial extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' At the base of the solar atmosphere, only remote-sensing data are available to assess the physical processes that might occur at the origin of the solar wind flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Current data quality is much higher than in previous decades, so with PSP flying so close to the Sun, linking the in situ measurements to remote-sensing ob- servations at much lower altitudes in the solar atmosphere is an exciting possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The data also show that the solar wind speed, as measured by PSP, is dominated by radial-velocity jets (Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019) superimposed on the background Parker-like wind (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The vertical bars represent the spread of the speeds of these plasma jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The red curve marks the low bound of these speeds, remarkably resembling a Parker-like solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The structuring of the inner solar wind and the dom- inance of in situ plasma jetting may also indicate the sig- nature of the small-scale magnetic reconnection and jetting at the base of the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Magnetic reconnection pro- duces Alfv´enic waves that eventually make their way to high altitudes and whose dissipation heats and accelerates the so- lar wind plasma (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Switchbacks are short magnetic field rotations that are ubiquitously observed in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' They are consis- tent with local folds in the magnetic field rather than changes in the magnetic connectivity to solar source regions (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The large number of the omnipresent eruptive jetting events observed at the base of the corona is a credible explanation of the source of the magnetic-field switchbacks observed by PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlets are the direct product of ubiquitous magnetic reconnection at small spatial and temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The EUV images from SDO and GOES-R/SUVI and the magnetic-field data from the BBSO/GST provide clear evidence for the preponder- ance of small-scale reconnection at these sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The EUV images, particularly in the solar polar regions, show a semi- regular spacing of brighter and darker coronal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The brighter areas exhibit a much higher jetlet occurrence than the darker ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Hence, the switchback patchiness could be explained by the varying magnetic connection of the space- craft to sites with different susceptibilities to reconnection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The quiet periods (dark in EUV) would correspond to locations with lower event rates, while the times of strong connectivity to regions with higher event rates characterized by more jetlets/switchbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Different models have been suggested to explain the for- mation of switchbacks: (1) interchange reconnection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', Fisk & Kasper 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Sterling & Moore 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Agapitov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (2) steep- ening of Alfv´en waves and/or Alfv´enic turbulence (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Mallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Shoda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (3) due to roll up from nonlinear Kelvin-Helmholtz instabili- MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND 7 ties (Ruffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' and (4) through magnetic field lines that stretch between sources of slower and faster wind (Schwadron & McComas 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We postulate that the om- nipresent magnetic reconnection and the resulting jetting in the corona satisfy most if not the proposed switchback mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Magnetic reconnection produces impulsive plasma jets and Alfv´en waves, which are the principal inputs for most switchback models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', Alfv´en waves, shear flows, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' CONCLUSIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' HOW IS THE SOLAR WIND BORN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The coronal holes where the fast solar wind originates are regions of open magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (Hassler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 1999) used coronal observations in the Ne7+ 770 ˚A spectral line to find evidence for strong outflows coinciding with the boundaries of the chromospheric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Although they did not discuss the physical mechanism generating these outflows, they sug- gested that the wind is rooted in the boundaries of this net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2005) suggested that these areas are open to the corona and could be the source of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The present data show the predominance of intermittently driven hot plasma outflows at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' These jetlets are om- nipresent, much like the solar wind, regardless of the phase of the sunspot cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Evidence for reconnection in the low solar atmosphere is present across the entire solar disk, particularly at the boundaries of the chromospheric network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', super- granules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Although these areas are typically dominated by unipolar fields, high-resolution magnetic-field measurements show the presence of minority-polarity intrusions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', the salt and pepper fields) actively moving amongst and cancel- ing with the dominant polarity field to drive the jetlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We believe that magnetic activity at small scales plays the dominant role in shaping the solar atmosphere, heating the corona, and driving the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We believe the jetlets analyzed here are part of a whole spectrum that extends to much smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' With higher spatial and temporal reso- lution and greater instrumental sensitivity, therefore, we ex- pect to detect more frequent signatures of magnetic recon- nection at finer scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For instance, DKIST will provide observations with spatial resolution three times better than BBSO/GST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' With these data, we expect to identify signif- icantly more fine-scale magnetic reconnection sites provid- ing hot and impulsive plasma jets to the corona and the solar wind, with significant implications for coronal heating and solar-wind acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Our proposed scenario applies most obviously to the fast solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This originates in coronal hole regions, where the magnetic field is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Therefore, jets on coronal hole open fields have a direct route to the heliosphere, and therefore can explain the fast solar wind in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' In contrast, the origin of the slow solar wind is not yet clear, but there is evidence that it originates in closed-field regions and/or at the boundaries between open- and closed-field re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Because jets/jetlets occur in the close-field regions also, we expect that they are source of not only the fast wind, but also the slow wind too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Further analyses are, however, required to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We hope to clarify these points with future PSP observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' One crucial aspect of PSP measurement close to the Sun is that almost all the observed solar wind is highly Alfv´enic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The observed Alfv´enicity of the wind seems independent of the wind regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' It might indicate a common physical pro- cess at the origin of the solar wind and that the difference between the slow and fast wind might result from evolution at higher altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Wang & Sheeley (1990) suggested that super-radial expansion of the coronal magnetic field can gen- erate a slow solar wind such as at the boundaries of coronal holes (see also Panasenco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Future PSP measure- ments, during the upcoming closest perihelia together with Solar Orbiter and DKIST observations, hold promise for con- firming the links between small-scale magnetic activity and the solar wind, hopefully by inferring direct connections be- tween small-scale reconnection or other magnetic events and small-scale structures in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Valentin Martinez Pillet for the con- structive comments and suggestions, which helped improve the quality of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Parker Solar Probe was designed, built, and is now oper- ated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s Living with a Star (LWS) program (contract NNN06AA01C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Support from the LWS management and technical team has played a critical role in the success of the Parker Solar Probe mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' SDO is the first mission to be launched for NASA’s Living With a Star (LWS) Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The SDO/AIA and SDO/HMI data are provided by the Joint Science Operations Center (JSOC) Science Data Processing (SDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Solar UltraViolet Imager (SUVI) product development, analysis, calibration, validation, and data stewardship by CIRES-affiliated authors within National Centers for Envi- ronmental Information (NCEI) was supported by National Oceanic and Atmospheric Administration cooperative agree- ment no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' NA17OAR4320101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We gratefully acknowledge the use of data from the Goode Solar Telescope (GST) of the Big Bear Solar Observatory (BBSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' BBSO operation is supported by US NSF AGS- 1821294 grant and New Jersey Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' GST operation is partly supported by the Korea Astronomy and Space Science Institute and the Seoul National University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' APPENDIX 8 RAOUAFI ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' BBSO/GST MAGNETIC FIELD DATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Taking advantage of high-order correction by the adaptive optics system with 308 sub-apertures (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2010) and the solar speckle interferometric data-reconstruction technique (W¨oger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2008), the observation during ∼16:34 – 18:38 UT achieved diffraction-limited resolution under a favorable seeing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' There is a ∼ 20-minute observation gap between 18:07 – 18:27 UT due to bad seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Spectroscopic polarization measurements of Fe I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='56 µm were taken by NIRIS with a 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='24 resolution and a 42 s cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Due to weak polarization signal in the quiet-Sun regions, line-of-sight (LOS) magnetograms are reduced by summing Stokes- V profiles from GST observations to enhance the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic field strength is scaled the contemporal HMI magnetic- field measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The small-scale magnetic elements are tracked with Southwest Automatic Magnetic Identification Suite (SWAMIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' DeForest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2007) based on similarity heuristics across a time series of magnetograms, by which the magnetic cancellation events are detected and their corresponding magnetic fluxes are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' ENERGY AND PARTICLE FLUXES FROM MAGNETIC RECONNECTION This section estimates the magnetic energy flux resulting from the small-scale reconnection episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This energy flux is transferred to the plasma in the form of bulk flows and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We start from the observed particle and kinetic energy ejection rates into the corona and wind, using the following average jetlet properties: Transverse scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', width): LJ ≈ 3′′ ≈ 2000 km Speed: VJ ≈ 150 km s−1 Lifetime: τ ≈ 5 min = 300 s Density: n ≈ 5 × 108 cm−3 The jetlet speed is determined from the time-distance diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' It is the projected speed on the plane of the sky, which should be considered as a lower limit on the real jetlet speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For the electron density, we used a typical plume density at the base of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The jetlets may be denser, perhaps by as much as a factor of 4 (Sterling & Moore 2020), but we employ the ambient coronal density to be conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' There are variations by at least a factor of 2 in the width (LJ) and lifetime (τ) of the jetlets that decrease or increase our estimated jetlet contributions to the wind and, therefore, the number of jetlets required to drive the entire solar-wind flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' It is not possible to be precise about these contributions beyond a factor of about 2 in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We have endeavored here to demonstrate that conservatively estimated jetlet contributions to the wind are comparable to the total estimated mass and energy fluxes from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The resulting values are given in the main text of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The reconnecting magnetic field strength in the corona is much smaller than the measured average photospheric flux density, Bph ≈ 100 G in quiet Sun and slightly higher in the coronal hole boundary region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Because we do not have direct coronal magnetic-field measurements, we must infer the strength of the reconnecting field, BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We do this by requiring the magnetic energy released by the reconnection to be sufficient to power the jetlet outflow, plus an assumed equivalent amount of plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For simplicity, we ignore the unknown but plausible contribution of released magnetic energy to accelerated nonthermal particles, which is a very important and well-known consequence of reconnection in large CMEs and flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The magnetic reconnection inflow speed, VR, is assumed to be a fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='1 of the Alfv´en speed, VA = BR/√4πρ, associated with the reconnecting field strength, BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This dimensionless reconnection rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', the inflow Alfv´en Mach number) is well established from numerical MHD simulations of fast reconnection, including those specifically of reconnection-driven coronal jets (Karpen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The reconnection occurs over a transverse scale LR that defines the width of the reconnection region in the corona, and which may be smaller than the width LJ of the jetlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' For further detail on the theory of magnetic reconnection, see Lin & Lee (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The kinetic energy flux density and total release rate into the bulk outflow are wKE = 1 2ρV 2 J VJ, (B1) WKE = wKEL2 J (B2) whence wKE ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='2 × 106 erg cm−2 s−1 and WKE ≈ 5 × 1022 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Assuming that half of the magnetic energy is transferred to the plasma in the form of heating, the total magnetic energy release rate during the reconnection, WME, satisfies WME = 2 WKE ≈ 1 × 1023 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' We have wME = 1 4π B2 R VR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content='1 4π B2 R BR √4πρ, (B3) WME = wME L2 R, (B4) wME = WME/L2 R = 2 WKE/L2 R = 2 wKE L2 J/L2 R, (B5) MAGNETIC RECONNECTION AS THE DRIVER OF THE SOLAR WIND 9 which can be solved for the reconnection field strength, B3 R = 80 π3/2 √ρ wME (B6) = 160 π3/2 L2 J L2 R wKE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' (B7) Substituting values from above, we obtain BR ≈ 5 (LJ/LR)2/3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The implied field strength depends, as is to be expected, on the ratio of characteristic scales in the reconnection region and in the resultant jetlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The minimum strength is about 5 G, 5% of the average photospheric flux density of ∼ 100 G, obtained for LR = LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' This is a reasonable value for the coronal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The implied field strength doubles to about 10 G if we assume that LR = LJ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL The supplementary material package contains five video files and can be downloaded HERE Movie M1: Composite of SDO/AIA and GOES-R/SUVI 171 ˚A image sequences showing the small-scale activity at the base of the solar corona and its extension to higher altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The movie begins on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Its real-time duration is 8 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Movie M2: SDO/AIA 171 ˚A image sequence showing the small-scale activity at the base of the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The movie starts on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Its real-time duration is 8 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Movie M3: Zoom on the SDO/AIA 171 ˚A image sequence of the northern solar polar region where small-scale activity is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The movie starts on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Its real-time duration is 8 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Movie M4: Zoom on the SDO/AIA 171 ˚A image sequence of the southern solar polar region where small-scale activity is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The movie starts on April 28th, 2021 at 00:00:09 and ends the same day 09:57:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Its real-time duration is 8 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' Movie M5: Animation of high-resolution magnetograms from the BBSO/GST-NIRIS instrument showing the highly-dynamic magnetic fields at small-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The number of the cancelling bipoles is much greater than that at 1′′ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' The movie begins on June 29th, 2018 at 16:32:18 UT and ends the same day 18:38:36 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', De Pontieu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', & Norton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2018, ApJL, 868, L27 Panesar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 2016, SSRv, 201, 1 Raouafi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', Petrie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=', Norton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQf-_r6/content/2301.00903v1.pdf'} +page_content=' 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0000000000000000000000000000000000000000..a1984dc670f9e50f8ab1e510361d9fd41373e9e3 --- /dev/null +++ b/jdE2T4oBgHgl3EQfIQZD/content/tmp_files/2301.03677v1.pdf.txt @@ -0,0 +1,2817 @@ +Draft version January 11, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +The fundamental signature of star formation quenching from AGN feedback: +A critical dependence of quiescence on supermassive black hole mass not accretion rate +Asa F. L. Bluck,1 Joanna M. Piotrowska,2, 3 and Roberto Maiolino2, 3, 4 +1Department of Physics, Florida International University,11200 SW 8th Street, Miami, 33199, Florida, USA +2Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK +3Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK +4Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK +(Accepted to ApJ on 16 December 2022) +ABSTRACT +We identify the intrinsic dependence of star formation quenching on a variety of galactic and envi- +ronmental parameters, utilizing a machine learning approach with Random Forest classification. We +have previously demonstrated the power of this technique to isolate causality, not mere correlation, in +complex astronomical data. First, we analyze three cosmological hydrodynamical simulations (Eagle, +Illustris, and IllustrisTNG), selecting snapshots spanning the bulk of cosmic history from comic noon +(z ∼ 2) to the present epoch, with stellar masses in the range 9 < log(M∗/M⊙) < 12. In the simu- +lations, black hole mass is unanimously found to be the most predictive parameter of central galaxy +quenching at all epochs. Perhaps surprisingly, black hole accretion rate (and hence the bolometric +luminosity of active galactic nuclei, AGN) is found to be of little predictive power over quenching. +This theoretical result is important for observational studies of galaxy quenching as it cautions against +using the current AGN state of a galaxy as a useful proxy for the cumulative impact of AGN feedback +on a galactic system. The latter is traced by black hole mass not AGN luminosity. Additionally, we +explore a sub-set of ‘observable’ parameters, which can be readily measured in extant wide-field galaxy +surveys targeting z = 0 − 2, at 9 < log(M∗/M⊙) < 12. All three simulations predict that in lieu of +black hole mass, the stellar gravitational potential will outperform the other parameters in predicting +quenching. We confirm this theoretical prediction observationally in the SDSS (at low redshifts) and +in CANDELS (at intermediate and high redshifts). +Keywords: Galaxies: formation, evolution, star formation, quenching, feedback +1. INTRODUCTION +Galaxies are observed to display profound bimodality in +key diagnostic diagrams, most notably in color - mag- +nitude (e.g., Strateva et al. 2001). Perhaps more phys- +ically, the star forming main sequence relation between +star formation rate (SFR) and stellar mass (M∗) exhibits +a tight star forming ridge line, with quiescent galax- +ies significantly offset to lower star formation rates at a +fixed stellar mass (e.g., Brinchmann et al. 2004). Con- +sequently, it is often stated that galaxies exhibit two +fundamental types: i) actively star forming, and ii) qui- +escent (or ‘quenched’) systems. Quenched systems tend +Corresponding author: Asa F. L. Bluck +abluck@fiu.edu +to have redder optical colors, older stellar populations, +more elliptical morphologies, more pressure supported +kinematics, higher masses, and reside in denser cosmic +environments compared to their actively star forming +counterparts (e.g., Baldry et al. 2006; Peng et al. 2010, +2012; Wuyts et al. 2011; Bluck et al. 2014, 2016; Brown- +son et al. 2022). Much of the field of galaxy evolution is +explicitly focused on explaining the origin of quiescent +galaxies, and hence accounting for the observations of +bimoodality in galactic properties. +Theoretically, the existence of quiescent galaxies is +translated to a problem of cosmological star formation +efficiency. More precisely, one of the main goals of mod- +ern simulations of galaxy formation is to reduce the effi- +ciency with which stars form within dark matter haloes +(i.e., ϵSF ≡ M∗/MHalo ≤ 0.1 Ωb/ΩM). The reason for +arXiv:2301.03677v1 [astro-ph.GA] 9 Jan 2023 + +2 +Bluck et al. +this is that simple models of galaxy formation (utilising +only gravitation and cooling) predict that the vast ma- +jority of baryons should reside in stars by the present +epoch (e.g., Cole et al. 2000; Bower et al. 2006, 2008; +Henriques et al. +2015, 2019). +However, observations +place the fraction of baryons in stars an order of magni- +tude lower (e.g., Fukugita & Peebles 2004; Shull 2012). +In the past decade or so, feedback from active galac- +tic nuclei (AGN) has become the favored mechanism +in modern hydrodynamical simulations to quench star +formation, and hence reproduce the observed galaxy bi- +modality (see, e.g., Sijacki et al. 2007; Vogelsberger et +al. 2014a,b; Schaye et al. 2015; Weinberger et al. 2018; +Zinger et al. 2020). There is abundant evidence that +AGN produce more than enough energy to quench star +formation (e.g., Silk & Rees 1998; Bluck et al. 2011; +Maiolino et al. 2012; Fabian 2012). Yet, direct obser- +vational evidence for AGN feedback causing quenching +within galaxies remains sparse, and hotly debated (al- +though see Hlavacek-Larrondo et al. 2012, 2017, 2018 +for perhaps the strongest direct evidence to date, pri- +marily within galaxy clusters and very massive groups). +Thus, it remains unclear whether the crucial ingredient +in cosmological models to quench galaxies is viable for +the vast majority of systems, or not. +Classification is the branch of data science that fo- +cuses on understanding the fundamental differences be- +tween types of objects, with the ultimate purpose of ac- +curate segregation between them. As such, classification +is at the heart of the scientific study of galaxy popula- +tions: If we can learn to accurately classify star form- +ing and quenched galaxies on the basis of their physical +properties, we can establish the underlying physics of +galaxy quenching. For example, we have demonstrated +the power of machine learning classification to ‘reverse +engineer’ cosmological simulations, revealing the input +physics and its observational consequences (see Bluck et +al. 2022; Piotrowska et al. 2022). But the real power +of machine learning in this application is apparent when +one compares the results of classification between simu- +lations and observational data. If there is strong agree- +ment, the models can be used as an explanatory tool +for the observations. Alternatively, if there is major dis- +agreement, the observations can be used to improve the +next generation of models by ruling out certain prescrip- +tions or processes. +The goal of this paper is to leverage the power of clas- +sification to determine what is intrinsically important +for central galaxy quenching in both simulations and +observations. To this end, we employ a Random Forest +classifier, which enables the identification of causality +through carefully controlling for nuisance variables (see +Bluck et al. 2022 for a detailed description of this tech- +nique including multiple tests). +Utilizing Random Forest classification, in Piotrowska +et al. (2022) we discovered that the most effective pa- +rameter for separating star forming and quenched cen- +tral galaxies in the local Universe is black hole mass. +Interestingly, this is a unanimous prediction from three +contemporary cosmological simulations (Eagle, Illustris, +and IllustrisTNG), which use very different AGN feed- +back prescriptions. +As such, this prediction is not +strongly model dependent and hence can be seen as a +ubiquitous consequence of AGN feedback (in almost any +mode). +We also considered stellar mass, dark matter +halo mass, and black hole accretion rate as potential +drivers of quenching in the simulations. Yet, all of these +parameters are of negligible importance to quenching +once black hole mass is made available to the classifier. +The results from Piotrowska et al. (2022) are highly +important for observational studies of AGN feedback be- +cause they clearly imply that ‘catching AGN quenching +in the act’ is not possible in the local Universe. Instead, +one must look for the fossil record of historic AGN feed- +back, encapsulated in the mass of the central black hole, +not the current accretion rate or AGN luminosity. We +also established that once black hole mass is controlled +for, neither stellar nor halo mass should be constraining +of central galaxy quenching. We confirmed this latter +prediction observationally through comparison with the +SDSS, which effectively rules out virial shocks or super- +nova feedback as significant mechanisms for quenching +central galaxies in the local Universe (see also Bluck et +al. 2016, 2020a for further evidence on this). +In this paper, we expand on the work of Piotrowska +et al. (2022) by identifying the key quenching predic- +tions from Eagle, Illustris, and IllustrisTNG at multiple +epochs from z = 0 to cosmic noon (where the number +density of quiescent galaxies becomes very low). This +is essential in order to establish whether there is red- +shift dependence on the predicted signature of AGN +feedback quenching or not. +For instance, it could be +that at high redshifts a strong dependence of quies- +cence on the current AGN state of a galaxy becomes +apparent, which would have profound implications for +how to search for observational evidence of AGN feed- +back (or lack thereof) in upcoming observational sur- +veys (e.g., with JWST and VLT-MOONS). Addition- +ally, we perform a preliminary test on the high-z predic- +tions from cosmological simulations, utilizing photomet- +ric data from HST-CANDELS at z = 0.5 - 2, and com- +pare this to a novel test of the simulations’ predictions +at low-z utilizing spectroscopic data from the SDSS. + +AGN feedback driven quenching +3 +The paper is structured as follows. In Section 2 we +give an overview of the simulations and observational +data. In Section 3 we present our method to identify qui- +escent systems and give a brief overview of our Random +Forest classification technique. In Section 4 we present +our results and discuss their importance for the field of +galaxy evolution. We summarize our contributions in +Section 5. +Additionally, in the Appendix, we present +numerous detailed tests on the Random Forest results, +which confirm our main conclusions. Throughout the +paper we assume a spatially flat ΛCDM cosmology, and +set h ≡ H0/(100 km/s/Mpc) = 0.7 consistently for all +physical representations of simulated and observational +data. +2. DATA SOURCES +2.1. Simulations +In this paper, we consider three cosmological hydrody- +namical simulations which incorporate AGN feedback to +quench central (and more generally, massive) galaxies. +Our goal is to identify the testable consequences of AGN +feedback driven quenching in these models. Explicitly, +we work with publicly available multi-epoch snapshot +data from: Eagle1 (Schaye et al. +2015; Crain et al. +2015; McAlpine et al. 2016); Illustris2 (Vogelsberger et +al. 2014a,b; Genel et al. 2014; Sijacki et al. 2015; Nel- +son et al. 2016); and IllustrisTNG3 (Marinacci et al. +2018; Naiman et al. 2018; Nelson et al. 2018; Springel +et al. 2018; Pillepich et al. 2018; Nelson et al. 2019). +Full details on the simulations are provided in the above +references, including information on data access. A de- +tailed description of the similarities and differences of +these three simulations is provided in Piotrowska et al. +(2022). Here we give a review of the most important de- +tails for this work (i.e. the black hole growth and AGN +feedback mechanisms). +In all simulations we select central galaxies as the most +massive systems in each (group) dark matter halo. Iso- +lated galaxies are treated as the centrals of their indi- +vidual dark matter haloes. We select galaxies to have +stellar masses in the range 9 < log(M∗/M⊙) < 12, resid- +ing in dark matter (group) haloes with Mhalo > 1011M⊙. +These cuts mitigate issues with mass and volume reso- +lution in the simulations, whilst still enabling a large +sampling of both star forming and quenched systems. +Even though the majority of quiescent galaxies in all +of the simulations and observations considered in this +1 Eagle Data Access: http://icc.dur.ac.uk/Eagle/ +2 Illustris Data Access: www.illustris-project.org +3 IllustrisTNG Data Access: www.tng-project.org/ +work have log(M∗/M⊙) > 10, it is vital to select a large +sample of both star forming and quiescent classes for +random forest classification. All specific parameters are +collated as in Piotrowska et al. +(2022), but here ex- +tracted for multiple snapshots (rather than just at z = +0). The identification of quiescent systems is discussed +in Section 3. +From Illustris and IllustrisTNG (which have the +same naming conventions) we take the following pa- +rameters +from +the +public +snapshot +data +at +z += +(0, +0.5, +1, +1.5, +2): +From the SubFind catalog - +SubhaloSFR, SubhaloMassStar, SubhaloBHMass, Sub- +haloBHmdot, SubhaloHalfmassRad; and from the FoF +catalog - Group M Crit200, Group R Crit200. +We +also extract the co-moving coordinates for each system +(galaxy and halo), as well as the FoF Halo ID and Sub- +Find sub-halo ID. From Eagle we take the following pa- +rameters from the public snapshot data at z = (0, 0.5, 1, +1.49, 2.01): From the SubFind catalog - StarFormation- +Rate, MassType Star, BlackHoleMass, BlackHoleAccre- +tionRate, HalfMassRad Star; and from the FoF catalog +- Group M Crit200, Group R Crit200. As with Illustris +and IllustrisTNG, we also extract halo and sub-halo co- +ordinates and IDs for each system in both the FoF and +SubFind catalogs. +A publicly available docker is pro- +vided with Piotrowska et al. +(2022) showing how to +extract these parameters from each simulation for the z += 0 snapshots. +2.1.1. Eagle +For Eagle, we utilize the EAGLE-RefL0100N1504 run +(Schaye et al. +2015). +This run has a box size of +∼100 cMpc3 and implements the most detailed feed- +back mechanisms of the Eagle simulation suite. Eagle +is performed utilizing a smoothed particle hydrodynam- +ics (SPH) code, explicitly GADGET-3 (Springel 2005). +Cosmological parameters are taken from Planck Collab- +oration I (2014), assuming a spatially flat ΛCDM cos- +mology. Black holes are seeded at MBH = 105M⊙h−1 in +all haloes once they reach MHalo = 1010M⊙h−1. Black +hole accretion is regulated by Bondi-Hoyle accretion, i.e. +˙MBH ∝ M 2 +BH (e.g., Hoyle & Lyttleton 1936; Bondi & +Hoyle 1944), and is Eddington limited. A single feed- +back mode is applied, which corresponds approximately +to a quasar wind, triggered primarily by cold-mode ac- +cretion (see Booth & Schaye 2009). +Energy injection +into the surrounding gas particles, in a given time step +∆t, is given explicitly by ∆EBH = ϵfϵr ˙MBHc2∆t, where +ϵr is the radiative efficiency (set equal to 0.1) and ϵf is +the fraction of energy which couples to the inter-stellar +medium (ISM) producing energetic feedback. Energy is +released thermally once ∆EBH is sufficient to induce a + +4 +Bluck et al. +temperature change of ∆T = 108.5K for at least one +neighboring gas particle. Hence, heating of gas parti- +cles near to the black hole is applied stochastically. The +large thermal injection, in essentially random directions, +is key to overcoming the numerical over-cooling problem +in this simulation. The Eagle AGN feedback mechanism +is effective at quenching massive galaxies; however, it +is less effective at keeping massive galaxies quenched +than the other simulations considered here (see, e.g., +Piotrowska et al. 2022). +2.1.2. Illustris +For Illustris, we utilize the full ILLUSTRIS-1 run with +box size ∼100 cMpc3 (Vogelsberger et al. 2014a,b). Il- +lustris is run utilizing the moving-mesh code AREPO +(Springel 2010). +Cosmological parameters are set as +in WMAP7 (Hinshaw et al. +2013), assuming a spa- +tially flat ΛCDM cosmology. +Black holes are seeded +at MBH = 105M⊙h−1 in all haloes once they reach +MHalo = 5×1010M⊙h−1. As in Eagle, black hole growth +is regulated via Bondi-Hoyle accretion, limited by the +Eddington rate. Illustris operates two distinct feedback +modes, although only one is effective at impacting star +formation within galaxies. The first is a ‘quasar’ mode +which operates using the same general principle as the +single mode in Eagle (outlined above). +However, the +energy injection is continuous, rather than bursty (see +Sijacki et al. 2007, 2015). This mode is uncorrelated +with quenching in the simulation. The second is a ‘ra- +dio’ mode, which aims to simulate the effects of rela- +tivistic jets on the circum-galactic medium (CGM). At +low accretion rates (χEdd = +˙MBH/ ˙MEdd < 0.05), once +a black hole increases its mass by 15% of its value, a +bubble is seeded in the host galaxy’s CGM, with energy +Ebubble = ϵmϵr∆MBHc2, where ϵm represents the cou- +pling efficiency of the mechanical feedback to the hot +gas halo. Heating is induced in the CGM through PdV +work as the bubble expands. This mode is partially ef- +fective at shutting down gas cooling from the CGM, and +hence reducing star formation in the galaxy via starva- +tion. However, as is now widely known, the jet bubbles +also have the deleterious effect of completely vacating +the CGM of gas in stark contrast to observations (e.g., +Nelson et al. 2018; Pillepich et al. 2018). +2.1.3. IllustrisTNG +For IllustrisTNG, we utilize the TNG-100-1 simulation +(Nelson et al. +2018; Pillepich et al. +2018), which +has an identical box size to our selected run in Illus- +tris. IllustrisTNG adopts cosmological parameters from +Planck Collaboration (2016). This simulation offers the +best compromise between resolution and volume for our +present study (see Piotrowska et al. 2022). TNG was +run with an updated version of AREPO, extended to add +magnetic fields to the implementation. Black holes are +seeded at a higher mass of MBH = 8 × 105M⊙h−1 in all +haloes once they reach MHalo = 5 × 1010M⊙h−1. As in +Eagle and Illustris, black hole growth is modeled sub- +grid via Eddington limited Bondi-Hoyle accretion. The +‘quasar’ mode feedback is left identical to Illustris, and it +still has very little impact on star formation or quench- +ing (see Weinberger et al. 2017). Alternatively, the ‘ra- +dio’ mode feedback of Illustris (which was over-zealous +in its removal of CGMs, though effective at quenching +galaxies) is replaced with a new ‘kinetic’ mode. When a +black hole is accreting at a ‘low’ level (defined relative to +the black hole mass, see Weinberger et al. 2017), energy +injection is applied kinetically to a group of neighbor- +ing gas cells in a stochastic manner. This occurs at a +threshold black hole mass of MBH ∼ 108M⊙, which is +set partly by the sub-grid AGN feedback prescription +and partly by relation to other evolving parameters in +the simulation (see Zinger et al. 2020 for a full discus- +sion on this). +The change in kinetic energy of a gas +cell is given by ˙Ekinetic = ϵk ˙MBHc2, where ϵk is the ef- +ficiency of energy transfer. +The efficiency itself is set +as a function of the gas density around the black hole +(see Weinberger et al. 2017). Ultimately, a momentum +kick is applied in a randomly chosen direction, such that +(integrated over time) isotropy is preserved. Unlike the +radio mode in Illustris, the kinetic mode in TNG im- +pacts the ISM as well as the CGM. In slightly more +detail, the TNG kinetic mode drives winds in the ISM +as well as jet-like features in the CGM, which simulta- +neously adds turbulence to the ISM and increases the +entropy (and hence cooling time) of the CGM. This oc- +curs without removing significant quantities of gas from +either, resolving the severe issues in Illustris (see Zinger +et al. 2020: Piotrowska et al. 2022). +2.1.4. Simulations Summary +Eagle, Illustris, and IllustrisTNG represent three con- +temporary galaxy evolution models which all quench +star formation in massive galaxies via AGN feedback. +It is important to appreciate that, although very differ- +ent in the details, all of the above AGN feedback models +extract energy from around the black hole (ultimately +some fraction of the accreted rest energy, MBHc2). +Hence, the total feedback energy is directly proportional +to black hole mass in each case, i.e. Efeedback ∝ MBH. +This suggests a way to test the entire paradigm of AGN +feedback quenching in an essentially model independent +manner (see Section 4.1, as well as Bluck et al. 2020a; +Piotrowska et al. 2022). Nonetheless, there are many +important differences between these three galaxy forma- + +AGN feedback driven quenching +5 +tion simulations in terms of quenching at a quantitative +level (see, e.g., Donnari et al. 2021; Piotrowska et al. +2022). +2.2. Observations +We compare the predictions for AGN feedback driven +quenching from simulations to the results from two ob- +servational galaxy surveys: the Sloan Digital Sky Sur- +vey - SDSS4 (Abazajian et al. 2009), and the Cosmic +Near-Infrared Deep Extragalactic Survey - CANDELS5 +(Grogin et al. 2011; Koekemoer et al. 2011). For CAN- +DELS, we additionally utilize value added catalogs from +Dimauro et al. (2018)6. A thorough description of the +parameter estimation of galaxies from these surveys is +provided in Bluck et al. (2022). Here we give only a brief +overview of the most important measurements used in +this work. +2.2.1. SDSS +We take star formation rates (SFRs) for SDSS galax- +ies from Brinchmann et al. +(2004), which are com- +puted from emission lines where possible, or else from +an empirical relationship between D4000 - sSFR for non- +emission line galaxies (or systems with a significant AGN +contribution). Fibre SFRs are converted to global SFRs +utilizing the colors of galaxies outside of the aperture +(see Brinchmann et al. +2004 for a full description). +We take stellar masses from the Mendel et al. (2014) +SDSS mass catalogs. +We take galaxy size estimates +from the Simard et al. (2011) morphological catalogs. +Additionally, we utilize volume weights from Bluck et +al. +(2014) to simulate the statistical appearance of a +volume complete sample, as appropriate for simulation +comparison. +Finally, we utilize the group catalogs of +Yang et al. (2007, 2009) to separate central and satel- +lite galaxies in the SDSS. Centrals are defined as the +most massive galaxy in the dark matter halo, with iso- +lated galaxies also counting as centrals for our purpose. +Satellites are defined to be any other galaxy contained +within the group (or cluster) halo. Extensive details on +all of these measurements, as well as numerous checks +on their reliability, are provided in Bluck et al. (2014, +2019, 2022). For our analysis, we select galaxies to be +centrals, have M∗ > 109M⊙, and reside within haloes +with masses Mhalo > 1011M⊙ (which is identical to our +simulations selection). +4 SDSS Data Access: https://classic.sdss.org/dr7/access/ +5 CANDELS Data Access: http://arcoiris.ucolick.org/candels/ +6 CANDELS +VAC: +https://mhuertascompany.weebly.com/data- +releases-and-codes.html +2.2.2. CANDELS +We utilize photometric star formation rates; stellar +masses of galaxies, bulges and disks; galaxy sizes; and +rest-frame/ dust corrected UVJ colors from SED fitting +of CANDELS galaxies provided in the value added cat- +alogs of Dimauro et al. (2018). The SED fitting is per- +formed utilizing the FAST code (Kriek et al. +2009), +with stellar population synthesis models taken from +Bruzual & Charlot (2003), assuming a Chabrier IMF +and Calzetti et al. (2000) extinction curve. Since CAN- +DELS is a photometric only survey, unlike the SDSS, it +is not possible to construct accurate central - satellite +segregation, or group determination, due to the lack of +precision in photometric redshift estimates compared to +spectroscopic redshift measurements. As such, we an- +alyze the full CANDELS galaxy sample, as opposed to +focusing only on centrals. This is unlikely to be a sig- +nificant problem since the vast majority of galaxies of +these masses are predicted to be centrals at all epochs +(∼80%, see, e.g., Henriques et al. 2015). Additionally, +we select galaxies to have stellar masses M∗ > 109M⊙ +(in line with the simulations and the SDSS), but do not +apply a halo mass cut (due to a lack of accurate halo +estimation in this survey). We note that in the SDSS +the impact of additionally applying a halo mass cut is +very small (< 5% of the sample is affected and all results +are invariant). Full details on the CANDELS measure- +ments are given in the above references, and an extensive +discussion on the reliability of this data is provided in +Bluck et al. (2022). +3. METHODS +3.1. Identifying Quenched Systems Throughout Cosmic +Time +In this paper we utilize a simple method to segregate +actively star forming and quenched galaxies at multiple +epochs. +We apply the exact same method in simula- +tions and observations to mitigate the potential for bias +through the use of heterogeneous approaches. However, +since the level of star formation in galaxies of a fixed +stellar mass varies as a strong function of redshift (e.g. +Madau & Dickinson 2014), it is necessary to allow the +method to reflect this. +Explicitly, we define quenched galaxies to be any sys- +tem with a specific star formation rate: +sSFR(z′) < sSFRpeak(z = z′) − 1 dex +(1) +where, sSFRpeak(z = z′) indicates the peak (mode) of +the sSFR distribution at a redshift equal to the galaxy in +question (i.e. z = z′). Hence, quenched galaxies are de- +fined to be forming stars at a rate less than one order of + +6 +Bluck et al. +11.5 +11.0 +10.5 +10.0 +9.5 +9.0 +8.5 +log(sSFR [yr +1]) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Normalised Counts +Eagle : sSFR Distributions +Eagle @ z=0 +Eagle @ z=1 +Eagle @ z=2 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +z +10.5 +10.0 +9.5 +9.0 +8.5 +8.0 +7.5 +7.0 +log(sSFRpeak [yr +1]) +CANDELS: Redshift Evolution in Peak sSFR +Best Fit sSFR(z) +Quenched Threshold : sSFRpeak(z) +1dex +1/tH(z) [yr +1] +Figure 1. Illustration of our method to select quenched and star forming galaxies in simulations and observations. Left Panel: +sSFR distributions for three redshift snapshots from the Eagle simulation. The peak sSFR is indicated by dashed lines, and +our adopted quenched threshold (at sSFR = sSFRpeak - 1dex) is indicated by dotted lines. Right Panel: +Evolution in peak +sSFR from z = 0.5 - 2 in CANDELS. Blue dots display the peak of the sSFR distribution, with x, y error bars indicating the +z-range applied and 1σ dispersion in sSFR, respectively. A linear best fit to the peak sSFR evolution is shown in magenta, with +our corresponding quenched threshold shown as a dashed red line. For comparison, the inverse of the Hubble time is displayed, +which provides a reasonable quenching threshold at these epochs. +magnitude below the star forming peak at the redshift of +each object (approximately 3σ below the main sequence +relation). In the simulations, all systems with SFR = +0 are defined as quenched and included in all analyses. +Star forming systems are considered to be any galax- +ies with sSFR values higher than the quenched limit. +This method is illustrated for one simulation (Eagle) in +Fig. 1 (left-hand panel). Note that we do not apply +a mass term to our sSFR limit unlike in some of our +prior work (e.g., Bluck et al. 2014, 2016). The reason +for this is two-fold. First, the impact of sSFR variation +with stellar mass is weak except at the high mass end, +where the majority of systems are quenched in any case. +Hence, this does not impact our classification results in +any significant manner (as we have checked). Second, +we choose this cut to be in line with our prior work +(Piotrowska et al. 2022), where this decision was made +to maximize the objectivity of the quenching criteria +between different models and observations, which have +different mass dependencies. Finally, the curve in the +sSFR - M∗ relation is likely due to high mass star form- +ing systems having quenched cores. Thus, it remains an +open question conceptually whether or not this ought to +be accounted for (see, e.g., Bluck et al. 2020a). +However, the CANDELS data spans a wide range of +redshifts. To combat this, we compute the peak sSFR +of star forming systems at multiple epochs, and fit a +simple redshift relation to this (as illustrated in Fig. 1, +right-hand panel). Additionally for CANDELS, we also +consider a check on the sSFR method utilizing UVJ col- +ors (exactly as in Bluck et al. 2022). The results from +these two approaches are essentially identical. +We have explored alternative definitions of quenching, +including an offset from the redshift evolving star form- +ing main sequence relation (∆SFR, which fully accounts +for potential variations in mass on the quenching defini- +tion), different threshold cuts, and excluding green val- +ley systems with intermediate levels of star formation. +The results in this paper are completely stable to these +alternative methods for identifying quenched systems. +3.2. Random Forest Classification: Revealing Causality +with Machine Learning +In this paper, we utilize a machine learning approach for +classifying star forming and quenched systems in simu- +lated and observed galaxy surveys. +More specifically, +we adopt a Random Forest classifier from the powerful +ScikitLearn Python package (Pedregosa et al. 2011)7. +This method is an ideal compromise between sophisti- +cation (the ability to account for highly non-linear and +non-monotonic relationships between multiple parame- +ters) and interpretability (the ability to extract mean- +ingful insights from the trained classifier). Full details +on this method are provided in Bluck et al. +(2022), +Brownson et al. (2022), and Piotrowska et al. (2022), +including numerous detailed tests. Of particular value +for our present application is the capacity of the Ran- +dom Forest classifier to separate causally related param- +eters from nuisance parameters in a classification prob- +lem (see Bluck et al. 2022 appendix B). Ultimately, this +is achieved through controlling for all other parameters +when ascertaining the importance of any given parame- +7 ScikitLearn: https://scikit-learn.org/stable/ + +AGN feedback driven quenching +7 +ter to the classification problem at hand. Here we give a +brief review of the most important aspects of our Ran- +dom Forest classification method for this work. +A Random Forest is a set of decision trees with dif- +ferences between them ensured through bootstrapped +random sampling. Two example decision trees from our +analysis are shown in the Appendix (see Figs. 8 & 9). +In training, the criteria for each decision branch (in each +decision tree) is set by selecting the optimal parameter +(and threshold) to most effectively separate the classes +of data, defined explicitly by the Gini impurity: +G(n) = 1 − +c=2 +� +i +� +pi(n)2� +(2) +where pi(n) indicates the probability of randomly se- +lecting class, i, at node, n. +The summation is made +over all classes (i.e. in this paper, over star forming and +quenched systems). For example, in our application, the +most useful observable (from a list containing black hole +mass, stellar mass, halo mass etc.) is selected to maxi- +mize the accuracy with which quenched and star forming +galaxies are separated at each branch. Then the optimal +threshold in that variable is found to minimize the Gini +impurity. The entire decision tree is filled out through +iteration, until no further improvements are possible (or +else a pre-defined threshold is reached). +Once the Random Forest classifier is trained, one can +extract how important each parameter is for solving the +classification problem. The relative importance of a fea- +ture (i.e. parameter under investigation), k, is defined +as: +IR(k) = +1 +Ntrees +� +trees +�� +nk N(nk)∆G(nk) +� +n N(n)∆G(n) +� +(3) +where the numerator in the above expression indicates +the the improvement in Gini impurity (∆G(n)) weighted +by the number of data points entering that node (N(n)), +summed over all nodes which utilize feature parameter +k (hence the additional subscripts). The denominator +provides a similar summation, but over all nodes, re- +gardless of the features used. +Hence, the ratio gives +the relative importance of the variable k for solving the +classification problem (i.e. +in this work, determining +whether galaxies will be star forming or quenched) in +a given decision tree. Finally, an average is taken over +all tress in the Random Forest to give the final relative +importance statistic of each training parameter. This is +the key statistic we use in this paper, and from now on +we will refer to it as the quenching importance (due to +our specific science application). +To assess the uncertainties on the Random Forest clas- +sification, we run 25 independent training and testing +runs of each analysis, taking the median importance as +the final result and the standard deviation as the sta- +tistical error. +We are careful to avoid over-fitting by +testing the performance of the Random Forest classifier +on unseen data. More specifically, we require a differ- +ence in performance (measured by the area under the +true positive - false positive curve, AUC) of ∆AUC < +0.02 (see Teimoorinia et al. +2016; Bluck et al. +2022 +for further details). +The required (very) small differ- +ence in performance between the training and testing +data sets prevents the classifier from learning patholog- +ical features in the training data which do not scale to +unseen data. +Before training the random forest, all parameters are +median subtracted and normalized by the interquar- +tile range, to avoid differences in parameter variance or +magnitude impacting the results. We train and test on a +balanced sample of 50% quiescent and 50% star forming +systems in each classification. This avoids weighting one +population as more important than the other. We also +reserve 50% of the data for performance testing, which is +unseen by the random forest classifier. This enables the +rigorous over-fitting testing, described above. The final +results are taken from the performance on the unseen +data. The above methodology is essentially standard in +the machine learning literature (see Teimoorinia et al. +2016; Bluck et al. 2019, 2020a,b and references therein +for further discussion). It is important to apply these +data preparation steps carefully to avoid biased results. +4. RESULTS & DISCUSSION +In this section we apply our Random Forest classi- +fication technique (discussed in Section 3.2) to three +cosmological hydrodynamical simulations, in order to +identify the key observable signature of AGN feedback +driven quenching across cosmic time (Section 4.1). Ad- +ditionally, we apply the Random Forest technique to +two observational wide-field galaxy surveys, in order to +test the model predictions across a wide range of epochs +(Section 4.2). We also discuss the implications of our +results within the context of the literature. +4.1. The Fundamental Signature of AGN Feedback +Driven Quenching in Simulations across Cosmic +Time +In Fig. 2 we present the results from a series of Random +Forest classification analyses applied to the problem of +identifying quiescent galaxies in the IllustrisTNG, Ea- +gle, and Illustris hydrodynamical simulations. Results + +8 +Bluck et al. +are presented separately for each of the three cosmo- +logical simulations (shown in separate panels) and for +five redshift snapshots spanning from z=0 to z=2 (as +labelled by the legend in each panel). The parameters +used to train the Random Forest are displayed along the +x-axis, and the relative importance for quenching is dis- +played as the bar height (on the y-axis). Uncertainties +on the quenching importance are estimated from the 1σ +dispersion of 25 parallel classification training and test- +ing runs, applying random sub-sampling of the data. +For each simulation, and at every redshift snapshot +considered, black hole mass is clearly identified as the +most predictive parameter for classifying star forming +and quenched galaxies. No other parameter reaches be- +yond ∼1/4 the relative importance of black hole mass at +any epoch, and most parameters are of negligible impor- +tance for quenching at all redshifts. Hence, the key pre- +diction of AGN feedback models is a clear dependence +of galaxy quenching on black hole mass. It is important +to highlight that this result is stable from cosmic noon +to the present epoch, unanimously predicted by three +independent simulations, and is (at least in principle) +observationally testable. +It is especially important to note the utter lack of im- +portance given to black hole accretion rate by the classi- +fier in all simulations and at all redshifts studied8. Given +that the total bolometric AGN luminosity is directly +proportional to accretion rate (i.e. LAGN = η RBH c2, +for an efficiency η ∼ 0.1), this further implies a negligi- +ble importance to quenching of contemporaneous AGN +luminosity (and hence also AGN identification). This re- +sult is in close agreement with Ward et al. (2022), who +find that the most luminous AGN in simulations are +found in the most gas rich (and hence star forming) sys- +tems. This demonstrates that there is expected to be no +instantaneous link to cessation of star formation in con- +temporary models, despite AGN being used to quench +galaxies in these simulations. Therefore, searching for +evidence of AGN feedback by looking for a difference in +(s)SFR between AGN and non-AGN systems is destined +to failure, if AGN feedback operates as it is predicted +to in modern hydrodynamical simulations. Nonetheless, +a great number of studies have proceeded in just this +manner (e.g., Nandra et al. 2007; Bundy et al. 2008; +Georgakakis et al. 2008; Silverman et al. 2008; Hickox +8 The only slight exception to this is for Eagle at the lowest red- +shift snapshot, where accretion rate performs as the second best +parameter. Interestingly, this is the ‘exception that proves the +rule’. The value of RBH actually turns out to be that it anti- +correlates with quenching. That is, higher accretion rates at low +redshifts in Eagle are associated with more star forming systems: +the opposite of catching AGN quenching in action! +et al. 2009; Xue et al. 2010; Aird et al. 2012; Rosario +et al. 2013; Heckman & Best 2014; Trump et al. 2015; +Ellison et al. 2016; Shimizo et al. 2017; Florez et al. +2020; and many more). Consequently, one must be cau- +tious in the interpretation of these prior results. Indeed, +from our analysis of hydrodynamical simulations, a lack +of clear connection between AGN luminosity and quies- +cence is actually a feature of the modern AGN feedback +quenching paradigm, not evidence against it. +We speculate that the above result reflects the impor- +tance of preventative feedback in these models. In such +a scenario, it is the energy released over long periods +from AGN which really matters for quenching (which is +clearly traced by MBH not RBH). Consequently, quies- +cence may emerge as a long term consequence of AGN +heating preventing gas cooling and accretion from the +CGM into massive galaxies, ultimately starving the sys- +tem of fuel needed for further star formation. This mode +of operation simultaneously resolves the cooling problem +in high mass groups and clusters (e.g., Fabian 2012), +explains the low cosmological star formation efficiency +(e.g., Fukugita & Peebles 2004), as well as accounting for +the observed demographics of star forming and quenched +galaxies (e.g., Peng et al. 2010; Bluck et al. 2014). In- +stantaneous feedback may still trigger quenching (e.g., +Zinger et al. +2020; Terrazas et al. +2020), but with- +out long term heating of the CGM, star formation will +inevitably reignite within massive galaxies, due to gas +cooling and condensation into the system, removing the +galaxy from our quenched sample. Since our machine +learning classification analysis is sensitive to what is fun- +damentally different between quenched and star forming +systems, our results pick up on the long term preven- +tative mode rather than the instantaneous trigger (by +design). +Additionally in Fig. +2, we present pie plot insets +showing the results from a simplified analysis, averag- +ing over all epochs in the simulations, and separating +parameters coarsely by whether they are black hole re- +lated (i.e., MBH & RBH) or non-black hole related (i.e., +M∗, Rh, MH, φ∗). It is clear that BH-parameters domi- +nate the predictive power in the Random Forest clas- +sification. +Within this set, as discussed above, it is +black hole mass not accretion rate which really mat- +ters for quenching. The rest of the parameters are all +of very little importance for predicting quiescence, offer- +ing on average <15% of the total quenching importance. +For straightforward theoretical reasons (see Bluck et al. +2020a; Piotrowska et al. 2022), stellar mass is closely re- +lated to the total energy released from supernovae, and +halo mass is closely related to the total energy released +from virial shocks. Hence, the simulations predict that + +AGN feedback driven quenching +9 +MBH +* +BH +M * +Rh +MH +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +TNG - All Parameters +TNG Centrals : z = 0.0 +TNG Centrals : z = 0.5 +TNG Centrals : z = 1.0 +TNG Centrals : z = 1.5 +TNG Centrals : z = 2.0 +BH 94% +6% +MBH +* +BH +M * +Rh +MH +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +Eagle - All Parameters +Eagle Centrals : z = 0.0 +Eagle Centrals : z = 0.5 +Eagle Centrals : z = 1.0 +Eagle Centrals : z = 1.5 +Eagle Centrals : z = 2.0 +BH 92% +8% +MBH +* +BH +M * +Rh +MH +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +Illustris - All Parameters +Illustris Centrals : z = 0.0 +Illustris Centrals : z = 0.5 +Illustris Centrals : z = 1.0 +Illustris Centrals : z = 1.5 +BH 87% +13% +Figure 2. Random Forest classification analysis to predict the existence of quiescent central galaxies in three cosmological +simulations. The quenching importance of black hole mass (MBH), the stellar potential (φ∗), black hole accretion rate (RBH ≡ +dMBH/dt), stellar mass (M∗), stellar half mass radius (Rh), and dark matter halo mass (MH) are displayed on the y-axis (as +labelled from left-to-right along the x-axis of each panel). The results are shown separately for IllustrisTNG (top panel), Eagle +(center panel), and Illustris (bottom panel), for a variety of snapshots spanning from cosmic noon (z ∼ 2) to the present epoch +(as shown on the legend of each panel). Note that the z=2 snapshot for Illustris is absent due to a lack of quenched galaxies at +that epoch. Uncertainties on the quenching importances are given from bootstrapped random sampling. It is clear that black +hole mass is overwhelmingly the most important parameter for predicting quiescence in central galaxies at all epochs (and for +every simulation) considered here. A pie plot is displayed on each panel, which averages the quenching importances across all +epochs, comparing two broad categories: BH-parameters (i.e. MBH and RBH); and non-BH-parameters (the remaining set). It +is clear that non-BH parameters engender only a very minor improvement in predicting quiescence over BH-parameters alone. + +10 +Bluck et al. +these important processes within galaxies have a negli- +gible impact on the quenching of centrals across cosmic +time. Again, this is a clear prediction of contemporary +AGN feedback driven quenching models, which can be +observationally tested (in principle at least). At low red- +shifts, the models clearly pass this test (see Piotrowska +et al. 2022), yet it remains to be seen whether this is so +at higher redshifts or not. +It is important to appreciate that even though Eagle, +Illustris, and IllustrisTNG utilize very different sub-grid +prescriptions for AGN feedback (including different pro- +cesses at a fundamental level, e.g. CGM jet bubbles vs. +SMBH-driven winds into the ISM; kinetic vs. thermal +energy injection), the fundamental qualitative quench- +ing predictions are identical between these simulations +at all epochs. +The reason for this is that black hole +mass traces the total energy released from black hole +accretion in a model independent manner (see Soltan +et al. 1982; Silk & Rees 1998; Bluck et al. 2020a; Ter- +razas et al. 2020; Zinger et al. 2020; Piotrowska et al. +2022). As such, these predictions are fundamental to +any quenching process with energy originating from the +central black hole. Note that this is true even though +there are many quantitative differences between these +simulations in terms of quenching (see, e.g., Donnari +et al. +2021; Piotrowska et al. +2022, as well as Ap- +pendix A1). This is a profound point for two reasons. +First, variation in the resolution, mechanism of AGN +- CGM energy/ momentum transfer, sub-grid recipe, +hydo-solver, or volume size of simulations are unable to +change the dependence of quiescence on black hole mass +without fundamentally abandoning AGN as the cause of +quenching. Second, due to the first point, this enables +an observational test of the entire paradigm of AGN +feedback driven quenching, not just one specific instan- +tiation of it. It is to this we turn in the next sub-section. +4.2. Observational Tests of AGN Feedback Driven +Quenching +Conceptually, it is straightforward to test the AGN feed- +back paradigm - simply measure the parameters in- +cluded in Fig. 2 for a representative sample of galaxies +at each epoch and run the Random Forest classification. +The problem is that many of these parameters, though +observable in principle, are difficult to measure in prac- +tice. This is especially the case for black hole mass, i.e. +the key observable in AGN feedback driven quenching +(see Fig. 1). At low redshifts, we were able to leverage +the power of statistical calibrations, e.g. the MBH − σ +relation, in order to make progress (see Piotrowska et al. +2022). Yet, at redshifts beyond z ∼ 0.2, even the proxy +variables needed for accurate statistical calibrations of +black hole mass (e.g. central velocity dispersion) are in- +frequently measured. This critical issue will be resolved +in the coming years through dedicated spectroscopic sur- +veys targeting intermediate-to-high redshifts, most no- +tably with VLT-MOONS and JWST. Consequently, a +robust test to the multi-epoch quenching predictions +from hydrodynamical simulations (discussed in the pre- +vious sub-section) can soon be made. +Although not ideal for the purpose of constraining +black hole mass, large photometric surveys exist with +HST rest-frame optical imaging at the epochs from +z = 0.5 − 2, the largest of which is HST-CANDELS +(Grogin et al. 2011; Koekemoer et al. 2022). As such, +it is interesting to attempt to find a useful proxy for +black hole mass in the simulations, which is relatively +easy to constrain in photometric data. +In lieu of a direct kinematic measurement of black hole +mass, or a measurement of central velocity dispersion, +one can still make progress by making several assump- +tions and leveraging the power of the virial theorem. +The gravitational potential (φg) for a system with dy- +namical mass, Mdyn, and gravitational radius, Rgrav., is +given by: +φg ≡ −GMdyn +Rgrav. +∼ Mdyn +Rgrav. +∼ M∗ +Rh +≡ φ∗ +(4) +where we make a series of assumptions intended to pro- +gressively simplify the measurements needed from left- +to-right in eq. 4 above, whilst preserving a strong re- +lation to the gravitational potential. Note that the use +of the tilda is intended to specify ‘scales with’ rather +than ‘of the order of’, since the latter is obviously not +true when changing units. In the first step, we drop the +constants (which are irrelevant for classification perfor- +mance). In the second step, we approximate the dynam- +ical mass with the stellar mass, and the gravitational ra- +dius with the half mass radius. Within ∼ 2 − 3 Rh, this +is a good approximation for most high mass systems, +although it is clearly imperfect (it ignores the contribu- +tion of gas and dark matter, as well as the structure of +the mass distribution). +For a virialised system, the gravitational potential is +related to the mean square velocity (vrms) as follows: +φg = v2 +rms ∼ σ2 ∼ M 1/2 +BH +(5) +where, for a pressure supported system, vrms ∼ σ (hence +the second step). Even for rotationally supported sys- +tems there is a strong relationship between σ and vrms +(e.g., Brownson et al. 2022), although the tightness of +the relation reduces. To arrive at the final step in eq. +5 above, we use the fact that black hole mass is empiri- + +AGN feedback driven quenching +11 +cally determined to scale with velocity dispersion to ap- +proximately the fourth power (e.g., Saglia et al. 2016). +Hence, we have found an approximate way to estimate +supermassive black hole mass in photometric data. Ex- +plicitly, +MBH ∼ φ2 +∗ +(6) +Or, in other words, there is at least some logic in suspect- +ing that φ∗ (which we can measure in pure photometric +data) may trace MBH. Yet, the above argument is little +more than a reasoned guess; the proof lies in the simu- +lations. It is important to appreciate that φ∗ is not the +only correlator to black hole mass that one can expect +from simple arguments of this type. For instance, bulge +mass (e.g., Bluck et al. 2014, 2022), central density (e.g., +Cheung et al. 2012; Fang et al. 2013) and light concen- +tration (e.g., Wuyts et al. 2011) are all also expected to +correlate strongly with central velocity dispersion and +hence black hole mass. Our purpose in utilizing φ∗ is +purely for convenience - it is easy to measure in both +the observational and simulated data sets used in this +work, without requiring significant further data process- +ing. +In Fig. 3 we present a series of Random Forest clas- +sification analyses to predict the existence of quenched +galaxies within the three simulations studied in this pa- +per, exactly as in Fig. +2. +However, here we restrict +the parameters used to train the classifier to: stellar +mass (M∗), half mass radius (Rh), and the stellar po- +tential (φ∗ = M∗/Rh). All of these parameters can be +estimated in a straightforward manner in extant pho- +tometric observations. It is clear from Fig. 3 that in +lieu of black hole mass, φ∗ is predicted to be the most +important parameter for constraining quenching, at all +epochs and for all simulations. This is as expected from +the simple argument sketched above. More physically, +in the simulations, this result emerges as a consequence +of supermassive black holes forming and growing in the +dense gravitational potentials at the center of galaxies +(e.g. Sijacki et al. 2007, 2015; Weinberger 2018). +In Fig. 4 we test the above prediction from hydrody- +namical simulations in observational data. At low red- +shifts we analyze the SDSS (blue bars), and at higher +redshifts we analyze CANDELS (shown in pink for inter- +mediate redshifts, and in red for high redshifts). Addi- +tionally, for the SDSS, we consider two subsamples: raw +data counts and a volume corrected sample. The latter +mimics a volume complete sample of galaxies, testing +whether Malmquist bias could impact the results. For +CANDELS, we consider the standard sSFR method for +identifying quenched systems (used throughout this pa- +per), as well as a UVJ color method (as in Bluck et al. +2022). +Since in CANDELS we do not have access to +spectroscopic star formation rates, we utilize this extra +test to ensure that the results of the Random Forest clas- +sification are stable to alternative methods for defining +quenching. +For all redshifts, galaxy surveys, and methodological +choices, in Fig. 4 we find that the stellar potential is by +far the most predictive parameter of quenching in ob- +servations, compared to stellar mass and galaxy size9. +Hence, there is remarkable agreement between the sim- +ulation predictions and the observational results. This +is a clear success of the models, and by extension the +paradigm of AGN feedback driven quenching. +The importance of the stellar potential for predicting +quiescence is also consistent with numerous other obser- +vational works investigating bulge mass, central mass +density, central velocity dispersion, and light concentra- +tion as phenomenological drivers of galaxy quenching +(see Bell et al. 2008, 2012; Wuyts et al. 2011; Wake et +al. 2012; Cheung et al. 2012; Fang et al. 2013; Omand +et al. 2014; Bluck et al. 2014, 2016, 2020a,b, 2022; Pi- +otrowska et al. 2022; Brownson et al. 2022; Varma et +al. 2022). The comparison with modern cosmological +simulations performed in this paper establishes a plau- +sible physical origin to these prior results. That is, AGN +feedback driven quenching is best predicted by the fossil +record of past accretion history (i.e. black hole mass), +which is itself highly correlated with the local poten- +tial well in which the black hole forms (approximately +given by φ∗). Naturally, the local potential also scales +with central mass density, stellar light concentration, +mass of the central bulge component, and central veloc- +ity dispersion, hence explaining these prior results as a +plausible consequence of historic AGN feedback. +Of course, whilst the above results are consistent with +the fundamental prediction from AGN feedback models, +it remains possible that other mechanisms could yield a +tight dependence of quenching on the stellar potential as +well. To fully rule out this possibility, dynamical mea- +surements of the masses of supermassive black holes are +needed on a large scale at intermediate-to-high redshifts. +9 Note that for the observational data we utilize half light radius +instead of half mass radius, choosing the reddest waveband avail- +able (with high quality data) in order to minimize the systematic +offset. Explicitly, we utilize r-band in the SDSS and H-band in +CANDELS, which both trace the rest-frame optical at the red- +shifts probed by each survey. This could be a source of difference +between observations and simulations, but in no way could this +ensure a greater similarity. + +12 +Bluck et al. +* +M * +Rh +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +TNG - Observable Parameters +TNG Centrals : z = 0.0 +TNG Centrals : z = 0.5 +TNG Centrals : z = 1.0 +TNG Centrals : z = 1.5 +TNG Centrals : z = 2.0 +* +M * +Rh +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +Eagle - Observable Prameters +Eagle Centrals : z = 0.0 +Eagle Centrals : z = 0.5 +Eagle Centrals : z = 1.0 +Eagle Centrals : z = 1.5 +Eagle Centrals : z = 2.0 +* +M * +Rh +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Quenching Importance +Illustris - Observable Parameters +Illustris Centrals : z = 0.0 +Illustris Centrals : z = 0.5 +Illustris Centrals : z = 1.0 +Illustris Centrals : z = 1.5 +Figure 3. Identical in structure to Fig. 2, except that fewer parameters are used to train the Random Forest classifier. Here +we focus on stellar mass, half mass radius, and the stellar potential. These parameters are relatively straightforward to measure +in extant wide-field galaxy surveys. Consequently, they offer an opportunity to test the predictions from the simulations. The +stellar potential is clearly identified as the most important predictor of quenching, in lieu of black hole mass. + +AGN feedback driven quenching +13 +* +M * +Rh +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Relative Quenching Importance + (Train,Test) = 0.91, 0.90 + (Train,Test) = 0.85, 0.83 +Quenching Classification for Observations +SDSS Centrals : z +0.1 (Raw Sample) +SDSS Centrals : z +0.1 (Volume Corrected) +CANDELS All Galaxies : z = 0.5 +1.0 (sSFR) +CANDELS All Galaxies : z = 0.5 +1.0 (UVJ) +CANDELS All Galaxies : z = 1.0 +2.0 (sSFR) +CANDELS All Galaxies : z = 1.0 +2.0 (UVJ) +Figure 4. Random Forest classification analysis to predict the existence of quiescent galaxies in observations. The relative +quenching importance of the stellar potential (φ∗), stellar mass (M∗), and rest-frame visible half light radius (Rh) are shown +on the y-axis (as labelled by the x-axis). Uncertainties are given by boot strapped random sampling. At low redshifts (blue +colored bars) we display results for the SDSS, using both raw galaxy counts and a 1/Vmax volume corrected sample. At higher +redshifts we utilize data from the CANDELS survey, shown separately for intermediate redshifts (displayed in pink) and high +redshifts (displayed in red). For the CANDELS data we consider both the sSFR approach (utilized throughout this paper) and +a UVJ color based approach to identify quiescence. At all epochs (and for all methodological sub-samples), the stellar potential +is clearly identified as the most predictive parameter for identifying quiescent systems. This is precisely as predicted by all three +cosmological simulations studied in this paper (compare to Fig. 3). + +14 +Bluck et al. +5. SUMMARY +In this paper we apply Random Forest classification to +three cosmological hydrodynamical simulations (Eagle, +Illustris, and IllustrisTNG) and two wide-field galaxy +surveys (SDSS & CANDELS), identifying the optimal +parameters for predicting whether central galaxies will +be star forming or quenched. +This work expands on +our initial low redshift analysis in Piotrowska et al. +(2022) by probing the dependence of quenching on +physical parameters out to cosmic noon. Throughout +this paper we focus on galaxies in the stellar mass range +9 < log(M∗/M⊙) < 12. +Our principal results are as follows: +1. Supermassive black hole mass is identified as the +most important parameter for predicting quies- +cence in all three simulations, and at every red- +shift snapshot analyzed (spanning ∼10 Gyr of cos- +mic history). This is the key testable prediction +from models of AGN feedback driven quenching. +2. Perhaps surprisingly, supermassive black hole ac- +cretion rate is not constraining of galaxy quench- +ing at any epoch, according to the simulations. +This further implies that AGN luminosity is not +predicted to correlate strongly with galaxy quies- +cence. Instead, it is the integrated effect of historic +AGN feedback which leaves a clear signature on +the galaxy population (traced by MBH not LAGN). +3. In lieu of a measurement of black hole mass, the +stellar potential (φ∗ ∼ M∗/Rh) is predicted to act +as a good proxy in simulations, outperforming all +other variables for predicting quenching. +4. We confirm the above prediction from simulations +in observations, utilizing the SDSS at low redshifts +and CANDELS at intermediate-to-high redshifts. +Hence, we find a remarkable consistency between simu- +lations which utilize AGN feedback to quench galaxies, +and multi-epoch observations. Further tests on the Ran- +dom Forest results, which confirm the above conclusions, +are provided in the Appendix. Ultimately, the key to +testing the paradigm of AGN quenching is to study the +fossil record of past accretion (i.e., black hole mass) or +its best available proxy, not a proxy of accretion rate +(such as AGN luminosity). More precise observational +tests of the key AGN quenching prediction from sim- +ulations will become feasible with VLT-MOONS and +JWST observations in the coming years. +ACKNOWLEDGEMENTS +AB acknowledges a faculty start-up grant at the Florida +International University. +JMP acknowledges funding +from the MERAC Foundation. RM acknowledges ERC +Advanced Grant 695671: ‘QUENCH’, support from the +Science and Technology Facilities Council (STFC), and +support from a Royal Society Research Professorship. +We thank the anonymous referee for a highly positive +and constructive report, which has helped to signifi- +cantly improve this publication. We are grateful to the +Illustris, Eagle, and IllustrisTNG teams for making their +simulations public. We are especially grateful to Paul +Torrey, Joop Schaye, Dylan Nelson, Annalisa Pillepich, +Lars Hernquist, and Rob Crain for many stimulating +and enlightening conversations on the simulations used +in this work. We thank the SDSS and CANDELS teams +for making their observational surveys public. We are +especially grateful to Luc Simard, Sara Ellison, Trevor +Mendel, Marc Huertas-Company, and Paula Dimauro +for much advise and support with these data prod- +ucts. We also thank Yingjie Peng, Emma Curtis-Lake, +Gareth Jones, Stephen Eales, Mirko Curti, Sara Ellison +and Christopher J. Conselice for many fruitful and en- +gaging conversations on this work. +DATA AVAILABILITY +All of the data used in this work is publicly available, +see the following links for access: +1. EAGLE: virgodb.dur.ac.uk/ +2. Illustris: www.illustris-project.org/ +3. 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As such, this appendix supports the main results of this paper. +A.1. Simulations: Area Statistics at z = 1 +The results from Random Forest classification in the simulations are extremely clear: black hole mass is predicted to +be the most important parameter regulating central galaxy quenching. In this part of the appendix, we show that +this conclusion can be arrived at through a different analysis, which has some advantages over the Random Forest +approach, and some disadvantages. Consequently, this provides a useful test on one of the main conclusions from this +paper. +We adopt our area statistics method (outlined in detail in Bluck et al. 2016, 2020a) to assess which parameters +engender the most significant impact on the quenched fraction, at fixed values of the other parameters. As in the main +body of this paper, we select central galaxies consistently at 9 < log(M∗/M⊙) < 12 and MHalo > 1011M⊙. Most of +the variables analyzed in the area statistics plots have similar (though not identical) ranges between the simulations. +The one exception is black hole accretion rate, where the lower limit varies significantly between the simulations. To +combat this, we present all panels with the same range (to aid in comparison) but compute the area statistics only +from the minimum to maximum value of each parameter in the source catalogs at this epoch, to avoid biasing this +statistic (as in Bluck et al. 2016). +One major advantage of the area statistics approach is that it is highly visual and intuitive. On the other hand, +this method only allows one variable to be controlled for at a time. Moreover, the area statistics approach is far less +efficient than the Random Forest classification. Due to the latter issue, we restrict our analysis here to just one redshift +snapshot (z = 1) and limit the parameters under consideration to: black hole mass, black hole accretion rate, stellar +mass, and halo mass. +In Fig. 5 we show the results from the area statistics approach for TNG galaxies at z = 1. The top set of panels shows +the results for a balanced sample (50% quenched; 50% star forming), as utilized in the Random Forest classification. +Additionally, we also present the results for the full volume complete sample, shown as the bottom set of panels. It +is clear that the fQ − MBH relation is by far the tightest of all of the relations considered in Fig. 5. This is true +for both a balanced and complete parent sample. Consequently, black hole mass engenders a much larger impact on +the fraction of quenched galaxies in the TNG simulation (at fixed other parameters) than any other parameter (at +fixed black hole mass). Therefore, black hole mass is clearly established as the most fundamental parameter governing +central galaxy quenching. This is in precise accord with the results from the top panel of Fig. 2. +In Figs. 6 & 7 we repeat the analysis for Eagle and Illustris (respectively) also selected at z = 1. Again, it is clear +that the fQ − MBH relation is the tightest and steepest of all of the relations. This confirms the results from the +center and bottom panels of Fig. 3. Additionally, we have explored the area statistics approach for all other redshift +bins, and the full list of parameters. Black hole mass is clearly established as the most fundamental parameter in all +of these extra tests as well. +As a result of the above analysis, it is evidently possible to arrive at the main conclusion of this paper for simulations +through area statistics. However, this is highly inefficient, as it would require ∼20 pages of plots, rather than a single +figure. More importantly, the Random Forest solves the full classification problem, controlling for all nuisance variables, +and accounting for both the steepness and tightness of the quenched fraction relationships simultaneously (see Bluck +et al. 2022). Hence, Random Forest classification is a much superior method to area statistics both conceptually and +practically. Nevertheless, the latter does offer further insight into why the Random Forest yields the results it gives, +as well as providing a simple visual check on the main conclusions. +Another advantage of the area statistics approach is that it allows us to probe some of the details of the process of +quenching in the simulations. Although not the primary focus of this work, it is clear from Figs. 5 - 7 that central +galaxies quench much more rapidly as a function of black hole mass (almost as a step function) in IllustrisTNG, but +quench much more gradually as a function of black hole mass in both Eagle and Illustris. Additionally, the quenching + +18 +Bluck et al. +threshold in black hole mass is over an order of magnitude higher in Illustris and IllustrisTNG, compared to Eagle. In +this work we have not attempted to estimate black hole masses for observed galaxies, so we are not in a position to +judge which is right. This notwithstanding, in Piotrowska et al. (2022) we do make a careful comparison of quenching +fractions between these three simulations and low-z observations. +No single simulation is an ideal march for the +observational data, but IllustrisTNG is the closest. This indicates that, at least at present, a step function quenching +threshold in black hole mass cannot be ruled out by the observational data. +A.2. Example Decision Trees +In Fig. 8 we present an example of one randomly selected decision tree from the Random Forest quenching classification +of the Eagle simulation at z = 1. This illustrates the structure of decision trees. Black hole mass is selected as the +parameter for the first split in the data. The reason for this is that a cut in black hole mass engenders the largest +reduction in Gini impurity, over any other parameter. Consequently, black hole mass acquires the highest weight, and +largest difference in Gini coefficient, leading to the highest importance in solving the classification problem (see eq. +3). Subsequent decision thresholds impact a smaller fraction of the data and have in general a lower reduction in Gini +Impurity. Thus, it is crucial which parameter is chosen first by the classifier. The stability of this choice is assured by +exploring 250 random generations of the parent sample, which is the fundamental advantage of a Random Forest over +a single decision tree (see, e.g., Pedregosa et al. 2011; Bluck et al. 2022a). +In Fig. 9 we show an alternative decision tree from a classification analysis of Eagle at z = 1, utilizing a volume +complete (as opposed to balanced) sample. Once again, black hole mass is chosen as the first parameter in the decision +tree, engendering a high importance of this parameter. However, in this example, the sample is overwhelmingly star +forming (due to the steepness of the mass function) and hence the classifier has to probe deeper to effectively separate +the data. This is not the preferred approach for using machine learning in classification for various technical reasons +(see, e.g., Teimoorinia et al. 2016). Nonetheless, this test clearly establishes the stability of the main results to the +manner in which the data is presented to the classifier. +A.3. Predicted MBH − φ∗ Dependence +In Fig. 10 we show the predicted MBH −φ∗ relations from TNG (left panels), Eagle (center panels), and Illustris (right +panels). It is clear that all three simulations predict that there should be a tight dependence of black hole mass on the +stellar potential. This provides further insight into why φ∗ is chosen by the Random Forest classifier in lieu of black +hole mass (see Fig. 3). We can leverage this theoretical result to test the quenching predictions of these simulations +in extant photometric observations. +In the SDSS, where we have the most complete suite of measurements on observational galaxy parameters, we have +additionally made a number of extra tests. We find that φ∗ is second only to central velocity dispersion for predicting +quiescence, comfortably beating bulge mass, total stellar mass, halo mass, B/T morphology, environmental parameters, +and disk mass. However, the real value of φ∗ for our analysis is that it can be straightforwardly estimated in extant +catalogs without further data processing, in both simulations and observations up to z = 2. Given that the simulations +unanimously predict that this parameter will act as a reasonable proxy for MBH, this is sufficient for our present +analysis. +A.4. Observations: Area Statistics in the SDSS & CANDELS +In Fig. 11 we apply our area statistics approach to the SDSS sample of central galaxies at low redshifts. It is clear +that the fQ − φ∗ relation is by far the tightest of the set. This is exactly as predicted by all three simulations (see +Fig. 3 & 10). In Fig. 12 we apply the area statistics technique to the full CANDELS galaxy sample at z = 0.5 - 2. +Again, the fQ − φ∗ relation is found to be the tightest (and steepest) of the observed relations, exactly as predicted by +the simulations. This confirms the results from the Random Forest classification (shown in Fig. 4 in the main body +of the paper). We have additionally tested the use of volume weighting, restricting to a balanced sample, including +more redshifts cuts, and utilizing alternative quenching definitions (and thresholds) on the observational results. The +results from this paper are extremely stable to these alternative analysis choices. + +AGN feedback driven quenching +19 +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.04 +A25 = 0.06 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.04 +A25 = 0.08 +TNG Centrals: z = 1 +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.10 +A25 = 0.17 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.17 +A25 = 0.27 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.19 +A25 = 0.31 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.35 +A25 = 0.55 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.04 +A25 = 0.06 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.02 +A25 = 0.04 +Volume Complete Sample +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.08 +A25 = 0.14 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.16 +A25 = 0.26 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.19 +A25 = 0.30 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.17 +A25 = 0.29 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +Figure 5. Top Panels: The quenched fraction relationship with black hole mass (top row) and stellar mass, halo mass, and +black hole accretion rate (from left-to-right along the bottom row) for the IllustrisTNG simulation at z = 1. Each quenched +fraction relationship is split into percentiles based on a third variable (as indicated on the legends). As in the Random Forest +analysis, we select a balanced sample (50% star forming; 50% quenched) for this test. The tightness of each of the quenched +fraction relationships is quantified by the area subtended by the upper and lower 50th (and 25th) percentiles in the third variable +(which is displayed on each panel). For example, in the top left panel, the quenched fraction - black hole mass relation is split +into ranges based on stellar mass, while the lower left panel inverts this. It is clear that the quenched fraction - black hole mass +relation is by far the tightest relationship, confirming the results of the Random Forest classification analysis at this epoch. +Bottom Panels: +Same as above, but for a volume complete sample (with a majority of star forming systems). Again, it is clear +that the quenched fraction relationship with black hole mass is by far the tightest of the set. + +20 +Bluck et al. +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.06 +A25 = 0.10 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.07 +A25 = 0.14 +Eagle Centrals: z = 1 +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.12 +A25 = 0.14 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.32 +A25 = 0.49 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.41 +A25 = 0.62 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.46 +A25 = 0.72 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.06 +A25 = 0.10 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.04 +A25 = 0.07 +Volume Complete Sample +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.06 +A25 = 0.10 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.11 +A25 = 0.16 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.07 +A25 = 0.13 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.07 +A25 = 0.12 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +Figure 6. The same in structure as Fig. 5, but here showing results for the Eagle simulation at z = 1. The top panels show +the results for a balanced sample, and the bottom panels show the results for the volume complete sample. In all cases, the +fQ − MBH relationship is the tightest relation (as quantified by the area statistics displayed on each panel). The only slight +exception to this is for MBH vs RBH in the volume complete sample, where the tightness of each relation is similar. Nonetheless, +the increase in quenched fraction is significantly larger as a function of black hole mass than accretion rate. Hence, black hole +mass is still a better predictor of quiescence than accretion rate, as seen clearly in the balanced sample (top panel). Moreover, +increasing accretion rate at a fixed black hole mass actually lowers the fraction of quenched galaxies. This is expected for a +co-fueling scenario but is the opposite of what is expected for ‘catching’ AGN quenching in action. + +AGN feedback driven quenching +21 +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.06 +A25 = 0.08 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.12 +A25 = 0.17 +Illustris Centrals: z = 1 +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.16 +A25 = 0.29 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.37 +A25 = 0.58 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.39 +A25 = 0.58 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.51 +A25 = 0.75 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +6 +7 +8 +9 +log(MBH/M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.05 +A25 = 0.07 +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.03 +A25 = 0.05 +Volume Complete Sample +lower 25% in MHalo +lower 50% in MHalo +ALL +upper 50% in MHalo +upper 25% in MHalo +6 +7 +8 +9 +log(MBH/M ) +A50 = 0.03 +A25 = 0.03 +lower 25% in +BH +lower 50% in +BH +ALL +upper 50% in +BH +upper 25% in +BH +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /M ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.09 +A25 = 0.15 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +11.0 +11.5 +12.0 +12.5 +13.0 +log(MHalo/M ) +A50 = 0.11 +A25 = 0.18 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +10 +8 +6 +4 +2 +0 +log( +BH [M yr +1]) +A50 = 0.10 +A25 = 0.15 +lower 25% in MBH +lower 50% in MBH +ALL +upper 50% in MBH +upper 25% in MBH +Figure 7. The same in structure as Fig. 5, but here showing results for the Illustris simulation at z = 1. The top panels show +the results for a balanced sample, and the bottom panels show the results for the volume complete sample. In all cases, the +fQ − MBH relationship is the tightest relation (as quantified by the area statistics displayed on each panel). + +22 +Bluck et al. +Figure 8. An example of one randomly selected decision tree (out of 250) for training in Eagle at z = 1. Here, as in the +main body of this paper, we restrict to a balanced sample of quenched and star forming systems. The parameter and threshold +chosen by the classifier at each of the decision forks is presented in each box, along with the Gini impurity and sample size (this +starts at Gini = 0.5 by design for a balanced sample). The color of boxes indicate whether the descendent sample is majority +quenched (red) or star forming (blue), with the depth of the hue indicating the purity of the sample. Colors shift from white to +deep blue (or red) down each branch of the decision tree, as the classifier solves the problem. Black hole mass is selected by the +classifier as the most effective first cut on the data, engendering the most dramatic reduction in Gini impurity for the largest +number of galaxies. The overall success of this parameter emerges from the stability of this choice to sample variation across +the entire Random Forest. + +MBH <= 7.127 +gini = 0.5 +samples = 472 +value = [236, 236] +class = SF +MBH <= 6.364 +Rh <= 0.743 +gini = 0.103 +gini = 0.192 +samples = 221 + samples = 251 +value = [209, 12] +value = [27, 224] +class = SF +class = Q +dM/dt <= -8.679 +dM/dt <= -3.532 +gini = 0.466 +gini = 0.453 +gini = 0.02 +gini = 0.147 +samples = 27 +samples = 26 +samples = 194 + samples = 225 +value = [17, 10] +value = [9, 17] +value = [192, 2] +value = [18, 207] +class = SF +class = Q +class = SF +class = Q +Rh <= 0.576 +MBH <= 7.582 +gini = 0.147 +gini = 0.0 +gini = 0.03 +gini = 0.28 +samples = 25 +samples = 169 +samples = 130 +samples = 95 +value = [23, 2] +value = [169, 0] +value = [2, 128] +value = [16, 79] +class = SF +class = SF +class = Q +class = Q +MBH <= 7.855 +gini = 0.0 +gini = 0.147 +gini = 0.488 +gini = 0.134 +samples = 105 + samples = 25 +samples = 26 +samples = 69 +value = [0, 105] +value = [2, 23] +value = [11, 15] +value = [5, 64] +class = Q +class = Q +class = Q +class = Q +gini = 0.269 +gini = 0.044 +samples = 25 + samples = 44 +value = [4, 21] +value = [1, 43] +class = Q +class = QAGN feedback driven quenching +23 +Figure 9. An alternative example of a randomly selected decision tree (out of 250) for training in Eagle at z = 1. Here we +analyze the raw sample of quenched and star forming systems (which is majority star forming, hence the increased blue shading +of this tree compared to Fig. 8). Consequently, the classification problem is more complex and the results require more care to +interpret. Nonetheless, black hole mass is still selected as the best parameter to make the first cut on the data, yielding a high +performance of black hole mass in this case as well. + +MBH <= 7.145 +gini = 0.056 + samples = 2467 +value = [3772, 113] +class = SF +dM/dt <= -10.208 +Rh <= 0.567 +gini = 0.001 +gini = 0.339 +samples = 2129 +samples = 338 +value = [3369, 2] +value = [403, 111] +class = SF +class = SF +MBH <= 5.487 +dM/dt <= -5.058 +MBH <= 7.627 +gini = 0.0 + gini = 0.046 + gini = 0.445 +gini = 0.141 + samples = 2075 +samples = 54 +samples = 178 +samples = 160 +value = [3286, 0] +value = [83, 2] +value = [185, 93] +value = [218, 18] +class = SF +class = SF +class = SF +class = SF +MBH <= 7.602 +Rh <= 0.705 + gini = 0.0 +gini = 0.091 + gini = 0.36 +gini = 0.0 +gini = 0.398 +gini = 0.204 +samples = 28 +samples = 26 +samples = 24 + samples = 50 + samples = 154 +samples = 110 +value = [43, 0] +value = [40, 2] +value = [8, 26] +value = [80, 0] + class = Q +value = [177, 67] +value =[138, 18] + class = SF +class = SF +class = SF +class = SF +class = SF +M* <= 10.313 +Mh <= 12.234 +dM/dt <= -3.464 + gini = 0.375 +gini = 0.248 + gini = 0.494 + gini = 0.105 + samples = 32 +samples = 90 + samples = 64 +samples = 78 +value = [36, 12] +value = [118, 20] +value = [59, 47] +value = [102, 6] +class = SF +class = SF +class = SF +class = SF +Mh <= 12.093 +MBH <= 8.004 +dM/dt <= -2.384 + gini = 0.456 +gini = 0.437 + gini = 0.198 +_gini = 0.024 + gini = 0.453 + gini = 0.054 + samples = 36 + samples = 21 +samples = 30 + samples = 54. + samples = 43 +samples = 48 +value = [35, 19] +value = [10, 21] +value = [32, 4] +value = [83, 1] +value = [49, 26] +value = [70, 2] +class = SF +class = Q +class = SF +class = SF +class = SF +class = SF + gini = 0.064 +gini = 0.0 +gini = 0.266 +gini = 0.497 + gini = 0.0 + gini = 0.137 +samples = 20 + samples = 34 +samples = 23 + samples = 20 +samples = 28 +samples = 20 +value = [29, 1] +value = [54, 0] +value = [32, 6] +value = [17, 20] +value = [45, 0] +value = [25, 2] +class = SF + class = SF + class = SF +class = Q +class = SF + class = SF24 +Bluck et al. +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.94 += 0.19 dex +TNG - Central Galaxies @ z = 0 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.81 += 0.32 dex +Eagle - Central Galaxies @ z = 0 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.85 += 0.29 dex +Illustris - Central Galaxies @ z = 0 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.94 += 0.20 dex +TNG - Central Galaxies @ z = 1 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.78 += 0.32 dex +Eagle - Central Galaxies @ z = 1 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.89 += 0.24 dex +Illustris - Central Galaxies @ z = 1 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.94 += 0.18 dex +TNG - Central Galaxies @ z = 2 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.72 += 0.32 dex +Eagle - Central Galaxies @ z = 2 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +log( +* [M kpc +1]) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +log(MBH [M ]) +S = 0.93 += 0.18 dex +Illustris - Central Galaxies @ z = 2 +Figure 10. Predicted relationships between black hole mass and the stellar potential for TNG (left panels), Eagle (middle +panels) and Illustris (right panels). Results are shown for z = 0 (top row), z = 1 (middle row), and z = 2 (bottom row). The +median relationship is indicated by a solid magenta line, and the spread of the data is indicated by linearly spaced contours. +The Spearman correlation statistic (ρS) and the dispersion about the median relation (σ) are displayed on each panel. There +is predicted to be a strong and reasonably tight relationship between black hole mass and the stellar potential by all three +simulations at all epochs. Hence, φ∗ is expected to act as a reasonable proxy for MBH in observational data as well. + +AGN feedback driven quenching +25 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +log( +* [M kpc +1]) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.03 +A25 = 0.04 +SDSS Centrals (z = 0.02 - 0.2) +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +log( +* [M kpc +1]) +A50 = 0.03 +A25 = 0.05 +lower 25% in Rh +lower 50% in Rh +ALL +upper 50% in Rh +upper 25% in Rh +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M* [M ]) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.23 +A25 = 0.35 +lower 25% in +* +lower 50% in +* +ALL +upper 50% in +* +upper 25% in +* +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +log(Rh [kpc]) +A50 = 0.50 +A25 = 0.70 +lower 25% in +* +lower 50% in +* +ALL +upper 50% in +* +upper 25% in +* +Figure 11. The quenched fraction relationship with φ∗ (top panels), M∗ (bottom-left panel), and Rh (bottom-right panel) in +the SDSS. As in Fig. 5, each quenched fraction relationship is displayed for percentile ranges of a third parameter, in order to +assess the impact on quenching of varying that parameter at fixed values of each of the others. The tightness of each relationship +is quantified by the area statistics displayed on each panel. It is clear that φ∗ engenders the tightest and steepest of the quenched +fraction relationships, exactly as predicted in models utilizing AGN feedback to quench galaxies. +. + +26 +Bluck et al. +8 +9 +10 +11 +log( +* [M kpc +1]) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.07 +A25 = 0.09 +CANDELS Galaxies (z = 0.5 - 2.0) +lower 25% in M * +lower 50% in M * +ALL +upper 50% in M * +upper 25% in M * +8 +9 +10 +11 +log( +* [M kpc +1]) +A50 = 0.11 +A25 = 0.16 +lower 25% in Rh +lower 50% in Rh +ALL +upper 50% in Rh +upper 25% in Rh +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M* [M ]) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fQ +A50 = 0.22 +A25 = 0.35 +lower 25% in +* +lower 50% in +* +ALL +upper 50% in +* +upper 25% in +* +1.0 +0.5 +0.0 +0.5 +1.0 +log(Rh [kpc]) +A50 = 0.22 +A25 = 0.34 +lower 25% in +* +lower 50% in +* +ALL +upper 50% in +* +upper 25% in +* +Figure 12. Identical in structure to Fig. 11, but here showing results from CANDELS. As at low redshifts, φ∗ exhibits the +tightest quenched fraction relationship, as quantified by the area statistics displayed on each panel. Note also that φ∗ exhibits +the steepest positive relationship with quiescence. + diff --git a/jdE2T4oBgHgl3EQfIQZD/content/tmp_files/load_file.txt b/jdE2T4oBgHgl3EQfIQZD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e1a7d6cb780bd54273a1738fac2a8cbd1de280d --- /dev/null +++ b/jdE2T4oBgHgl3EQfIQZD/content/tmp_files/load_file.txt @@ -0,0 +1,2012 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf,len=2011 +page_content='Draft version January 11, 2023 Typeset using LATEX twocolumn style in AASTeX63 The fundamental signature of star formation quenching from AGN feedback: A critical dependence of quiescence on supermassive black hole mass not accretion rate Asa F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck,1 Joanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3 and Roberto Maiolino2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Florida International University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='11200 SW 8th Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Miami,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 33199,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' USA 2Kavli Institute for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' UK 3Cavendish Laboratory - Astrophysics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 19 JJ Thomson Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' UK 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Gower Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' London WC1E 6BT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' UK (Accepted to ApJ on 16 December 2022) ABSTRACT We identify the intrinsic dependence of star formation quenching on a variety of galactic and envi- ronmental parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' utilizing a machine learning approach with Random Forest classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We have previously demonstrated the power of this technique to isolate causality, not mere correlation, in complex astronomical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' First, we analyze three cosmological hydrodynamical simulations (Eagle, Illustris, and IllustrisTNG), selecting snapshots spanning the bulk of cosmic history from comic noon (z ∼ 2) to the present epoch, with stellar masses in the range 9 < log(M∗/M⊙) < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the simu- lations, black hole mass is unanimously found to be the most predictive parameter of central galaxy quenching at all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Perhaps surprisingly, black hole accretion rate (and hence the bolometric luminosity of active galactic nuclei, AGN) is found to be of little predictive power over quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This theoretical result is important for observational studies of galaxy quenching as it cautions against using the current AGN state of a galaxy as a useful proxy for the cumulative impact of AGN feedback on a galactic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The latter is traced by black hole mass not AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, we explore a sub-set of ‘observable’ parameters, which can be readily measured in extant wide-field galaxy surveys targeting z = 0 − 2, at 9 < log(M∗/M⊙) < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' All three simulations predict that in lieu of black hole mass, the stellar gravitational potential will outperform the other parameters in predicting quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We confirm this theoretical prediction observationally in the SDSS (at low redshifts) and in CANDELS (at intermediate and high redshifts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Keywords: Galaxies: formation, evolution, star formation, quenching, feedback 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' INTRODUCTION Galaxies are observed to display profound bimodality in key diagnostic diagrams, most notably in color - mag- nitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Strateva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Perhaps more phys- ically, the star forming main sequence relation between star formation rate (SFR) and stellar mass (M∗) exhibits a tight star forming ridge line, with quiescent galax- ies significantly offset to lower star formation rates at a fixed stellar mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Con- sequently, it is often stated that galaxies exhibit two fundamental types: i) actively star forming, and ii) qui- escent (or ‘quenched’) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Quenched systems tend Corresponding author: Asa F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck abluck@fiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='edu to have redder optical colors, older stellar populations, more elliptical morphologies, more pressure supported kinematics, higher masses, and reside in denser cosmic environments compared to their actively star forming counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2010, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Wuyts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Brown- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Much of the field of galaxy evolution is explicitly focused on explaining the origin of quiescent galaxies, and hence accounting for the observations of bimoodality in galactic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Theoretically, the existence of quiescent galaxies is translated to a problem of cosmological star formation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More precisely, one of the main goals of mod- ern simulations of galaxy formation is to reduce the effi- ciency with which stars form within dark matter haloes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', ϵSF ≡ M∗/MHalo ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1 Ωb/ΩM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The reason for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03677v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='GA] 9 Jan 2023 2 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' this is that simple models of galaxy formation (utilising only gravitation and cooling) predict that the vast ma- jority of baryons should reside in stars by the present epoch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2006, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, observations place the fraction of baryons in stars an order of magni- tude lower (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Fukugita & Peebles 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Shull 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the past decade or so, feedback from active galac- tic nuclei (AGN) has become the favored mechanism in modern hydrodynamical simulations to quench star formation, and hence reproduce the observed galaxy bi- modality (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' There is abundant evidence that AGN produce more than enough energy to quench star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Silk & Rees 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Maiolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Yet, direct obser- vational evidence for AGN feedback causing quenching within galaxies remains sparse, and hotly debated (al- though see Hlavacek-Larrondo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012, 2017, 2018 for perhaps the strongest direct evidence to date, pri- marily within galaxy clusters and very massive groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Thus, it remains unclear whether the crucial ingredient in cosmological models to quench galaxies is viable for the vast majority of systems, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Classification is the branch of data science that fo- cuses on understanding the fundamental differences be- tween types of objects, with the ultimate purpose of ac- curate segregation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, classification is at the heart of the scientific study of galaxy popula- tions: If we can learn to accurately classify star form- ing and quenched galaxies on the basis of their physical properties, we can establish the underlying physics of galaxy quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For example, we have demonstrated the power of machine learning classification to ‘reverse engineer’ cosmological simulations, revealing the input physics and its observational consequences (see Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' But the real power of machine learning in this application is apparent when one compares the results of classification between simu- lations and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' If there is strong agree- ment, the models can be used as an explanatory tool for the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Alternatively, if there is major dis- agreement, the observations can be used to improve the next generation of models by ruling out certain prescrip- tions or processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The goal of this paper is to leverage the power of clas- sification to determine what is intrinsically important for central galaxy quenching in both simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To this end, we employ a Random Forest classifier, which enables the identification of causality through carefully controlling for nuisance variables (see Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022 for a detailed description of this tech- nique including multiple tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Utilizing Random Forest classification, in Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) we discovered that the most effective pa- rameter for separating star forming and quenched cen- tral galaxies in the local Universe is black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Interestingly, this is a unanimous prediction from three contemporary cosmological simulations (Eagle, Illustris, and IllustrisTNG), which use very different AGN feed- back prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, this prediction is not strongly model dependent and hence can be seen as a ubiquitous consequence of AGN feedback (in almost any mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also considered stellar mass, dark matter halo mass, and black hole accretion rate as potential drivers of quenching in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Yet, all of these parameters are of negligible importance to quenching once black hole mass is made available to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The results from Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) are highly important for observational studies of AGN feedback be- cause they clearly imply that ‘catching AGN quenching in the act’ is not possible in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Instead, one must look for the fossil record of historic AGN feed- back, encapsulated in the mass of the central black hole, not the current accretion rate or AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also established that once black hole mass is controlled for, neither stellar nor halo mass should be constraining of central galaxy quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We confirmed this latter prediction observationally through comparison with the SDSS, which effectively rules out virial shocks or super- nova feedback as significant mechanisms for quenching central galaxies in the local Universe (see also Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016, 2020a for further evidence on this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In this paper, we expand on the work of Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) by identifying the key quenching predic- tions from Eagle, Illustris, and IllustrisTNG at multiple epochs from z = 0 to cosmic noon (where the number density of quiescent galaxies becomes very low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is essential in order to establish whether there is red- shift dependence on the predicted signature of AGN feedback quenching or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For instance, it could be that at high redshifts a strong dependence of quies- cence on the current AGN state of a galaxy becomes apparent, which would have profound implications for how to search for observational evidence of AGN feed- back (or lack thereof) in upcoming observational sur- veys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', with JWST and VLT-MOONS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Addition- ally, we perform a preliminary test on the high-z predic- tions from cosmological simulations, utilizing photomet- ric data from HST-CANDELS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 - 2, and com- pare this to a novel test of the simulations’ predictions at low-z utilizing spectroscopic data from the SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' AGN feedback driven quenching 3 The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Section 2 we give an overview of the simulations and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Section 3 we present our method to identify qui- escent systems and give a brief overview of our Random Forest classification technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Section 4 we present our results and discuss their importance for the field of galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We summarize our contributions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, in the Appendix, we present numerous detailed tests on the Random Forest results, which confirm our main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Throughout the paper we assume a spatially flat ΛCDM cosmology, and set h ≡ H0/(100 km/s/Mpc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='7 consistently for all physical representations of simulated and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' DATA SOURCES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Simulations In this paper, we consider three cosmological hydrody- namical simulations which incorporate AGN feedback to quench central (and more generally, massive) galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Our goal is to identify the testable consequences of AGN feedback driven quenching in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Explicitly, we work with publicly available multi-epoch snapshot data from: Eagle1 (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Illustris2 (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nel- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and IllustrisTNG3 (Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Full details on the simulations are provided in the above references, including information on data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A de- tailed description of the similarities and differences of these three simulations is provided in Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here we give a review of the most important de- tails for this work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' the black hole growth and AGN feedback mechanisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In all simulations we select central galaxies as the most massive systems in each (group) dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Iso- lated galaxies are treated as the centrals of their indi- vidual dark matter haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We select galaxies to have stellar masses in the range 9 < log(M∗/M⊙) < 12, resid- ing in dark matter (group) haloes with Mhalo > 1011M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' These cuts mitigate issues with mass and volume reso- lution in the simulations, whilst still enabling a large sampling of both star forming and quenched systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Even though the majority of quiescent galaxies in all of the simulations and observations considered in this 1 Eagle Data Access: http://icc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='dur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='uk/Eagle/ 2 Illustris Data Access: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='illustris-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='org 3 IllustrisTNG Data Access: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='tng-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='org/ work have log(M∗/M⊙) > 10, it is vital to select a large sample of both star forming and quiescent classes for random forest classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' All specific parameters are collated as in Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022), but here ex- tracted for multiple snapshots (rather than just at z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The identification of quiescent systems is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' From Illustris and IllustrisTNG (which have the same naming conventions) we take the following pa- rameters from the public snapshot data at z = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5, 2): From the SubFind catalog - SubhaloSFR, SubhaloMassStar, SubhaloBHMass, Sub- haloBHmdot, SubhaloHalfmassRad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and from the FoF catalog - Group M Crit200, Group R Crit200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also extract the co-moving coordinates for each system (galaxy and halo), as well as the FoF Halo ID and Sub- Find sub-halo ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' From Eagle we take the following pa- rameters from the public snapshot data at z = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='49, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='01): From the SubFind catalog - StarFormation- Rate, MassType Star, BlackHoleMass, BlackHoleAccre- tionRate, HalfMassRad Star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and from the FoF catalog Group M Crit200, Group R Crit200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As with Illustris and IllustrisTNG, we also extract halo and sub-halo co- ordinates and IDs for each system in both the FoF and SubFind catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A publicly available docker is pro- vided with Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) showing how to extract these parameters from each simulation for the z = 0 snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Eagle For Eagle, we utilize the EAGLE-RefL0100N1504 run (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This run has a box size of ∼100 cMpc3 and implements the most detailed feed- back mechanisms of the Eagle simulation suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Eagle is performed utilizing a smoothed particle hydrodynam- ics (SPH) code, explicitly GADGET-3 (Springel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Cosmological parameters are taken from Planck Collab- oration I (2014), assuming a spatially flat ΛCDM cos- mology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black holes are seeded at MBH = 105M⊙h−1 in all haloes once they reach MHalo = 1010M⊙h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black hole accretion is regulated by Bondi-Hoyle accretion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' ˙MBH ∝ M 2 BH (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Hoyle & Lyttleton 1936;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bondi & Hoyle 1944), and is Eddington limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A single feed- back mode is applied, which corresponds approximately to a quasar wind, triggered primarily by cold-mode ac- cretion (see Booth & Schaye 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Energy injection into the surrounding gas particles, in a given time step ∆t, is given explicitly by ∆EBH = ϵfϵr ˙MBHc2∆t, where ϵr is the radiative efficiency (set equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1) and ϵf is the fraction of energy which couples to the inter-stellar medium (ISM) producing energetic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Energy is released thermally once ∆EBH is sufficient to induce a 4 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' temperature change of ∆T = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5K for at least one neighboring gas particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, heating of gas parti- cles near to the black hole is applied stochastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The large thermal injection, in essentially random directions, is key to overcoming the numerical over-cooling problem in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The Eagle AGN feedback mechanism is effective at quenching massive galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' however, it is less effective at keeping massive galaxies quenched than the other simulations considered here (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Illustris For Illustris, we utilize the full ILLUSTRIS-1 run with box size ∼100 cMpc3 (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Il- lustris is run utilizing the moving-mesh code AREPO (Springel 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Cosmological parameters are set as in WMAP7 (Hinshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2013), assuming a spa- tially flat ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black holes are seeded at MBH = 105M⊙h−1 in all haloes once they reach MHalo = 5×1010M⊙h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As in Eagle, black hole growth is regulated via Bondi-Hoyle accretion, limited by the Eddington rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Illustris operates two distinct feedback modes, although only one is effective at impacting star formation within galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The first is a ‘quasar’ mode which operates using the same general principle as the single mode in Eagle (outlined above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, the energy injection is continuous, rather than bursty (see Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2007, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This mode is uncorrelated with quenching in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The second is a ‘ra- dio’ mode, which aims to simulate the effects of rela- tivistic jets on the circum-galactic medium (CGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At low accretion rates (χEdd = ˙MBH/ ˙MEdd < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='05), once a black hole increases its mass by 15% of its value, a bubble is seeded in the host galaxy’s CGM, with energy Ebubble = ϵmϵr∆MBHc2, where ϵm represents the cou- pling efficiency of the mechanical feedback to the hot gas halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Heating is induced in the CGM through PdV work as the bubble expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This mode is partially ef- fective at shutting down gas cooling from the CGM, and hence reducing star formation in the galaxy via starva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, as is now widely known, the jet bubbles also have the deleterious effect of completely vacating the CGM of gas in stark contrast to observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' IllustrisTNG For IllustrisTNG, we utilize the TNG-100-1 simulation (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2018), which has an identical box size to our selected run in Illus- tris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' IllustrisTNG adopts cosmological parameters from Planck Collaboration (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This simulation offers the best compromise between resolution and volume for our present study (see Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' TNG was run with an updated version of AREPO, extended to add magnetic fields to the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black holes are seeded at a higher mass of MBH = 8 × 105M⊙h−1 in all haloes once they reach MHalo = 5 × 1010M⊙h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As in Eagle and Illustris, black hole growth is modeled sub- grid via Eddington limited Bondi-Hoyle accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The ‘quasar’ mode feedback is left identical to Illustris, and it still has very little impact on star formation or quench- ing (see Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Alternatively, the ‘ra- dio’ mode feedback of Illustris (which was over-zealous in its removal of CGMs, though effective at quenching galaxies) is replaced with a new ‘kinetic’ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' When a black hole is accreting at a ‘low’ level (defined relative to the black hole mass, see Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2017), energy injection is applied kinetically to a group of neighbor- ing gas cells in a stochastic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This occurs at a threshold black hole mass of MBH ∼ 108M⊙, which is set partly by the sub-grid AGN feedback prescription and partly by relation to other evolving parameters in the simulation (see Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020 for a full discus- sion on this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The change in kinetic energy of a gas cell is given by ˙Ekinetic = ϵk ˙MBHc2, where ϵk is the ef- ficiency of energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The efficiency itself is set as a function of the gas density around the black hole (see Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ultimately, a momentum kick is applied in a randomly chosen direction, such that (integrated over time) isotropy is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Unlike the radio mode in Illustris, the kinetic mode in TNG im- pacts the ISM as well as the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In slightly more detail, the TNG kinetic mode drives winds in the ISM as well as jet-like features in the CGM, which simulta- neously adds turbulence to the ISM and increases the entropy (and hence cooling time) of the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This oc- curs without removing significant quantities of gas from either, resolving the severe issues in Illustris (see Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020: Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Simulations Summary Eagle, Illustris, and IllustrisTNG represent three con- temporary galaxy evolution models which all quench star formation in massive galaxies via AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is important to appreciate that, although very differ- ent in the details, all of the above AGN feedback models extract energy from around the black hole (ultimately some fraction of the accreted rest energy, MBHc2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, the total feedback energy is directly proportional to black hole mass in each case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Efeedback ∝ MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This suggests a way to test the entire paradigm of AGN feedback quenching in an essentially model independent manner (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1, as well as Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nonetheless, there are many important differences between these three galaxy forma- AGN feedback driven quenching 5 tion simulations in terms of quenching at a quantitative level (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Observations We compare the predictions for AGN feedback driven quenching from simulations to the results from two ob- servational galaxy surveys: the Sloan Digital Sky Sur- vey - SDSS4 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2009), and the Cosmic Near-Infrared Deep Extragalactic Survey - CANDELS5 (Grogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For CAN- DELS, we additionally utilize value added catalogs from Dimauro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2018)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A thorough description of the parameter estimation of galaxies from these surveys is provided in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here we give only a brief overview of the most important measurements used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' SDSS We take star formation rates (SFRs) for SDSS galax- ies from Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2004), which are com- puted from emission lines where possible, or else from an empirical relationship between D4000 - sSFR for non- emission line galaxies (or systems with a significant AGN contribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Fibre SFRs are converted to global SFRs utilizing the colors of galaxies outside of the aperture (see Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2004 for a full description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We take stellar masses from the Mendel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2014) SDSS mass catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We take galaxy size estimates from the Simard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2011) morphological catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, we utilize volume weights from Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2014) to simulate the statistical appearance of a volume complete sample, as appropriate for simulation comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Finally, we utilize the group catalogs of Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2007, 2009) to separate central and satel- lite galaxies in the SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Centrals are defined as the most massive galaxy in the dark matter halo, with iso- lated galaxies also counting as centrals for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Satellites are defined to be any other galaxy contained within the group (or cluster) halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Extensive details on all of these measurements, as well as numerous checks on their reliability, are provided in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2014, 2019, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For our analysis, we select galaxies to be centrals, have M∗ > 109M⊙, and reside within haloes with masses Mhalo > 1011M⊙ (which is identical to our simulations selection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 SDSS Data Access: https://classic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='org/dr7/access/ 5 CANDELS Data Access: http://arcoiris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='ucolick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='org/candels/ 6 CANDELS VAC: https://mhuertascompany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='weebly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='com/data- releases-and-codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='html 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' CANDELS We utilize photometric star formation rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' stellar masses of galaxies, bulges and disks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' galaxy sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and rest-frame/ dust corrected UVJ colors from SED fitting of CANDELS galaxies provided in the value added cat- alogs of Dimauro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The SED fitting is per- formed utilizing the FAST code (Kriek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2009), with stellar population synthesis models taken from Bruzual & Charlot (2003), assuming a Chabrier IMF and Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2000) extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Since CAN- DELS is a photometric only survey, unlike the SDSS, it is not possible to construct accurate central - satellite segregation, or group determination, due to the lack of precision in photometric redshift estimates compared to spectroscopic redshift measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, we an- alyze the full CANDELS galaxy sample, as opposed to focusing only on centrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is unlikely to be a sig- nificant problem since the vast majority of galaxies of these masses are predicted to be centrals at all epochs (∼80%, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, we select galaxies to have stellar masses M∗ > 109M⊙ (in line with the simulations and the SDSS), but do not apply a halo mass cut (due to a lack of accurate halo estimation in this survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We note that in the SDSS the impact of additionally applying a halo mass cut is very small (< 5% of the sample is affected and all results are invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Full details on the CANDELS measure- ments are given in the above references, and an extensive discussion on the reliability of this data is provided in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' METHODS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Identifying Quenched Systems Throughout Cosmic Time In this paper we utilize a simple method to segregate actively star forming and quenched galaxies at multiple epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We apply the exact same method in simula- tions and observations to mitigate the potential for bias through the use of heterogeneous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, since the level of star formation in galaxies of a fixed stellar mass varies as a strong function of redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Madau & Dickinson 2014), it is necessary to allow the method to reflect this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Explicitly, we define quenched galaxies to be any sys- tem with a specific star formation rate: sSFR(z′) < sSFRpeak(z = z′) − 1 dex (1) where, sSFRpeak(z = z′) indicates the peak (mode) of the sSFR distribution at a redshift equal to the galaxy in question (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' z = z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, quenched galaxies are de- fined to be forming stars at a rate less than one order of 6 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(sSFR [yr 1]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='25 Normalised Counts Eagle : sSFR Distributions Eagle @ z=0 Eagle @ z=1 Eagle @ z=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 z 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(sSFRpeak [yr 1]) CANDELS: Redshift Evolution in Peak sSFR Best Fit sSFR(z) Quenched Threshold : sSFRpeak(z) 1dex 1/tH(z) [yr 1] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Illustration of our method to select quenched and star forming galaxies in simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Left Panel: sSFR distributions for three redshift snapshots from the Eagle simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The peak sSFR is indicated by dashed lines, and our adopted quenched threshold (at sSFR = sSFRpeak - 1dex) is indicated by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Right Panel: Evolution in peak sSFR from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 - 2 in CANDELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Blue dots display the peak of the sSFR distribution, with x, y error bars indicating the z-range applied and 1σ dispersion in sSFR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A linear best fit to the peak sSFR evolution is shown in magenta, with our corresponding quenched threshold shown as a dashed red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For comparison, the inverse of the Hubble time is displayed, which provides a reasonable quenching threshold at these epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' magnitude below the star forming peak at the redshift of each object (approximately 3σ below the main sequence relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the simulations, all systems with SFR = 0 are defined as quenched and included in all analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Star forming systems are considered to be any galax- ies with sSFR values higher than the quenched limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This method is illustrated for one simulation (Eagle) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 1 (left-hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Note that we do not apply a mass term to our sSFR limit unlike in some of our prior work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The reason for this is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' First, the impact of sSFR variation with stellar mass is weak except at the high mass end, where the majority of systems are quenched in any case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, this does not impact our classification results in any significant manner (as we have checked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Second, we choose this cut to be in line with our prior work (Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022), where this decision was made to maximize the objectivity of the quenching criteria between different models and observations, which have different mass dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Finally, the curve in the sSFR - M∗ relation is likely due to high mass star form- ing systems having quenched cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Thus, it remains an open question conceptually whether or not this ought to be accounted for (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, the CANDELS data spans a wide range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To combat this, we compute the peak sSFR of star forming systems at multiple epochs, and fit a simple redshift relation to this (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 1, right-hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally for CANDELS, we also consider a check on the sSFR method utilizing UVJ col- ors (exactly as in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The results from these two approaches are essentially identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We have explored alternative definitions of quenching, including an offset from the redshift evolving star form- ing main sequence relation (∆SFR, which fully accounts for potential variations in mass on the quenching defini- tion), different threshold cuts, and excluding green val- ley systems with intermediate levels of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The results in this paper are completely stable to these alternative methods for identifying quenched systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Random Forest Classification: Revealing Causality with Machine Learning In this paper, we utilize a machine learning approach for classifying star forming and quenched systems in simu- lated and observed galaxy surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More specifically, we adopt a Random Forest classifier from the powerful ScikitLearn Python package (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This method is an ideal compromise between sophisti- cation (the ability to account for highly non-linear and non-monotonic relationships between multiple parame- ters) and interpretability (the ability to extract mean- ingful insights from the trained classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Full details on this method are provided in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022), Brownson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022), and Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022), including numerous detailed tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Of particular value for our present application is the capacity of the Ran- dom Forest classifier to separate causally related param- eters from nuisance parameters in a classification prob- lem (see Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022 appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ultimately, this is achieved through controlling for all other parameters when ascertaining the importance of any given parame- 7 ScikitLearn: https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='org/stable/ AGN feedback driven quenching 7 ter to the classification problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here we give a brief review of the most important aspects of our Ran- dom Forest classification method for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A Random Forest is a set of decision trees with dif- ferences between them ensured through bootstrapped random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Two example decision trees from our analysis are shown in the Appendix (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 8 & 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In training, the criteria for each decision branch (in each decision tree) is set by selecting the optimal parameter (and threshold) to most effectively separate the classes of data, defined explicitly by the Gini impurity: G(n) = 1 − c=2 � i � pi(n)2� (2) where pi(n) indicates the probability of randomly se- lecting class, i, at node, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The summation is made over all classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' in this paper, over star forming and quenched systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For example, in our application, the most useful observable (from a list containing black hole mass, stellar mass, halo mass etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=') is selected to maxi- mize the accuracy with which quenched and star forming galaxies are separated at each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Then the optimal threshold in that variable is found to minimize the Gini impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The entire decision tree is filled out through iteration, until no further improvements are possible (or else a pre-defined threshold is reached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Once the Random Forest classifier is trained, one can extract how important each parameter is for solving the classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The relative importance of a fea- ture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' parameter under investigation), k, is defined as: IR(k) = 1 Ntrees � trees �� nk N(nk)∆G(nk) � n N(n)∆G(n) � (3) where the numerator in the above expression indicates the the improvement in Gini impurity (∆G(n)) weighted by the number of data points entering that node (N(n)), summed over all nodes which utilize feature parameter k (hence the additional subscripts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The denominator provides a similar summation, but over all nodes, re- gardless of the features used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, the ratio gives the relative importance of the variable k for solving the classification problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' in this work, determining whether galaxies will be star forming or quenched) in a given decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Finally, an average is taken over all tress in the Random Forest to give the final relative importance statistic of each training parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is the key statistic we use in this paper, and from now on we will refer to it as the quenching importance (due to our specific science application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To assess the uncertainties on the Random Forest clas- sification, we run 25 independent training and testing runs of each analysis, taking the median importance as the final result and the standard deviation as the sta- tistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We are careful to avoid over-fitting by testing the performance of the Random Forest classifier on unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More specifically, we require a differ- ence in performance (measured by the area under the true positive - false positive curve, AUC) of ∆AUC < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='02 (see Teimoorinia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The required (very) small differ- ence in performance between the training and testing data sets prevents the classifier from learning patholog- ical features in the training data which do not scale to unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Before training the random forest, all parameters are median subtracted and normalized by the interquar- tile range, to avoid differences in parameter variance or magnitude impacting the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We train and test on a balanced sample of 50% quiescent and 50% star forming systems in each classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This avoids weighting one population as more important than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also reserve 50% of the data for performance testing, which is unseen by the random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This enables the rigorous over-fitting testing, described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The final results are taken from the performance on the unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The above methodology is essentially standard in the machine learning literature (see Teimoorinia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2019, 2020a,b and references therein for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is important to apply these data preparation steps carefully to avoid biased results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' RESULTS & DISCUSSION In this section we apply our Random Forest classi- fication technique (discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2) to three cosmological hydrodynamical simulations, in order to identify the key observable signature of AGN feedback driven quenching across cosmic time (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ad- ditionally, we apply the Random Forest technique to two observational wide-field galaxy surveys, in order to test the model predictions across a wide range of epochs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also discuss the implications of our results within the context of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The Fundamental Signature of AGN Feedback Driven Quenching in Simulations across Cosmic Time In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2 we present the results from a series of Random Forest classification analyses applied to the problem of identifying quiescent galaxies in the IllustrisTNG, Ea- gle, and Illustris hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Results 8 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' are presented separately for each of the three cosmo- logical simulations (shown in separate panels) and for five redshift snapshots spanning from z=0 to z=2 (as labelled by the legend in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The parameters used to train the Random Forest are displayed along the x-axis, and the relative importance for quenching is dis- played as the bar height (on the y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Uncertainties on the quenching importance are estimated from the 1σ dispersion of 25 parallel classification training and test- ing runs, applying random sub-sampling of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For each simulation, and at every redshift snapshot considered, black hole mass is clearly identified as the most predictive parameter for classifying star forming and quenched galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' No other parameter reaches be- yond ∼1/4 the relative importance of black hole mass at any epoch, and most parameters are of negligible impor- tance for quenching at all redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, the key pre- diction of AGN feedback models is a clear dependence of galaxy quenching on black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is important to highlight that this result is stable from cosmic noon to the present epoch, unanimously predicted by three independent simulations, and is (at least in principle) observationally testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is especially important to note the utter lack of im- portance given to black hole accretion rate by the classi- fier in all simulations and at all redshifts studied8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Given that the total bolometric AGN luminosity is directly proportional to accretion rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' LAGN = η RBH c2, for an efficiency η ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1), this further implies a negligi- ble importance to quenching of contemporaneous AGN luminosity (and hence also AGN identification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This re- sult is in close agreement with Ward et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022), who find that the most luminous AGN in simulations are found in the most gas rich (and hence star forming) sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This demonstrates that there is expected to be no instantaneous link to cessation of star formation in con- temporary models, despite AGN being used to quench galaxies in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Therefore, searching for evidence of AGN feedback by looking for a difference in (s)SFR between AGN and non-AGN systems is destined to failure, if AGN feedback operates as it is predicted to in modern hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nonetheless, a great number of studies have proceeded in just this manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Nandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Georgakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Silverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hickox 8 The only slight exception to this is for Eagle at the lowest red- shift snapshot, where accretion rate performs as the second best parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Interestingly, this is the ‘exception that proves the rule’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The value of RBH actually turns out to be that it anti- correlates with quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' That is, higher accretion rates at low redshifts in Eagle are associated with more star forming systems: the opposite of catching AGN quenching in action!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Rosario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Heckman & Best 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Trump et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Shimizo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Florez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and many more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, one must be cau- tious in the interpretation of these prior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Indeed, from our analysis of hydrodynamical simulations, a lack of clear connection between AGN luminosity and quies- cence is actually a feature of the modern AGN feedback quenching paradigm, not evidence against it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We speculate that the above result reflects the impor- tance of preventative feedback in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In such a scenario, it is the energy released over long periods from AGN which really matters for quenching (which is clearly traced by MBH not RBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, quies- cence may emerge as a long term consequence of AGN heating preventing gas cooling and accretion from the CGM into massive galaxies, ultimately starving the sys- tem of fuel needed for further star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This mode of operation simultaneously resolves the cooling problem in high mass groups and clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Fabian 2012), explains the low cosmological star formation efficiency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Fukugita & Peebles 2004), as well as accounting for the observed demographics of star forming and quenched galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In- stantaneous feedback may still trigger quenching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Terrazas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020), but with- out long term heating of the CGM, star formation will inevitably reignite within massive galaxies, due to gas cooling and condensation into the system, removing the galaxy from our quenched sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Since our machine learning classification analysis is sensitive to what is fun- damentally different between quenched and star forming systems, our results pick up on the long term preven- tative mode rather than the instantaneous trigger (by design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2, we present pie plot insets showing the results from a simplified analysis, averag- ing over all epochs in the simulations, and separating parameters coarsely by whether they are black hole re- lated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', MBH & RBH) or non-black hole related (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', M∗, Rh, MH, φ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that BH-parameters domi- nate the predictive power in the Random Forest clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Within this set, as discussed above, it is black hole mass not accretion rate which really mat- ters for quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The rest of the parameters are all of very little importance for predicting quiescence, offer- ing on average <15% of the total quenching importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For straightforward theoretical reasons (see Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022), stellar mass is closely re- lated to the total energy released from supernovae, and halo mass is closely related to the total energy released from virial shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, the simulations predict that AGN feedback driven quenching 9 MBH BH M * Rh MH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance TNG - All Parameters TNG Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 TNG Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 TNG Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 TNG Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 TNG Centrals : z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 BH 94% 6% MBH BH M * Rh MH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance Eagle - All Parameters Eagle Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Eagle Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Eagle Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Eagle Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Eagle Centrals : z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 BH 92% 8% MBH BH M * Rh MH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance Illustris - All Parameters Illustris Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Illustris Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Illustris Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Illustris Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 BH 87% 13% Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Random Forest classification analysis to predict the existence of quiescent central galaxies in three cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The quenching importance of black hole mass (MBH), the stellar potential (φ∗), black hole accretion rate (RBH ≡ dMBH/dt), stellar mass (M∗), stellar half mass radius (Rh), and dark matter halo mass (MH) are displayed on the y-axis (as labelled from left-to-right along the x-axis of each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The results are shown separately for IllustrisTNG (top panel), Eagle (center panel), and Illustris (bottom panel), for a variety of snapshots spanning from cosmic noon (z ∼ 2) to the present epoch (as shown on the legend of each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Note that the z=2 snapshot for Illustris is absent due to a lack of quenched galaxies at that epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Uncertainties on the quenching importances are given from bootstrapped random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that black hole mass is overwhelmingly the most important parameter for predicting quiescence in central galaxies at all epochs (and for every simulation) considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A pie plot is displayed on each panel, which averages the quenching importances across all epochs, comparing two broad categories: BH-parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' MBH and RBH);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' and non-BH-parameters (the remaining set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that non-BH parameters engender only a very minor improvement in predicting quiescence over BH-parameters alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 10 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' these important processes within galaxies have a negli- gible impact on the quenching of centrals across cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Again, this is a clear prediction of contemporary AGN feedback driven quenching models, which can be observationally tested (in principle at least).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At low red- shifts, the models clearly pass this test (see Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022), yet it remains to be seen whether this is so at higher redshifts or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is important to appreciate that even though Eagle, Illustris, and IllustrisTNG utilize very different sub-grid prescriptions for AGN feedback (including different pro- cesses at a fundamental level, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' CGM jet bubbles vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' SMBH-driven winds into the ISM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' kinetic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' thermal energy injection), the fundamental qualitative quench- ing predictions are identical between these simulations at all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The reason for this is that black hole mass traces the total energy released from black hole accretion in a model independent manner (see Soltan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Silk & Rees 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ter- razas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, these predictions are fundamental to any quenching process with energy originating from the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Note that this is true even though there are many quantitative differences between these simulations in terms of quenching (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022, as well as Ap- pendix A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is a profound point for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' First, variation in the resolution, mechanism of AGN CGM energy/ momentum transfer, sub-grid recipe, hydo-solver, or volume size of simulations are unable to change the dependence of quiescence on black hole mass without fundamentally abandoning AGN as the cause of quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Second, due to the first point, this enables an observational test of the entire paradigm of AGN feedback driven quenching, not just one specific instan- tiation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is to this we turn in the next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Observational Tests of AGN Feedback Driven Quenching Conceptually, it is straightforward to test the AGN feed- back paradigm - simply measure the parameters in- cluded in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2 for a representative sample of galaxies at each epoch and run the Random Forest classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The problem is that many of these parameters, though observable in principle, are difficult to measure in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is especially the case for black hole mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' the key observable in AGN feedback driven quenching (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At low redshifts, we were able to leverage the power of statistical calibrations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' the MBH − σ relation, in order to make progress (see Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Yet, at redshifts beyond z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2, even the proxy variables needed for accurate statistical calibrations of black hole mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' central velocity dispersion) are in- frequently measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This critical issue will be resolved in the coming years through dedicated spectroscopic sur- veys targeting intermediate-to-high redshifts, most no- tably with VLT-MOONS and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, a robust test to the multi-epoch quenching predictions from hydrodynamical simulations (discussed in the pre- vious sub-section) can soon be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Although not ideal for the purpose of constraining black hole mass, large photometric surveys exist with HST rest-frame optical imaging at the epochs from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 − 2, the largest of which is HST-CANDELS (Grogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, it is interesting to attempt to find a useful proxy for black hole mass in the simulations, which is relatively easy to constrain in photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In lieu of a direct kinematic measurement of black hole mass, or a measurement of central velocity dispersion, one can still make progress by making several assump- tions and leveraging the power of the virial theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The gravitational potential (φg) for a system with dy- namical mass, Mdyn, and gravitational radius, Rgrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', is given by: φg ≡ −GMdyn Rgrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' ∼ Mdyn Rgrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' ∼ M∗ Rh ≡ φ∗ (4) where we make a series of assumptions intended to pro- gressively simplify the measurements needed from left- to-right in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 above, whilst preserving a strong re- lation to the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Note that the use of the tilda is intended to specify ‘scales with’ rather than ‘of the order of’, since the latter is obviously not true when changing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the first step, we drop the constants (which are irrelevant for classification perfor- mance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the second step, we approximate the dynam- ical mass with the stellar mass, and the gravitational ra- dius with the half mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Within ∼ 2 − 3 Rh, this is a good approximation for most high mass systems, although it is clearly imperfect (it ignores the contribu- tion of gas and dark matter, as well as the structure of the mass distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For a virialised system, the gravitational potential is related to the mean square velocity (vrms) as follows: φg = v2 rms ∼ σ2 ∼ M 1/2 BH (5) where, for a pressure supported system, vrms ∼ σ (hence the second step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Even for rotationally supported sys- tems there is a strong relationship between σ and vrms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Brownson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022), although the tightness of the relation reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To arrive at the final step in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5 above, we use the fact that black hole mass is empiri- AGN feedback driven quenching 11 cally determined to scale with velocity dispersion to ap- proximately the fourth power (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Saglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, we have found an approximate way to estimate supermassive black hole mass in photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ex- plicitly, MBH ∼ φ2 ∗ (6) Or, in other words, there is at least some logic in suspect- ing that φ∗ (which we can measure in pure photometric data) may trace MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Yet, the above argument is little more than a reasoned guess;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' the proof lies in the simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is important to appreciate that φ∗ is not the only correlator to black hole mass that one can expect from simple arguments of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For instance, bulge mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014, 2022), central density (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Cheung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2013) and light concen- tration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Wuyts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011) are all also expected to correlate strongly with central velocity dispersion and hence black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Our purpose in utilizing φ∗ is purely for convenience - it is easy to measure in both the observational and simulated data sets used in this work, without requiring significant further data process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3 we present a series of Random Forest clas- sification analyses to predict the existence of quenched galaxies within the three simulations studied in this pa- per, exactly as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, here we restrict the parameters used to train the classifier to: stellar mass (M∗), half mass radius (Rh), and the stellar po- tential (φ∗ = M∗/Rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' All of these parameters can be estimated in a straightforward manner in extant pho- tometric observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3 that in lieu of black hole mass, φ∗ is predicted to be the most important parameter for constraining quenching, at all epochs and for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is as expected from the simple argument sketched above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More physically, in the simulations, this result emerges as a consequence of supermassive black holes forming and growing in the dense gravitational potentials at the center of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2007, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Weinberger 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 we test the above prediction from hydrody- namical simulations in observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At low red- shifts we analyze the SDSS (blue bars), and at higher redshifts we analyze CANDELS (shown in pink for inter- mediate redshifts, and in red for high redshifts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Addi- tionally, for the SDSS, we consider two subsamples: raw data counts and a volume corrected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The latter mimics a volume complete sample of galaxies, testing whether Malmquist bias could impact the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For CANDELS, we consider the standard sSFR method for identifying quenched systems (used throughout this pa- per), as well as a UVJ color method (as in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Since in CANDELS we do not have access to spectroscopic star formation rates, we utilize this extra test to ensure that the results of the Random Forest clas- sification are stable to alternative methods for defining quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For all redshifts, galaxy surveys, and methodological choices, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 we find that the stellar potential is by far the most predictive parameter of quenching in ob- servations, compared to stellar mass and galaxy size9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, there is remarkable agreement between the sim- ulation predictions and the observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is a clear success of the models, and by extension the paradigm of AGN feedback driven quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The importance of the stellar potential for predicting quiescence is also consistent with numerous other obser- vational works investigating bulge mass, central mass density, central velocity dispersion, and light concentra- tion as phenomenological drivers of galaxy quenching (see Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2008, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Wuyts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Wake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Cheung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Omand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2014, 2016, 2020a,b, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Pi- otrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Brownson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Varma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The comparison with modern cosmological simulations performed in this paper establishes a plau- sible physical origin to these prior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' That is, AGN feedback driven quenching is best predicted by the fossil record of past accretion history (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' black hole mass), which is itself highly correlated with the local poten- tial well in which the black hole forms (approximately given by φ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Naturally, the local potential also scales with central mass density, stellar light concentration, mass of the central bulge component, and central veloc- ity dispersion, hence explaining these prior results as a plausible consequence of historic AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Of course, whilst the above results are consistent with the fundamental prediction from AGN feedback models, it remains possible that other mechanisms could yield a tight dependence of quenching on the stellar potential as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To fully rule out this possibility, dynamical mea- surements of the masses of supermassive black holes are needed on a large scale at intermediate-to-high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 9 Note that for the observational data we utilize half light radius instead of half mass radius, choosing the reddest waveband avail- able (with high quality data) in order to minimize the systematic offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Explicitly, we utilize r-band in the SDSS and H-band in CANDELS, which both trace the rest-frame optical at the red- shifts probed by each survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This could be a source of difference between observations and simulations, but in no way could this ensure a greater similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 12 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' M * Rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance TNG - Observable Parameters TNG Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 TNG Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 TNG Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 TNG Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 TNG Centrals : z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 M * Rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance Eagle - Observable Prameters Eagle Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Eagle Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Eagle Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Eagle Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Eagle Centrals : z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 M * Rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Quenching Importance Illustris - Observable Parameters Illustris Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Illustris Centrals : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Illustris Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Illustris Centrals : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Identical in structure to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2, except that fewer parameters are used to train the Random Forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here we focus on stellar mass, half mass radius, and the stellar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' These parameters are relatively straightforward to measure in extant wide-field galaxy surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, they offer an opportunity to test the predictions from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The stellar potential is clearly identified as the most important predictor of quenching, in lieu of black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' AGN feedback driven quenching 13 M * Rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 Relative Quenching Importance (Train,Test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='91, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='90 (Train,Test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='83 Quenching Classification for Observations SDSS Centrals : z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1 (Raw Sample) SDSS Centrals : z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1 (Volume Corrected) CANDELS All Galaxies : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 (sSFR) CANDELS All Galaxies : z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 (UVJ) CANDELS All Galaxies : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 (sSFR) CANDELS All Galaxies : z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 (UVJ) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Random Forest classification analysis to predict the existence of quiescent galaxies in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The relative quenching importance of the stellar potential (φ∗), stellar mass (M∗), and rest-frame visible half light radius (Rh) are shown on the y-axis (as labelled by the x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Uncertainties are given by boot strapped random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At low redshifts (blue colored bars) we display results for the SDSS, using both raw galaxy counts and a 1/Vmax volume corrected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At higher redshifts we utilize data from the CANDELS survey, shown separately for intermediate redshifts (displayed in pink) and high redshifts (displayed in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For the CANDELS data we consider both the sSFR approach (utilized throughout this paper) and a UVJ color based approach to identify quiescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' At all epochs (and for all methodological sub-samples), the stellar potential is clearly identified as the most predictive parameter for identifying quiescent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is precisely as predicted by all three cosmological simulations studied in this paper (compare to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 14 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' SUMMARY In this paper we apply Random Forest classification to three cosmological hydrodynamical simulations (Eagle, Illustris, and IllustrisTNG) and two wide-field galaxy surveys (SDSS & CANDELS), identifying the optimal parameters for predicting whether central galaxies will be star forming or quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This work expands on our initial low redshift analysis in Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) by probing the dependence of quenching on physical parameters out to cosmic noon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Throughout this paper we focus on galaxies in the stellar mass range 9 < log(M∗/M⊙) < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Our principal results are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Supermassive black hole mass is identified as the most important parameter for predicting quies- cence in all three simulations, and at every red- shift snapshot analyzed (spanning ∼10 Gyr of cos- mic history).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is the key testable prediction from models of AGN feedback driven quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Perhaps surprisingly, supermassive black hole ac- cretion rate is not constraining of galaxy quench- ing at any epoch, according to the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This further implies that AGN luminosity is not predicted to correlate strongly with galaxy quies- cence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Instead, it is the integrated effect of historic AGN feedback which leaves a clear signature on the galaxy population (traced by MBH not LAGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In lieu of a measurement of black hole mass, the stellar potential (φ∗ ∼ M∗/Rh) is predicted to act as a good proxy in simulations, outperforming all other variables for predicting quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We confirm the above prediction from simulations in observations, utilizing the SDSS at low redshifts and CANDELS at intermediate-to-high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, we find a remarkable consistency between simu- lations which utilize AGN feedback to quench galaxies, and multi-epoch observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Further tests on the Ran- dom Forest results, which confirm the above conclusions, are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Ultimately, the key to testing the paradigm of AGN quenching is to study the fossil record of past accretion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', black hole mass) or its best available proxy, not a proxy of accretion rate (such as AGN luminosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More precise observational tests of the key AGN quenching prediction from sim- ulations will become feasible with VLT-MOONS and JWST observations in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' ACKNOWLEDGEMENTS AB acknowledges a faculty start-up grant at the Florida International University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' JMP acknowledges funding from the MERAC Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' RM acknowledges ERC Advanced Grant 695671: ‘QUENCH’, support from the Science and Technology Facilities Council (STFC), and support from a Royal Society Research Professorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We thank the anonymous referee for a highly positive and constructive report, which has helped to signifi- cantly improve this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We are grateful to the Illustris, Eagle, and IllustrisTNG teams for making their simulations public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We are especially grateful to Paul Torrey, Joop Schaye, Dylan Nelson, Annalisa Pillepich, Lars Hernquist, and Rob Crain for many stimulating and enlightening conversations on the simulations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We thank the SDSS and CANDELS teams for making their observational surveys public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We are especially grateful to Luc Simard, Sara Ellison, Trevor Mendel, Marc Huertas-Company, and Paula Dimauro for much advise and support with these data prod- ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We also thank Yingjie Peng, Emma Curtis-Lake, Gareth Jones, Stephen Eales, Mirko Curti, Sara Ellison and Christopher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Conselice for many fruitful and en- gaging conversations on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' DATA AVAILABILITY All of the data used in this work is publicly available, see the following links for access: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' EAGLE: virgodb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='dur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='ac.' 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number of detailed tests on the Random Forest classification results from the main body of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' All of the conclusions from the Random Forest analyses are consistent with the conclusions drawn from the alternative methods outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As such, this appendix supports the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Simulations: Area Statistics at z = 1 The results from Random Forest classification in the simulations are extremely clear: black hole mass is predicted to be the most important parameter regulating central galaxy quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In this part of the appendix, we show that this conclusion can be arrived at through a different analysis, which has some advantages over the Random Forest approach, and some disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, this provides a useful test on one of the main conclusions from this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We adopt our area statistics method (outlined in detail in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016, 2020a) to assess which parameters engender the most significant impact on the quenched fraction, at fixed values of the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As in the main body of this paper, we select central galaxies consistently at 9 < log(M∗/M⊙) < 12 and MHalo > 1011M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Most of the variables analyzed in the area statistics plots have similar (though not identical) ranges between the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The one exception is black hole accretion rate, where the lower limit varies significantly between the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' To combat this, we present all panels with the same range (to aid in comparison) but compute the area statistics only from the minimum to maximum value of each parameter in the source catalogs at this epoch, to avoid biasing this statistic (as in Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' One major advantage of the area statistics approach is that it is highly visual and intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' On the other hand, this method only allows one variable to be controlled for at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Moreover, the area statistics approach is far less efficient than the Random Forest classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Due to the latter issue, we restrict our analysis here to just one redshift snapshot (z = 1) and limit the parameters under consideration to: black hole mass, black hole accretion rate, stellar mass, and halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5 we show the results from the area statistics approach for TNG galaxies at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The top set of panels shows the results for a balanced sample (50% quenched;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 50% star forming), as utilized in the Random Forest classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, we also present the results for the full volume complete sample, shown as the bottom set of panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that the fQ − MBH relation is by far the tightest of all of the relations considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is true for both a balanced and complete parent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, black hole mass engenders a much larger impact on the fraction of quenched galaxies in the TNG simulation (at fixed other parameters) than any other parameter (at fixed black hole mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Therefore, black hole mass is clearly established as the most fundamental parameter governing central galaxy quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is in precise accord with the results from the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 6 & 7 we repeat the analysis for Eagle and Illustris (respectively) also selected at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Again, it is clear that the fQ − MBH relation is the tightest and steepest of all of the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This confirms the results from the center and bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, we have explored the area statistics approach for all other redshift bins, and the full list of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black hole mass is clearly established as the most fundamental parameter in all of these extra tests as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As a result of the above analysis, it is evidently possible to arrive at the main conclusion of this paper for simulations through area statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, this is highly inefficient, as it would require ∼20 pages of plots, rather than a single figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' More importantly, the Random Forest solves the full classification problem, controlling for all nuisance variables, and accounting for both the steepness and tightness of the quenched fraction relationships simultaneously (see Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, Random Forest classification is a much superior method to area statistics both conceptually and practically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nevertheless, the latter does offer further insight into why the Random Forest yields the results it gives, as well as providing a simple visual check on the main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Another advantage of the area statistics approach is that it allows us to probe some of the details of the process of quenching in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Although not the primary focus of this work, it is clear from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5 - 7 that central galaxies quench much more rapidly as a function of black hole mass (almost as a step function) in IllustrisTNG, but quench much more gradually as a function of black hole mass in both Eagle and Illustris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Additionally, the quenching 18 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' threshold in black hole mass is over an order of magnitude higher in Illustris and IllustrisTNG, compared to Eagle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In this work we have not attempted to estimate black hole masses for observed galaxies, so we are not in a position to judge which is right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This notwithstanding, in Piotrowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' (2022) we do make a careful comparison of quenching fractions between these three simulations and low-z observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' No single simulation is an ideal march for the observational data, but IllustrisTNG is the closest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This indicates that, at least at present, a step function quenching threshold in black hole mass cannot be ruled out by the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Example Decision Trees In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 8 we present an example of one randomly selected decision tree from the Random Forest quenching classification of the Eagle simulation at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This illustrates the structure of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black hole mass is selected as the parameter for the first split in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The reason for this is that a cut in black hole mass engenders the largest reduction in Gini impurity, over any other parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, black hole mass acquires the highest weight, and largest difference in Gini coefficient, leading to the highest importance in solving the classification problem (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Subsequent decision thresholds impact a smaller fraction of the data and have in general a lower reduction in Gini Impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Thus, it is crucial which parameter is chosen first by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The stability of this choice is assured by exploring 250 random generations of the parent sample, which is the fundamental advantage of a Random Forest over a single decision tree (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 9 we show an alternative decision tree from a classification analysis of Eagle at z = 1, utilizing a volume complete (as opposed to balanced) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Once again, black hole mass is chosen as the first parameter in the decision tree, engendering a high importance of this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, in this example, the sample is overwhelmingly star forming (due to the steepness of the mass function) and hence the classifier has to probe deeper to effectively separate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is not the preferred approach for using machine learning in classification for various technical reasons (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=', Teimoorinia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nonetheless, this test clearly establishes the stability of the main results to the manner in which the data is presented to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Predicted MBH − φ∗ Dependence In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 10 we show the predicted MBH −φ∗ relations from TNG (left panels), Eagle (center panels), and Illustris (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that all three simulations predict that there should be a tight dependence of black hole mass on the stellar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This provides further insight into why φ∗ is chosen by the Random Forest classifier in lieu of black hole mass (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We can leverage this theoretical result to test the quenching predictions of these simulations in extant photometric observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In the SDSS, where we have the most complete suite of measurements on observational galaxy parameters, we have additionally made a number of extra tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We find that φ∗ is second only to central velocity dispersion for predicting quiescence, comfortably beating bulge mass, total stellar mass, halo mass, B/T morphology, environmental parameters, and disk mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' However, the real value of φ∗ for our analysis is that it can be straightforwardly estimated in extant catalogs without further data processing, in both simulations and observations up to z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Given that the simulations unanimously predict that this parameter will act as a reasonable proxy for MBH, this is sufficient for our present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Observations: Area Statistics in the SDSS & CANDELS In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 11 we apply our area statistics approach to the SDSS sample of central galaxies at low redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that the fQ − φ∗ relation is by far the tightest of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is exactly as predicted by all three simulations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 3 & 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 12 we apply the area statistics technique to the full CANDELS galaxy sample at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Again, the fQ − φ∗ relation is found to be the tightest (and steepest) of the observed relations, exactly as predicted by the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This confirms the results from the Random Forest classification (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 4 in the main body of the paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' We have additionally tested the use of volume weighting, restricting to a balanced sample, including more redshifts cuts, and utilizing alternative quenching definitions (and thresholds) on the observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The results from this paper are extremely stable to these alternative analysis choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' AGN feedback driven quenching 19 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='08 TNG Centrals: z = 1 lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='17 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='17 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='27 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='19 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='31 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='35 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='55 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='02 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 Volume Complete Sample lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='08 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='14 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='16 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='26 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='19 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='30 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='17 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='29 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Top Panels: The quenched fraction relationship with black hole mass (top row) and stellar mass, halo mass, and black hole accretion rate (from left-to-right along the bottom row) for the IllustrisTNG simulation at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Each quenched fraction relationship is split into percentiles based on a third variable (as indicated on the legends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As in the Random Forest analysis, we select a balanced sample (50% star forming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 50% quenched) for this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The tightness of each of the quenched fraction relationships is quantified by the area subtended by the upper and lower 50th (and 25th) percentiles in the third variable (which is displayed on each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' For example, in the top left panel, the quenched fraction - black hole mass relation is split into ranges based on stellar mass, while the lower left panel inverts this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that the quenched fraction - black hole mass relation is by far the tightest relationship, confirming the results of the Random Forest classification analysis at this epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Bottom Panels: Same as above, but for a volume complete sample (with a majority of star forming systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Again, it is clear that the quenched fraction relationship with black hole mass is by far the tightest of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 20 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='14 Eagle Centrals: z = 1 lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='12 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='14 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='32 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='49 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='41 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='62 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='46 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='72 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 Volume Complete Sample lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='11 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='16 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='13 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='12 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The same in structure as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5, but here showing results for the Eagle simulation at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The top panels show the results for a balanced sample, and the bottom panels show the results for the volume complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In all cases, the fQ − MBH relationship is the tightest relation (as quantified by the area statistics displayed on each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The only slight exception to this is for MBH vs RBH in the volume complete sample, where the tightness of each relation is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nonetheless, the increase in quenched fraction is significantly larger as a function of black hole mass than accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, black hole mass is still a better predictor of quiescence than accretion rate, as seen clearly in the balanced sample (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Moreover, increasing accretion rate at a fixed black hole mass actually lowers the fraction of quenched galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' This is expected for a co-fueling scenario but is the opposite of what is expected for ‘catching’ AGN quenching in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' AGN feedback driven quenching 21 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='06 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='08 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='12 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='17 Illustris Centrals: z = 1 lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='16 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='29 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='37 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='58 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='39 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='58 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='51 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='75 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 6 7 8 9 log(MBH/M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='05 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='05 Volume Complete Sample lower 25% in MHalo lower 50% in MHalo ALL upper 50% in MHalo upper 25% in MHalo 6 7 8 9 log(MBH/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 lower 25% in BH lower 50% in BH ALL upper 50% in BH upper 25% in BH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M * /M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='09 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='15 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MHalo/M ) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='11 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='18 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH 10 8 6 4 2 0 log( BH [M yr 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='10 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='15 lower 25% in MBH lower 50% in MBH ALL upper 50% in MBH upper 25% in MBH Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The same in structure as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5, but here showing results for the Illustris simulation at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The top panels show the results for a balanced sample, and the bottom panels show the results for the volume complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' In all cases, the fQ − MBH relationship is the tightest relation (as quantified by the area statistics displayed on each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 22 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' An example of one randomly selected decision tree (out of 250) for training in Eagle at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here, as in the main body of this paper, we restrict to a balanced sample of quenched and star forming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The parameter and threshold chosen by the classifier at each of the decision forks is presented in each box, along with the Gini impurity and sample size (this starts at Gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 by design for a balanced sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The color of boxes indicate whether the descendent sample is majority quenched (red) or star forming (blue), with the depth of the hue indicating the purity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Colors shift from white to deep blue (or red) down each branch of the decision tree, as the classifier solves the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Black hole mass is selected by the classifier as the most effective first cut on the data, engendering the most dramatic reduction in Gini impurity for the largest number of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The overall success of this parameter emerges from the stability of this choice to sample variation across the entire Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='127 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 samples = 472 value = [236, 236] class = SF MBH <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='364 Rh <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='743 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='103 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='192 samples = 221 samples = 251 value = [209, 12] value = [27, 224] class = SF class = Q dM/dt <= -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='679 dM/dt <= -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='532 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='466 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='453 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='02 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='147 samples = 27 samples = 26 samples = 194 samples = 225 value = [17, 10] value = [9, 17] value = [192, 2] value = [18, 207] class = SF class = Q class = SF class = Q Rh <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='576 MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='582 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='147 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='28 samples = 25 samples = 169 samples = 130 samples = 95 value = [23, 2] value = [169, 0] value = [2, 128] value = [16, 79] class = SF class = SF class = Q class = Q MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='855 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='147 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='488 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='134 samples = 105 samples = 25 samples = 26 samples = 69 value = [0, 105] value = [2, 23] value = [11, 15] value = [5, 64] class = Q class = Q class = Q class = Q gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='269 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='044 samples = 25 samples = 44 value = [4, 21] value = [1, 43] class = Q class = QAGN feedback driven quenching 23 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' An alternative example of a randomly selected decision tree (out of 250) for training in Eagle at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Here we analyze the raw sample of quenched and star forming systems (which is majority star forming, hence the increased blue shading of this tree compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Consequently, the classification problem is more complex and the results require more care to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Nonetheless, black hole mass is still selected as the best parameter to make the first cut on the data, yielding a high performance of black hole mass in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='145 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='056 samples = 2467 value = [3772, 113] class = SF dM/dt <= -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='208 Rh <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='567 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='001 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='339 samples = 2129 samples = 338 value = [3369, 2] value = [403, 111] class = SF class = SF MBH <= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='487 dM/dt <= -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='058 MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='627 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='046 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='445 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='141 samples = 2075 samples = 54 samples = 178 samples = 160 value = [3286, 0] value = [83, 2] value = [185, 93] value = [218, 18] class = SF class = SF class = SF class = SF MBH <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='602 Rh <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='705 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='091 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='36 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='398 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='204 samples = 28 samples = 26 samples = 24 samples = 50 samples = 154 samples = 110 value = [43, 0] value = [40, 2] value = [8, 26] value = [80, 0] class = Q value = [177, 67] value =[138, 18] class = SF class = SF class = SF class = SF class = SF M* <= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='313 Mh <= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='234 dM/dt <= -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='464 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='375 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='248 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='494 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='105 samples = 32 samples = 90 samples = 64 samples = 78 value = [36, 12] value = [118, 20] value = [59, 47] value = [102, 6] class = SF class = SF class = SF class = SF Mh <= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='093 MBH <= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='004 dM/dt <= -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='384 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='456 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='437 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='198 _gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='024 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='453 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='054 samples = 36 samples = 21 samples = 30 samples = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' samples = 43 samples = 48 value = [35, 19] value = [10, 21] value = [32, 4] value = [83, 1] value = [49, 26] value = [70, 2] class = SF class = Q class = SF class = SF class = SF class = SF gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='064 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='266 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='497 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 gini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='137 samples = 20 samples = 34 samples = 23 samples = 20 samples = 28 samples = 20 value = [29, 1] value = [54, 0] value = [32, 6] value = [17, 20] value = [45, 0] value = [25, 2] class = SF class = SF class = SF class = Q class = SF class = SF24 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='94 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='19 dex TNG - Central Galaxies @ z = 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='81 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='32 dex Eagle - Central Galaxies @ z = 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='85 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='29 dex Illustris - Central Galaxies @ z = 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='94 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='20 dex TNG - Central Galaxies @ z = 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='78 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='32 dex Eagle - Central Galaxies @ z = 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='89 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='24 dex Illustris - Central Galaxies @ z = 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='94 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='18 dex TNG - Central Galaxies @ z = 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='72 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='32 dex Eagle - Central Galaxies @ z = 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log( [M kpc 1]) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(MBH [M ]) S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='93 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='18 dex Illustris - Central Galaxies @ z = 2 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Predicted relationships between black hole mass and the stellar potential for TNG (left panels), Eagle (middle panels) and Illustris (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Results are shown for z = 0 (top row), z = 1 (middle row), and z = 2 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The median relationship is indicated by a solid magenta line, and the spread of the data is indicated by linearly spaced contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The Spearman correlation statistic (ρS) and the dispersion about the median relation (σ) are displayed on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' There is predicted to be a strong and reasonably tight relationship between black hole mass and the stellar potential by all three simulations at all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Hence, φ∗ is expected to act as a reasonable proxy for MBH in observational data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' AGN feedback driven quenching 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log( [M kpc 1]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='04 SDSS Centrals (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='02 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2) lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log( [M kpc 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='03 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='05 lower 25% in Rh lower 50% in Rh ALL upper 50% in Rh upper 25% in Rh 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M* [M ]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='23 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='35 lower 25% in lower 50% in ALL upper 50% in upper 25% in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='50 log(Rh [kpc]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='50 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='70 lower 25% in lower 50% in ALL upper 50% in upper 25% in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The quenched fraction relationship with φ∗ (top panels), M∗ (bottom-left panel), and Rh (bottom-right panel) in the SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 5, each quenched fraction relationship is displayed for percentile ranges of a third parameter, in order to assess the impact on quenching of varying that parameter at fixed values of each of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' The tightness of each relationship is quantified by the area statistics displayed on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' It is clear that φ∗ engenders the tightest and steepest of the quenched fraction relationships, exactly as predicted in models utilizing AGN feedback to quench galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 26 Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 8 9 10 11 log( [M kpc 1]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='07 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='09 CANDELS Galaxies (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0) lower 25% in M * lower 50% in M * ALL upper 50% in M * upper 25% in M * 8 9 10 11 log( [M kpc 1]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='11 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='16 lower 25% in Rh lower 50% in Rh ALL upper 50% in Rh upper 25% in Rh 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 log(M* [M ]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 fQ A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='22 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='35 lower 25% in lower 50% in ALL upper 50% in upper 25% in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='0 log(Rh [kpc]) A50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='22 A25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content='34 lower 25% in lower 50% in ALL upper 50% in upper 25% in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Identical in structure to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' 11, but here showing results from CANDELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' As at low redshifts, φ∗ exhibits the tightest quenched fraction relationship, as quantified by the area statistics displayed on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} +page_content=' Note also that φ∗ exhibits the steepest positive relationship with quiescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE2T4oBgHgl3EQfIQZD/content/2301.03677v1.pdf'} diff --git a/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/2301.08355v1.pdf.txt b/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/2301.08355v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1823740c898e2e8a961d6602a5c8dc5f789b3af5 --- /dev/null +++ b/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/2301.08355v1.pdf.txt @@ -0,0 +1,8753 @@ +RAA Vol.22 (2022) No.9, 000–000 +© 2022 National Astronomical Observatories, CAS and IOP Publishing Ltd. +http://www.raa-journal.org +http://iopscience.iop.org/raa +Research in +Astronomy and +Astrophysics +A Catalog of 59 Thousand δ Scuti Stars and A Dozen Discoveries of New +Variables with TESS Data +A.-Y. Zhou +National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China; aiying@nao.cas.cn +Abstract We present discoveries of stellar pulsation, variability, binarity and multiperiodicity among a +sample of 50 stars including types of DSCT, HADS, SX Phe, EB, and photometric standards. We initially +aimed at checking the known δ Scuti star HD 52788 and its field stars with TESS data and found that +the previously reported complex light variations with uncertain frequency solutions were partly caused by +the two comparison stars, which turn out to be pulsating variable stars. HD 52788 exhibits 135 pulsation +frequencies in a small domain in 4–12 d−1 based on the non-differential Pre-search Data Conditioning +Simple Aperture Photometry results of TESS. The record high rich frequency solution turns HD 52788 into +a distinctive and very interesting object among δ Sct stars for testing current stellar evolution and pulsation +models. Inspired by the discoveries around HD 52788, we extended our exploration to a small group of +interested stars and resulted in discovery of 20 new variables including 5 δ Sct stars, 4 eclipsing binaries, +and other kinds of pulsating variable stars. In addition, based on existing sources, we have compiled a new +comprehensive catalog of 59350 δ Sct stars, which is by far the largest collection of DSCT with TESS Input +Catalog and Gaia DR3 cross-identifiers and a number of astronomical parameters extracted from TIC and +Gaia archives. With the new catalog covering almost a hundred times the earlier list, the δ Sct domain on +the pulsating H-R diagram is largely extended, which would impact the theoretical borders. +Key words: +stars: oscillation (pulsation) — stars: binaries: eclipsing: — stars: variables: δ Scuti, γ +Doradus, HADS, SX Phe, β Cephei, Solar-like — stars: variables: general — methods: statistical — +techniques: photometric — catalogues — stars: individual: HD 52788, HD 53166, HD 53349, AD Ari, +HD 191025, HD 227658, HD 227647, HD 227681, CD-58 1608, TIC 41195818, TIC 309661100, TYC +2671-577-1, AL Tri, IT Dra, BR Cnc, BU Cnc, BV Cnc, BN Cnc, EX Cnc, HD 73712, V1821 Cyg, V2238 +Cyg, V2245 Cyg, V2455 Cyg, BL Cam, ι Boo, TYC 6672-772-1, 21 Com +1 INTRODUCTION +HD +52788 +(=V383 +Carinae += +TIC +279361762 += +SAO 234839 = HIP 33616 = ASASSN-V J065904.08- +583054.1, V = 8.m37, α2000 = 06h59m04s [104◦.7667], +δ2000 = −58◦30′54′′ [−58◦.5147] ) was announced to be +a δ Scuti star by Kurtz (1979). +Based on a total of 104 hours of differential +photometric observations obtained in 1978–1980, both +Kurtz (1981) and Zhou (2004) failed to establish a +consistent frequency solution for the pulsational behaviour +of the star. The pulsation frequency spectrum of HD 52788 +changed yearly with somewhat complex light variability, +makes it a much interesting δ Sct target than others, +referring to reviews of δ Sct variables by Handler (2009); +Rodr´ıguez & Breger (2001); Breger (2000). +What makes the author interested is the cause of the +unstable pulsation frequency spectrum of HD 52788. The +poor frequency resolution could result from inadequate +0 Submitted to RAA on 2022 October 26. This work is dedicated to +my wife Jingyun Zhang who has been supporting my research all along. +time series data, intrinsic rapid changing, and data issues +using wrong stars as comparison among others. Which +is the true source? Potential comparison stars in the +target field were often discovered to be new variables, +for example, Du et al. (1999); Zhou et al. (2006); Jurcsik +et al. (2012). Due to the star in the southern sky, while +the interested observers in the northern hemisphere, the +question is could we have a chance of obtaining adequate +data to derive a unique set of frequencies that represents +the light variations of this star? Fortunately, the Transiting +Exoplanet Survey Satellite (TESS, Ricker et al. 2015) data +provide us with the best opportunity to check any star’s +light variations more precisely. We are delighted with +our unexpected good fortune when checking TESS data. +We do find rich intrinsic pulsation of HD 52788 based +on the perfect data sets, and serendipitously we reveal +that the two comparison stars used in Kurtz’s differential +photometry turned out to be pulsating variable stars. This +fact of using the wrong comparison would have added +uncertainties in resolving the target’s pulsation. Inspired +by the stellar variability detection results of HD 52788 +arXiv:2301.08355v1 [astro-ph.SR] 19 Jan 2023 + +2 +A.-Y. Zhou +and its comparisons, we decided to extend our studies to a +group of interested stars in a couple of sky fields. Now +we report all our discoveries and preliminary results in +this paper. Meanwhile, we describe the data reduction with +newly developed Python-based tools for the undergoing +project. Furthermore, we will report a series of results +showing that TESS full-sky photometry survey has a huge +space for serendipity and discovery even only examining +a few selected known pulsating stars’ fields. +TESS is an MIT-led NASA mission dedicated to +discovering transiting exoplanets orbiting nearby bright +stars by an all-sky photometry survey. TESS rotates every +∼13.7 days per cycle along a unique highly elliptical +lunar-resonant orbit around the Earth (about 600 km from +Earth). TESS is equipped with four identical refractive +cameras with a combined field-of-view (FOV) of 24x96 +degrees (a segment of sky, known as an observing sector). +The lens assembly in each camera has a 10.5 cm diameter +entrance pupil (aperture) and a focal ratio f/1.4. Each +of the four cameras has four 2048x2048 CCD detectors +(i.e. total 16 CCDs). Each detector pixel sized 15x15 +micron corresponds to 21 arcseconds in the sky. TESS +collects light in 600–1000 nm centered on the traditional +Cousins I-band (IC, central wavelength = 786.5 nm). +Please refer to the details to TESS Science Data Products +Description Document +1 and ”Characteristics of the TESS +space telescope” web page +2 . TESS filter basically matches +with Gaia Red Photometer with bandpass spanning in +630–1050 nm (Gaia Collaboration et al. 2016). +TESS cameras actually expose at a cadence of 2 +seconds. That is, the CCDs take images and read them +out continuously at 2-second intervals. However, the 2- +second frames are used for spacecraft guiding, they were +not downloaded to the ground. The images are processed +on the spacecraft by the data handling unit (DHU, a Space +Micro Image Processing Computer). The DHU stacks +2-second images in groups of 60 to produce 2-minute +cadence images (in cut Target Pixel File, TPF) or 30- +minute cadence for general observations. High cadence is +needed for the detection of exoplanets, so exposures of +planet search targets and other stars of particular interest +(cataloged 200,000 primary stars) are obtained every 2 +minutes while the Full-Frame Images (FFIs) of the entire +field of view are returned every 30 minutes. Finally, pixels +in postage stamps around TESS mission target stars will +be downloaded at a 2-minute cadence, while FFIs will +be downloaded at the 30-minute cadence. These two sets +of data will allow general variability studies for the vast +majority of stars. The data on spacecraft are transmitted +to Earth when the spacecraft reaches orbital perigee every +∼13.7 days. Each sector of the sky will be observed twice +1 https://archive.stsci.edu/missions-and-data/tess; https://archive. +stsci.edu/files/live/sites/mast/files/home/missions-and-data/active- +missions/tess/ documents/EXP-TESS-ARC-ICD-TM-0014-Rev-F.pdf +2 https://heasarc.gsfc.nasa.gov/docs/tess/the-tess-space- +telescope.html +or https://tess.mit.edu/science/ +with 27.4 days observing period. So the TESS observation +is most sensitive to exoplanets with periods of less than 13 +days so that at least two transits are used for discovery. +In the meantime, TESS high-precision uninterrupted +photometry and time resolution (2-minute and 30- +minute cadence) are perfect for stellar pulsation and +asteroseismology (e.g. the works by Lund et al. 2021; +Handberg et al. 2021). Furthermore, TESS produces +additional pixels in little postage stamps surrounding a few +(1000) bright asteroseismology targets and downloads at +20-second cadence (e.g. the works by Huber et al. 2022). +After TESS first two-year prime mission, started in July +2020, TESS was revisiting the sky in an extended ongoing +mission that records full-frame images at a fast ten-minute +cadence. +With the advantage of TESS along with Kepler data, +we are about to re-visit those special targets, including the +selected High-Amplitude δ Scuti stars (HADS), SX Phe- +type stars, pulsating subdwarf B stars (sdBV), pulsating +white dwarfs, eclipsing binary systems with pulsating +components, exoplanet-host pulsating stars, photometric +standard stars, and so on. We are going to characterize +their true variability and complete pulsation contents, +binarity, multiperiodicity, and whether or not they are +hosts of exoplanets in a high precision level of less than +a few parts per million (ppm). We check typical stars +over the pulsational H-R diagram and even photometric +standard stars over the full spectral series from O to M +types. We will first pay close attention to those highly cited +stars and poorly studied stars in the literature that either +suffered from unresolved puzzles in light variations due +to poor ground observations or were neglected. The aim +is to refine our current understanding of variable stars by +resolving high-precision light variations data from space. +2 KEPLER K2 DATA +The Kepler second mission (K2, Howell et al. 2014) had +released its data for open access +3 . Each K2 Campaign has +a duration of approximately 80 days. Light curve data +are usually provided in a 30-minute cadence. Long-time +coverage improves the frequency resolution of variability +detection and is suitable for both stable and unstable +periodic and non-periodic variations of a wide range of +variable star studies. +There are three kinds of light curve data available +to the public: (1) K2 Systematics Correction (K2SC +4 ): +The K2SC-detrended light curves are especially suited for +studying variable stars in K2 photometry (by allowing +us to remove the position-dependent systematics while +3 K2: Extending Kepler’s Power to the Ecliptic, The Ecliptic +Plane +Input +Catalog +(EPIC) +for +Kepler’s +K2 +mission, +see +http://keplerscience.arc.nasa.gov/K2/ +4 K2SC (K2 Systematics Correction) is a K2 light curve detrending +tool that uses Gaussian processes to robustly model the systematics due +to the Kepler telescope pointing jitter together with the astrophysical +variability. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +3 +keeping time-dependent variability intact), and searches +for transiting planets (by allowing us to remove both +the systematics and astrophysical variability). (2) K2 +Extracted Lightcurves (K2SFF): The lightcurves from +K2 contain larger systematics than the original Kepler +mission, due to the reduction in pointing precision as +a result of having to rely on only two reaction wheels. +Vanderburg & Johnson have created a technique to +correct for the pointing-dependent nature of the pixel- +level fluxes. This correction improves the photometric +precision by typical factors of 2-5, and results in the +median photometric performance of K2 targets to within +a factor of two of the original, 4-wheeled mission. Such +extracted lightcurves (using a variety of photometric +apertures), as well as diagnostic plots, for each target, +have been released. The FITS lightcurves include all data +points (with flags to indicate thruster firing), as well as +multiple versions (each FITS extension is an extracted +lightcurve for a different aperture). There are a total of +twenty apertures provided: ten circular and ten based on +the pixel response function. The final extension contains +the summed image from all the postage stamp frames. (3) +EVEREST (EPIC Variability Extraction and Removal for +Exoplanet Science Targets) light curves, these are products +from an open-source pipeline for removing instrumental +systematics from K2 light curves, using a combination of +pixel-level decorrelations to remove spacecraft pointing +error and Gaussian processes to capture astrophysical +variability. Light curves from campaigns 0 through 8, 102, +111, 112, 12, and 13 are currently available. +Either one of K2SFF, K2SC, and EVEREST will +be used depending upon whichever looks better. For our +current work, we used K2SFF light curves. However, the +K2SFF data is still not flat, we have to further remove a +profile by the polynomial fitting of orders between 4 and +24 to the data. Each campaign data set might be divided +into 2 or 3 portions for better fitting and detrending. Then +we put the residuals together for pulsation analysis. +3 TESS DATA +We first checked the TESS archive at the MAST Portal +(Mikulski Archive for Space Telescopes +5 ) for available +data of the interested target star HD 52788 (=V383 +Carinae =TIC 279361762 = ASASSN-V J065904.08- +583054.1, α2000=06:59:04, δ2000=−58:30:53 [104.7667◦, +−58.5147◦], V =8.m40) and the two comparison stars used +in literature C1=HD 53166 (=TIC 279431011, V =8.m1) +and C2=HD 53349 (=TIC 279476396, V =6.m01). We +downloaded the NASA’s Science Processing Operations +Centre (SPOC) generated files, including both the +extracted light curves (*-s lc.fits) and Target Pixel files +(*-s tp.fits). The light curve files *-s lc.fits will be read +directly with ASTROPY and LIGHTKURVE packages (The +5 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html; +or +https://archive.stsci.edu/tess/ +A +s +t +r +o +p +y +C +o +l +l +a +b +o +r +a +t +i +o +n +e +t +a +l +. +2 +0 +1 +8 +a +, +b +) +i +n +a +P +y +t +h +o +n +s +c +r +i +p +t +f +o +l +l +o +w +i +n +g +t +h +e +TESS Archive Manual +6 . The TESS data +information for these three stars is listed in Table 1. +There are two kinds of SPOC-extracted light curves: +one is Simple Aperture Photometry flux (SAP flux), +which is the flux after summing the calibrated pixels +within the TESS optimal photometric aperture, and the +other is Pre-search Data Conditioned Simple Aperture +Photometry (PDCSAP flux), it is the SAP flux from which +long-term trends have been removed using so-called Co- +trending Basis Vectors (CBVs) and nominally corrected +for instrumental variations and excessive scattered light +removed. PDCSAP flux is usually cleaner data than the +SAP flux and will have fewer systematic trends. Thus +PDCSAP flux is widely used for final analysis without +further processing (e.g. Dumusque et al. 2019; Astudillo- +Defru et al. 2020; Demory et al. 2020; Battley et al. +2021). However, the PDCSAP flux might suffer from +loss or wrong-deletion of long-term and transient burst +variability intrinsic to stars, for instance, as that pointed +out by Hill et al. 2022; Littlefield et al. 2021). With these +concerns, SAP flux is used by some authors (e.g.von Essen +et al. 2020; Steindl et al. 2021; Hon et al. 2021; Prˇsa +et al. 2022; Southworth & Van Reeth 2022) accompanied +with additional custom processing such as detrendings +depending on science goals. +In the current work, we first did graphic screenings on +each sector’s light curves for choosing SAP or PDCSAP +flux. A close visual inspection of both light curves plotted +up and down and overlapping helped the author compare +and judge which one is better for undergoing pulsation +detection. If the SAP flux looks no evident difference +with PDCSAP flux, then use SAP flux. If the PDCSAP +flux data lost too many points while the SAP flux data +are flat, then we choose SAP flux, otherwise, PCDSAP +is used. In case of severe systematic trends (slope and +steep brightness down or up in most cases), PDCSAP +flux is chosen. In any SAP or PDCSAP flux, all data +points flagged as ”bad quality” were removed. During +a course of surveying a large amount of TESS light +curves the author found that both PDCSAP and SAP +fluxes could be sometimes suffered from a systematic +shift in brightness between two orbits of a TESS sector, +which would result in abnormal discontinuity and actual +displacement of two orbits’ light curves. This is quite +frequently occurred in HLSP-QLP (uncorrected SAP) +light curves for long-period variables, for instance, Mira +variables TIC 329891910, TIC 457021714, and TIC +359401246 on sector 38. Displacements of a segment light +curve also occurred within individual orbits, e.g. RRLyr +star TIC 127088233 on sector 38 and so on. Thus, a +custom processing of correction to the displacement is +6 The Beginner Tutorial Notebooks: https://outerspace.stsci.edu/ +display/TESS/TESS+Archive+Manual; https://github.com/spacetelescope/ +notebooks/blob/master/notebooks/MAST/TESS/beginner how to use lc/ +beginner how to use lc.ipynb + +4 +A.-Y. Zhou +applied whenever needed. With the above criteria, these +final used fluxes of PDCSAP and SAP mixture are ensured +the TESS best estimate of the intrinsic variability of a +target. One thing was kept in mind that low-frequency +instrumental systematics, which is either not removed by +the TESS science pipeline, or introduced by the TESS +reduction pipeline (Cunha et al. 2019). Caution will +be taken in picking up a frequency when strong low- +frequency instrumental noise is present. The final derived +periods have ejected all possible aliases. +We then checked the two comparison stars in General +Catalog of Variable Stars (GCVS database, Version 2022 +Jun. and 5.1, Samus et al. 2017) +7 and Revised Version +of the New Catalogue of Suspected Variable Stars (NSV +Release 2, 1982, 1998, Kazarovets et al. 2022) +8 , and +found that C2=HD 53349 =HR 2661 =NSV 3349 is a +dwarf (DM, spectral type F0V) and a new suspected +variable star but without further information. We further +looked for TESS Input Catalog (TIC) v8.2 and CTL v8.01 +catalog (Stassun et al. 2018, 2019; Paegert et al. 2021), and +adopted the basic astronomical parameters in Table 2. No +variability type is provided in TIC. +Table 1 TESS observations (at two-minute cadence) of +HD 52788, HD 53166 and HD 53349. +Star Identifiers +V +Sp. +V =HD 52788 = TIC 279361762 +8.m37 +Fm dD +C1=HD 53166 = TIC 279431011 +8.m1 +A1V +C2=HD 53349 = TIC 279476396 +6.m0 +A8III +Orbits; Sector; (Camera,CCD) +11–25, 61–85;2–10,12,13, +27–39;(4,4),(4,1),(4,3),(4,2) +S0036 started on 2021 Mar 1 +Time span +BJD 2458354.107–2459389.716 +(2018 Aug 22 23:50 – 2021 May 26 02:43 UT, 1035.6 days) +Data length +24 sectors * 13.7 days * 2 += 657.599 days +Frequency resolution +0.0009656 d−1 +Number of data points analyzed +419,915 +(outliers and unusual data points excluded) +Total number of data +422,046 +Last, we proceed to process the TESS data. In a +few cases, we used fluxes counts directly. However, +for comparison with literature data, we prefer to use +magnitude. Using pixels counts, the fluxes (SAP Flux and +PDCSAP Flux), corresponding magnitudes (SAP mag +and PDCSAP mag) are calculated using the following for- +mula PDCSAP mag = −2.5 log(PDCSAP Flux) + +20.4436. The magnitude zero point of 20.4436 mag was +selected for a resulting magnitude matching with the value +in existing catalogs (Simbad/CDS) and especially with +Tmag of TESS CTL v8.01 or TIC v8.2. By arbitrarily +selecting a star, TIC 41196013 (Tmag=11.m4745) which +was observed in TESS sector 41 as non-variables in a time +scale of days for calibrating the magnitude zero point, +7 online URL: http://www.sai.msu.su/gcvs/cgi-bin/search.htm#cat +8 https://heasarc.gsfc.nasa.gov/W3Browse/all/gcvsnsvars.html +Table 2 +Astronomical Parameters of HD 52788, HD +53166 and HD 53349. +Para. +V=HD 52788 +C1=HD 53166 +C2=HD 53349 +B +8.788±0.028 +8.27±0.027 +6.283±0.023 +V +8.39±0.03 +8.15±0.03 +6.01±0.03 +J +7.683±0.024 +8.07±0.026 +5.422±0.018 +H +7.521±0.027 +8.045±0.024 +5.296±0.023 +K +7.473±0.029 +8.006±0.023 +5.245±0.016 +Tmag +8.0396 +8.17 +5.722 +Sp. +Fm dD +A1V +A8III +Teff +6838±134.87 +9562±186.18 +7118.48±111.54 +log g +3.319±0.092 +4.21±0.06 +4.039±0.079 +R/R⊙ +4.412±0.216 +2.026±0.053 +1.988±0.069 +M/M⊙ +1.48±0.25 +2.43±0.32 +1.59±0.25 +and using the average magnitude in a sector or an orbit, +we found 20.4436 and 20.2531 mag for converting SAP +and PDCSAP fluxes into magnitudes, respectively. The +data in a TESS sector which consists of two successive +orbits observations (each ∼13.7 days) have sometimes +slightly different mean values or zero-point offsets, so +the average of each orbit’s magnitudes or fluxes was +subtracted individually to obtain a flattened final set of +data with unique zero mean or zero-offset in order for +Fourier analysis and least-squares fitting. Such flattening +at a uniform level by subtracting individual means of each +part was applied whenever necessary. A similar aligning- +up also applied to the uneven cases from sector to sector. +Therefore light curves between two orbits and among +sectors are always lined up before doing Fourier analysis. +This adjustment of magnitude/flux zero points ensures the +elimination of externally-entailed additional noise in the +low-frequency domain. Concerning no atmospheric effect +on the TESS photometry, it is not necessary to establish +differential magnitude as in traditional ground astronomy. +Thus the above-calculated magnitudes were used in our +analyses, this way also allows us to diagnose each star +without effects brought in by comparison stars. +4 SAMPLE OF TARGETS +Besides the initial target HD 52788 and its two comparison +stars, an extended small sample of 50 stars were selected +in following categories (refer to Table 43): +– Poorly studied,neglected known pulsating variable +stars and eclipsing binaries. For those with a few +identifiers and linked references in Simbad/CDS, we +annotate them ’plain stars’ — stars mostly with object +types of ’* and IR’ in Simbad/CDS, the author’s +notation hereafter; +– Stars well studied and highly cited, but still not +completely clear for its pulsation or light variations; +– The stars that the author personally observed but not +well characterized together with their neighborhoods; +– Photometric standard stars. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +5 +5 CATALOGS FOR CHECKING NEW VARIABLES +For a given star with one of its identifiers, we first +check it in popular existing catalogs for astronomical +parameters and variability properties. Upon a finding of +light variations, one needs to know whether the result +is already reported in a published article, on the web +pages of any known sky surveys or not ever, and make +sure that the discovery of variability is not yet listed +in existing catalogs. However, the situation of checking +whether a suspected variable star was already reported in +a catalog or not, it is just a recovered variable, becomes +more challenging and more time-consuming than years +ago due to the vast outputs of various surveys in the past +two decades. As various-purpose sky surveys continue +to evolve, the amount of newly discovered variable stars +including eclipsing binary stars databases are increasing +rapidly, some still need to be sifted through to identify a +class of variabilities. The author attempted to find a com- +plementary way for checking a star whose variability type +is not classified in Simbad/VizieR/CDS. After a survey of +various sources (catalogs and websites) providing online +databases of variables, especially those with dynamic +updating, we established that for identifying variability +types and classifications of newly discovered variables, a +safe checking procedure should be first conducted against +those well-known primary databases and catalogs of +Simbad/VizieR at CDS, GCVC, the International Variable +Stars Index (VSX) database of AAVSO (Watson et al. +2022, 2006) +9 , key surveys including ASAS-SN Catalog +of Variable Stars X (Christy et al. 2022) +10 , OGLE- +IV (Soszy´nski et al. 2021; Pietrukowicz et al. 2020) +11 , +ZTF (Bellm et al. 2019; Masci et al. 2019; Ofek et al. +2020; Chen et al. 2020) +12 , SuperWASP (Pollacco et al. +2006; McMaster et al. 2021) +13 , etc., see Table 3. +First, we scan all variable stars which have been +reported or cataloged in literature and websites, then we +focus on δ Sct stars mainly. How many δ Sct stars now? +What is the number of pulsators in binary systems reported +so far? These are two tough questions in the author’s +mind all the time. Finding answers to these questions may +be beyond the scope of the present work. In the earlier +works (Zhou 2010, 2015) the author initiated a statistical +exploration on pulsating binaries and hybrid pulsators, one +of the results is a web-version catalog consisting of 4697 δ +Sct stars extracted from Simbad/CDS as of the time 2014 +October 26. As an update, the author has made an effort +to combine the known δ Sct stars based on the existing +catalogs. +9 https://www.aavso.org/vsx/ +10 Christy +et +al., +2022, +MNRAS, +in +press +:https://asas- +sn.osu.edu/variables +11 https://ogledb.astrouw.edu.pl/ ogle/OCVS/ +12 Zwicky Transient Facility: https://www.ztf.caltech.edu/ +13 https://www.superwasp.org/vespa/; https://www.zooniverse.org/ +projects/ajnorton/superwasp-variable-stars/about/research +Meanwhile, several surveys operated in the past +two decades served as DSCT secondary sources were +also checked: (1) ASAS-3 (Pojmanski 2002; Pojmanski +et al. 2005) +14 . (2) SuperWASP did not classify it +discovered pulsators into different types. By plotting +them in the Hertzsprung-Russell diagrams using the +parameters extracted from Gaia DR3 and TIC, luminosity, +effective temperature and surface gravity, along with data +from LAMOST, compared with those known DSCT, we +would sift a good number of δ Sct candidates from +the SuperWASP catalog of 24667 pulsators. (3) Pan- +STARRS (Kaiser et al. 2010) +15 +surveys all the sky +north of declination 47.5◦, about three-quarters of the +entire sky on a short duty cycle of days, down to 23 +magnitude, it had discovered numerous Cepheid variables +and eclipsing binary stars. An extremely large number +of variable stars are expected from Pan-STARRS DR2. +(4) ROTSE +16 , (5) WISE/AllWISE (Wright et al. 2010) +17 , +(6) NSVS (Wo´zniak et al. 2004) +18 , (7) CRTS/CSS +19 , (8) +WFCAM +20 , etc. +On the other hand, the catalog of Chang et al. +(2013) was carefully checked and found that multiple +stars cross-matching failed. No stars within 2′ radius, at +23:51:24 −25:45:00, the nearest star is BY Scl (SX Phe- +type) apart from 120′′; The star [KPA2010] 36 (22:01:08 ++24:44:33, B=13.m42, V=13.m01) and HD 5076 (00:52:40 ++06:39:55, [KPA2010] 1) (Kim et al. 2010) compiled in +the table of Chang et al. (2013) without discovery source +ID; No source ID for the star at 23:43:00 −29:52:00 +(V=13.m51), and within 67′′ radius no star in Simbad, +in 24′′ no star in TIC, can be identified, and others. +At 16:27:51 −49:07:36 in 9.39′′ identified to be Gaia +DR3 5941411883325639296 (G=16.m23). At 19:38:06 ++30:51:54, nearest star is [MPC98] V798 8, but as a DSCT +it should be identified to be V2116 Cyg, 17′′ apart. At +19:53:46 +18:46:42 is PSR J1953+1846A, but it should be +identified to NGC 6838 1084, 4.21′′ apart, multiple stars +share the same coordinats. Some duplicate coordinates +with no source identifiers for variables in stellar clusters +such as NGCA 288, 2099, 3201, 4590, 5139, 6809, etc. +MACHO and OGLE ID were not provided either. In +a few cases, we can not simply take the nearest star +14 1275+2263 rows/DSCT from The All Sky Automated Survey +(ASAS): +h +t +t +p +: +/ +/ +w +w +w +. +a +s +t +r +o +u +w +. +e +d +u +. +p +l +/ +a +s +a +s +/ +? +p +a +g +e += +a +s +a +s +3 +15 The Panoramic Survey Telescope And Rapid Response System: +https://panstarrs.ifa.hawaii.edu/pswww/ +16 http://www.rotse.net/transients/; The Robotic Optical Transient +Search Experiment (ROTSE) is a multi-telescope experiment designed +to observe the optical afterglow of gamma-ray bursts. Most ROTSE- +discovered variable stars have been reported in VSX. +17 The +Wide-field +Infrared +Survey +Explorer +mission: +https://wise2.ipac.caltech.edu/docs/release/allwise/ +18 Northern +Sky +Variability +Survey +(NSVS): +http://skydot.lanl.gov/nsvs/nsvs.php +19 The Catalina Real-Time Transient Survey, Catalina Sky Survey +(CSS): http://crts.caltech.edu/ +20 The Wide Field Camera (WFCAM) on UKIRT – The United +Kingdom Infrared Telescope (a 3.8m IR-dedicated telescope, Mauna +Kea, Hawaii): http://wsa.roe.ac.uk/ + +6 +A.-Y. Zhou +Table 3 Primary Catalogs for Checking Existence of a +New Variable Star†. +Project/Catalog VarType +Number of Stars +Notes +GSC 2.4.2 * +3,485,671,481 +2020 +pan-STARAS * +1,919,106,885 +2016 +Gaia DR3 * +1,811,709,771 +2022 +TIC v8.2 * +1,727,987,580 +2021 +Gaia DR2 * +1,692,919,135 +2018 +USNO-B 1.0 * +1,045,913,669 +2003 +GSC 2.3.2 * +945,592,683 +2006 +AllWISE * +747,634,026 +2013.11.13 +WISE * +563,921,584 +2012.03.14 +USNO-A 2.0 * +526,280,881 +1998 +2MASS * +470,992,970 +2003 +GSC 2.2.1 * +455,851,237 +2001 +UCAC4 * +113,780,093 +2012 +APASS * +61,176,401 +2022.05 +LAMOST DR9 * +10,907,516 +2021.06 +LAMOST DR8 * +10,351,254 +2020.05 +LAMOST DR7 * +9,846,793 +2019.06 +Simbad * +8,322,647 +2022.09.26 +LAMOST DR4 * +4,132,782 +2016.06 +VSX-AAVSO V* +2,234,703 +2022 Sep 24 +Simbad V* +1,299,113 +Variable stars +ASAS-SN V* +378,861 +of 687,696 stars +Simbad Pu* +227,130 +Pulsators +SuperWASP Pu* +24,667 +Pulsators +OGLE-V dS* +24,488 +2022.03.30 +ZTF dS* +16,709 +2020.07.17 +VSX-DSCT +15,308 +2022.09.10 +VSX-HADS +10,896 +2022.09.10 +Simbad dS* +9,287 +2022.09.26 +ASAS-SN DSCT +3,939 +2022.09 +ASAS-3 DSCT +3,538 +2022.09 +ASAS-SN HADS +2,231 +2022.09 +Zhou (2015) +4,697 +2015.01 +Chang et al. (2013) +1,578 +2013 +GCVS DSCT +1,018 +2022.09 V5.2 +Rodr´ıguez et al. (2000) +636 +2000 +Simbad SX* +609 +2022.09.26 +VSX-DSCTC +425 +2022.09.10 +VSX-HADSB +273 +2022.09.10 +VSX-SXPHE +292 +2022.09.10 +GCVS SXPHE +267 +2022.09.10 +ROTSE-I +61 +2022.09 +†: Simbad/CDS and GCVS Classification Labels Used: *: +general stars; V *: variable stars; dS*/DSCT/DSCTC: δ Sct +stars; Pu*: pulsating variable stars; SX*/SXPHE: SX Phe-type; +HADS/HADSB: high-amplitude δ Sct stars. +around coordinates but considering variability property +and use human inspection sometimes. Therefore, some +stars were double-checked manually through Simbad/CDS +and MAST Portal web queries. +After putting the δ Sct stars listed by Simbad, OGLE, +VSX, ASAS-SN, ZTF, GCVS, and other sources together +(see Table 3), we reached a sum of over 84,375 entries. +Last but not least, we added three dozen sporadic new +DSCTs picked up from recent publications based on ADS +and IBVS (e.g. Kirmizitas et al. 2022; Shi et al. 2021) and +18 new discoveries in present work. +Surely, a known variable might be compiled in several +catalogs such as GCVS, VSX, and Simbad, therefore there +are lots of duplications in the combination of multiple +source catalogs. A computer program was written to +check duplication and found more than 23,228 stars are +duplicated in the combined list of published catalogs, +for instance TIC 322378080 = OGLE GD-DSCT-6707 = +ZTFJ183900.91+074800.7, TIC 307185343 = OGLE GD- +DSCT-7318 = ZTFJ190220.03–040019.2, TIC 282617026 += OGLE GD-DSCT-0591 = ZTFJ065358.89–034623.2, +TIC 51835214 = ROTSE1 J223159.77+135641.9 = +ZTFJ223159.76+135641.8, TIC 57987993 = ROTSE1 +J002103.71+304215.2 += +ZTFJ002103.69+304215.0, +TIC +301909021 += +ZTFJ024113.04+485846.9 += +UCAC4 695-018090 , IBVS No.6300 (detected on +10 November 2015, so IBVS re-reported) DW Psc = +TIC 365157858 = ZTFJ013026.96+084133.5 =Gaia DR3 +2566925171965557632, in table of Chang et al. (2013), +and so on. Probably later catalogs (e.g. ZTF) could have +re-reported discoveries regarding their publication dates. +For GSC 0762-2924 (2017, IBVS Vol.63 No.6300), the +Dec value suffered from a typo error, it should be 07:49:01 +rather than 07:40:01. +Duplication largely attributes to combining the col- +lection by Simbad and VSX, no attempt was made to +distinguish whether the duplications are re-reported as +new discoveries among multiple surveys, but for those +duplicate entries, one of the sources is imported into the +new catalog in the order of ID-selection priority: GCVS +designations, the original discoverer used ID, Simbad +main ID (HR, HD, 2MASS, etc.), OGLE, ASAS, ZTF, +and other surveys/project names. By and large, the actual +origin of variability discoverers and identification should +be reflected in source identifiers. +Then the duplicated entries were removed, last form +an up-to-date comprehensive catalog of δ Sct stars that +consists of 59,350 individual stars. The number is almost +93 times of 636 DSCT in the catalog of Rodr´ıguez +et al. (2000) and 37 times 1578 DSCT from Chang +et al. (2013). This new catalog has both TIC v8.2 +and Gaia DR3 identifiers cross-matched with source +ID within a distance radius of 5.5 arcseconds, a few +cases with larger separation from given coordinates up +to 2′. The present catalog of δ Sct stars focused on +cross-identifications and parameters from the two space +projects, such that did not gather all stellar parameters +available in source catalogs but include four identifiers (the +source catalog ID, TIC, Gaia DR3 and 2MASS/AllWISE +or coordinates); parallax, distance, radial velocity, Gaia +photometric magnitudes (G, BP, RP, BP −RP, G−RP) +extracted from Gaia DR3; stellar atmospheric parameters +effective temperature, luminosity, surface gravity, and +radius retrieved from both TIC v8.2 and Gaia DR3, +magnitudes in B and V and mass from TIC v8.2, and +absolute magnitude, which was calculated using Gaia +distance directly according to definition MV += V + +5.0 − 5 ∗ log(d). It is known that distance is not +simply the inversion of the observed parallax, though +they are reciprocal in the definition. The naive approach +of inverting parallax is a blind use and is just a biased +estimator as noted in Gaia DR3 data release (Gaia + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +7 +C +o +l +l +a +b +o +r +a +t +i +o +n +e +t +a +l +. +2 +0 +2 +2 +, +2 +0 +1 +6 +) +. +T +h +e +r +e +l +i +a +b +i +l +i +t +y +o +f +t +h +e +c +a +t +a +l +o +g +w +a +s +a +c +h +i +e +v +e +d +b +y +t +h +r +e +e +f +a +c +t +o +r +s +: +( +1 +) +e +x +t +r +a +c +t +i +n +g +s +o +u +r +c +e +s +g +i +v +e +n +i +n +r +e +f +e +r +e +e +d +p +u +b +l +i +c +a +t +i +o +n +s +a +n +d +w +e +l +l +- +r +e +c +o +g +n +i +s +e +d +s +u +r +v +e +y +s +; +( +2 +) +c +r +o +s +s +- +m +a +t +c +h +i +n +g +b +y +T +I +C +a +n +d +Gaia DR3 +identifiers; (3) duplicate checking using TIC, Gaia ID, +and coordinates by a computer program. The advantage +of the catalog is cross-identifying all source IDs with the +two largest catalogs of celestial objects TIC and Gaia DR3 +along with up-to-date parameters. +Nevertheless, there must be some stars missing. +Further updates will be complemented in the next version +of the catalog with the strategy of automatedly dynamical +updating +21 . Statistics of contribution from various sources +are shown in Figure 1. +Fig. 1 Source Contributions to the 59350 δ Sct stars. +Each entry in the catalog table is comprised of +more than 26 columns: TIC, source ID, Gaia DR3 ID, +RA Dec/2MASS ID, B TIC, V TIC, Teff TIC, Teff Gaia, +log g TIC, log g Gaia, Mass TIC, R TIC, R Gaia, L TIC, +L Gaia, parallax(mas), distance(pc), MV , log Teff TIC, +log L TIC, G mag, BP mag, RP mag, BP − RP, +BP − G, G − RP, Radial Velocity Gaia and errors +for a few parameters. It is inconvenient to illustrate +such a wide catalog table, it is too wide to be printed +in A4-paper layout. Table 4 demonstrated a simplified +version of the table. The whole catalog in its entirety is +provided as online materials in both machine-readable text +format (CSV file) and human-friendly HTML version +22 . +Interested readers may enquire it from the author. More +importantly, in the HTML version, each TIC ID is +hyperlinked to STScI archive MAST in a default radius of +0.00033◦ of the target, each Gaia DR3 ID with hyperlink +to Gaia data at VizieR/CDS, and main IDs or coordinates +are linked to CDS Portal. The HTML version makes +21 Automation is imperatively needed. Since the day commenced +retrieving Simbad, the number of DSCTs in Simbad has increased by +585, from 8702 to 9287 by the time of writing the manuscript. The current +version used source data as of September 26, 2022 +22 https://deltascuti.wixsite.com/delta/dsct-catalog +consulting, checking and downloading TESS/Kepler/K2 +and Gaia DR3 data unprecedentedly convenient and +efficient. It is useful for follow-up. +Recalling the release dates interval of TIC v8.2 and +Gaia DR3, on September 2021 and 13 June 2022, respec- +tively, there are discrepancies between TIC v8.2 (where +parallax adopted from Gaia DR2 by the time of this writ- +ing) and Gaia DR3 for several parameters, for instance, +TIC 213014556 = Gaia DR3 2524140189527507968, +TIC v8.2 gives plx=0.127042±0.048995, Teff=6558±247 +(DR2: 6529.3335), log g=3.82235, d=4063.85±630.11, +whereas DR3 updated them to plx=0.075798, d=2618 pc, +Teff=6318, log g=4.01. We first queried TIC v8.2 with +a Python program, which was then modified to retrieve +parameters from Gaia DR3 archive. An additional work +in progress is adding results of LAMOST (spectral type, +effective temperature, gravity and metallicity) and other +stellar parameters, as reported for about 525 of 766 +LAMOST-observed δ Sct variable stars by Qian et al. +(2018), who checked among 3,689 known δ Sct stars +in VSX. Now VSX has indexed more than 27,194 +DSCT, LAMOST should have observed more DSCT since +then (Qian et al. 2019). Moreover, new DSCT from TESS, +K2 and Gaia will be particularly gathered from a series of +publications, for examples several in progress: Shi et al. +2022; Kahraman Alic¸avus¸ et al. 2022; Fetherolf et al. +2022; Kahraman Alic¸avus¸ et al. 2022; Gaia Collaboration +et al. 2021, 2019). Benefited from the TIC v8.2 and Gaia +DR3 identifiers being cross-matched with source IDs in +the new catalog, observers are able to easily examine a star +manually or a number of stars programmatically. With the +catalog, it is thus convenient to draw our new discoveries +against the known ones in various parameters spaces (see +Figs.2-6). +When retrieving parameters from TIC v8.2, some +fainter stars were ignored. A few querying program +exceptions are: +– Eight δ Sct stars [MHN98] V1, V2,V3,V6–8,V11 and +V19 were ignored due to fainter than V =22.m5 and no +IDs of either TIC or Gaia DR3. +– V4317 Sgr was listed in R00 but no any other ID given +in Simbad (not in TIC and Gaia DR3 either), no color +magnitude info. +– [KPA2010] 36, B=13.m42, V=13.m01 only ID should +correspond to TIC 283408312, Tmag=13.m13 +– [BMG2010] V046, V055–V159 are fainter than +V =26.m0 were ignored. +– [VM2013] DC-1, DC-10 – DC-99, 89 stars fainter +than V =22.m0 were ignored. +– Cl* NGC 7245 VHB 456 – no coord in Simbad except +V =14.m7: TIC query could not resolve the ID to a sky +position +As querying Gaia DR3 over such number of stars, we +experienced a few failure cases and solved with human +intervention, reported below: + +SE +52'800 +4e'S +Js'eo +OCE +2A2A +.20 +400 +J0'α0 +22AMS +Voso +1AC +EbIC +OfUGL2(CBI2'NCVC+'AWS0I3'COBOI*..")8 +A.-Y. Zhou +– TIC 63371156 = BD+41 3376: 2 stars within +1′′ +sky +region, +identified +to +be +Gaia +DR3 +2077694728707664128. +– TIC +122717173 += +TYC +3134-1328-1: +2 +stars +within 1′′ sky region, identified to be Gaia DR3 +2052844254090672640. Actually no more data except +G and BP RP. +– OGLE LMC-DSCT-164 = TIC 295070037: no Gaia +DR3 designation star within 3′′ sky region +– TIC 153264713 = OGLE-GD-DSCT-1418: 2 stars in +1′′ circle, discriminated the variable to be Gaia DR3 +5698385763057621376. +– alf Lyr: has four TIC IDs: TIC 471012052, TIC +70257116, TIC 471012017, TIC 157587146, no Gaia +DR3 ID +– bet Leo: =TIC 14725877, no Gaia DR3 identifier; +– V2988 Cyg: not resolved by Sesame Strasbourg, Gaia +DR3 query cross-identified its TIC 278355617 to DR3 +2180055374306851968. +– CoRoT 101436549 = TIC 223887633 = UCAC4 453- +099296, no coordinates, 2 stars in 1′′ region, cross- +identified DR3 4287505636453818752. +– TIC 319438599 = [VM2013] DC-86: no Gaia DR3 +designation star within 3′′ sky region +– TIC 82359935 = V1233 Her (EB) = ROTSE1 +J165852.87+391421.7: 2 DR3 designations in 1′′ sky +region, to be Gaia DR3 1352082539737123840. +Finally, we show three plots of the collected 59350 +δ stars in multiple observational Hertzsprung-Russell +diagrams together with our newly discovered variable stars +in Figures 2-6. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +9 +Table 4 A Combined Catalogs of 59,350 δ Sct Stars (reduced demo version). +Table 4 A Combined Catalogs of 59,350 δ Sct Stars (reduced demo version). +TIC +Source ID +Gaia DR3 +2MASS/RA Dec +Bmag +V mag +Teff +log g +M/M⊙ +R/R⊙ +Luminosity +Parallax(mas) +Distance(pc) +MV +log Teff +log L/L⊙ +TIC 273871163 +J19525891+4636506 +2085540087774214656 +J19525891+4636506 +14.4 +13.944 +7021.38 +3.81715 +1.55 +2.54464 +14.1786737 +0.444449 +2113.5 +2.319 +3.846 +1.152 +TIC 273875324 +J19530663+4736579 +2086440961449151488 +J19530663+4736579 +13.731 +13.6 +7318.0 +3.53668 +1.66 +3.63703 +34.17894 +0.296576 +3070.58 +1.164 +3.864 +1.534 +TIC 273875361 +J19531465+4736026 +2086440171175169664 +J19531465+4736026 +11.974 +11.856 +7692.0 +3.69583 +1.81 +3.16197 +31.5330982 +0.705406 +1363.86 +1.182 +3.886 +1.499 +TIC 274019642 +TYC 3558-2397-1 +2079453706798300416 +J19532728+4543155 +12.126 +11.967 +8629.0 +4.23425 +2.16 +1.85838 +17.25065 +1.04374 +932.891 +2.118 +3.936 +1.237 +TIC 274021225 +J19532329+4511434 +2079401681853657088 +J19532329+4511434 +12.532 +11.92 +7088.0 +3.75867 +1.57 +2.73937 +17.0643559 +1.0732 +908.1 +2.129 +3.851 +1.232 +TIC 274023013 +J19530742+4438086 +2079290974772684672 +J19530742+4438086 +13.251 +12.628 +7319.0 +4.13157 +1.66 +1.8336 +8.691827 +1.01207 +961.389 +2.714 +3.864 +0.939 +TIC 274023141 +HD 188524 +2079102412837482752 +J19534271+4435426 +10.842 +10.578 +8852.0 +3.98402 +2.23 +2.5187 +35.0922852 +1.41195 +697.431 +1.36 +3.947 +1.545 +TIC 274118915 +J19540320+4205467 +2075439355495147008 +J19540320+4205467 +13.831 +12.966 +7654.0 +3.77826 +1.79 +2.85977 +25.2878036 +0.514102 +1845.35 +1.636 +3.884 +1.403 +TIC 274119690 +HD 226454 +2075445570294475520 +J19540498+4217299 +10.796 +10.506 +7814.0 +3.72119 +1.86 +3.11312 +32.5520973 +1.38594 +707.123 +1.259 +3.893 +1.513 +TIC 274120294 +TYC 3145-1509-1 +2078455345942029952 +J19535176+4226365 +12.149 +12.142 +7431.0 +4.07847 +1.7 +1.97253 +10.6888838 +1.17078 +834.049 +2.536 +3.871 +1.029 +TIC 274120769 +HD 188611A +2075455534618595840 +J19541213+4233399 +9.306 +9.092 +8200.0 +3.6578 +2.01 +3.48122 +49.36443 +2.04321 +482.825 +0.673 +3.914 +1.693 +TIC 274126038 +UCAC4 670-080618 +2079047987010137984 +J19541191+4359198 +14.413 +13.846 +7738.49 +3.84405 +1.82 +2.67329 +23.089325 +0.42208 +2219.73 +2.114 +3.889 +1.363 +TIC 274126342 +UCAC4 671-079817 +2079051766581362560 +J19535326+4405088 +14.226 +13.543 +7260.7 +4.04326 +1.64 +2.01755 +10.1919632 +0.641725 +1491.93 +2.674 +3.861 +1.008 +TIC 274126712 +J19535544+4411490 +2079059016486241280 +J19535544+4411490 +13.63 +13.039 +7283.46 +4.05798 +1.64 +1.98366 +9.976589 +0.81746 +1182.82 +2.674 +3.862 +0.999 +TIC 274126864 +J19540707+4414453 +2079058810327878912 +J19540707+4414453 +13.412 +12.982 +8220.0 +4.19663 +2.01 +1.87205 +14.4150085 +0.793595 +1217.32 +2.555 +3.915 +1.159 +TIC 274127939 +TYC 3149-863-1 +2079100145094706816 +J19542256+4434199 +10.976 +10.796 +9143.0 +4.02359 +2.31 +2.44932 +37.76949 +1.24426 +786.457 +1.318 +3.961 +1.577 +TIC 274128600 +UCAC4 674-078010 +2079108288351818496 +J19541545+4447053 +15.019 +14.466 +7540.99 +4.16597 +1.74 +1.80436 +9.485419 +0.498925 +1896.32 +3.076 +3.877 +0.977 +TIC 274129089 +UCAC4 675-078916 +2079302072968160384 +J19534759+4456521 +12.803 +12.242 +7521.0 +4.23489 +1.74 +1.66672 +8.007922 +1.41334 +693.529 +3.037 +3.876 +0.904 +TIC 274129832 +J19535272+4512358 +2079400105605767936 +J19535272+4512358 +14.578 +13.944 +7419.32 +3.95293 +1.7 +2.27925 +14.18192 +0.497181 +1902.56 +2.547 +3.87 +1.152 +TIC 274130413 +BD+45 3006 +2079410379167694720 +J19540713+4524067 +10.918 +10.514 +7384.48 +3.40273 +1.68 +4.26896 +48.82244 +1.24938 +782.775 +1.046 +3.868 +1.689 +TIC 274508912 +ROTSE1 J165536.79+522244.6 +1413637534282804352 +J16553687+5222438 +13.881 +13.283 +6272.0 +4.31566 +1.22 +1.2717 +2.25470519 +1.29345 +756.22 +3.89 +3.797 +0.353 +TIC 274510661 +V* V927 Her +1412476759241299584 +J16561799+5007358 +10.451 +10.08 +7062.0 +3.60602 +1.56 +3.25528 +23.7455215 +1.76967 +556.056 +1.354 +3.849 +1.376 +TIC 274674793 +V* MR Dra +1419000505326243584 +J17042563+5249065 +8.451 +8.2 +7768.0 +3.95776 +1.837 +2.35619 +18.2117558 +4.87496 +203.929 +1.653 +3.89 +1.26 +TIC 275487196 +J19322594+4734242 +2128439939109963136 +J19322594+4734242 +13.62 +13.098 +7113.71 +4.00421 +1.58 +2.07137 +9.899067 +0.707841 +1357.86 +2.434 +3.852 +0.996 +TIC 275488065 +TYC 3560-2538-1 +2128387708006918144 +J19322537+4658286 +12.423 +12.451 +8364.15 +3.75954 +2.07 +3.14232 +43.5393677 +0.530723 +1787.25 +1.19 +3.922 +1.639 +TIC 275488154 +TYC 3560-2590-1 +2128381248376108288 +J19321423+4654209 +11.048 +10.805 +7216.35 +3.97183 +1.62 +2.1771 +11.5803051 +1.99568 +494.026 +2.336 +3.858 +1.064 +TIC 275488593 +BD+46 2714 +2128282915105870720 +J19322177+4635297 +10.188 +10.018 +7859.06 +nan +nan +nan +nan +nan +383.595 +2.099 +3.895 +0 +TIC 275488733 +J19321828+4629529 +2128277829864563712 +J19321828+4629529 +13.298 +12.873 +7517.36 +4.29032 +1.74 +1.56368 +7.0348 +0.870377 +1112.61 +2.641 +3.876 +0.847 +TIC 275488931 +BOKS 691 +2128270545599999872 +J19322728+4622304 +17.311 +17.2455 +5574.0 +4.77915 +0.98 +0.668455 +0.38860333 +0.550827 +1733.78 +6.051 +3.746 +-0.41 +TIC 275489178 +BOKS 413 +2128079986492339072 +J19322519+4612391 +18.185 +16.492 +3825.0 +4.67687 +0.563 +0.569851 +0.06262433 +2.77375 +356.905 +8.729 +3.583 +-1.203 +TIC 275493853 +HD 184449 +2125886971888144000 +J19323912+4326112 +9.062 +8.864 +8367.0 +3.80714 +2.068 +2.97331 +39.0349464 +2.46155 +401.769 +0.844 +3.923 +1.591 +TIC 275495261 +TYC 3143-951-1 +2077805672010904448 +J19322836+4241403 +12.617 +11.595 +7553.15 +nan +nan +1.84435 +nan +nan +687.301 +2.409 +3.878 +0 +TIC 275496829 +TYC 3143-1362-1 +2077673597472239104 +J19323593+4153323 +11.409 +11.234 +7323.0 +4.05215 +1.66 +2.00916 +10.4587107 +1.57316 +624.388 +2.257 +3.865 +1.019 + +10 +A.-Y. Zhou +6 FREQUENCY ANALYSIS +Fourier analysis was carried out for each data set by using +PERIOD04 (Lenz & Breger 2005). The results are summa- +rized in frequency solution tables for each star. However, +it is well known that stellar activities may not be periodic, +and even irregular. Periodic signals including pulsation are +not strictly sinusoidal. We analyze pulsation contents by +resolving light curves into a sum of multiple sine or cosine +waveforms with different frequencies and amplitudes. +Ground astronomy has various sampling intervals and +gaps without data in time-space. Uninterrupted space +observations never mean no intervals of observation. TESS +exposed every two seconds, HDU stacked images into +groups of 2-minute cadence and 30-minute cadence for +different science goals. together with data-downloading +periods and orbits or sectors switch, these are actual TESS +data sampling intervals involved. Because the observations +even from space are still true discrete sampling in time- +space, aliasing, as a data acquisition effect caused by +sampling intervals is not yet completely eliminated. A +discussion on aliasing involved in multiple sites observing +campaigns can be referred to Breger et al. (2005). In +this reference, 75+ frequencies for FG Virginis were +resolved using extensive photometric multisite campaign +data. Guzik et al. (2022) has re-visited FG Vir using +Kepler K2 and TESS data and found around 30 significant +frequencies in the K2 data, and more than 100 significant +frequencies in the TESS data. With TESS data, this number +75 will be easily overpassed as shown below. +Here we explored the TESS data aliasing for keeping +in mind when picking up a peak frequency. The gaps in +the middle of each sector’s time series are due to the +data downlink separating the two physical orbits within +each TESS sector. The points showing severe residual +uncorrected systematics in the fits are removed in the +joint analysis presented in this paper, similar case as +Shporer et al. (2019). Under Nyquist sampling theorem, +frequencies less than half the sampling rate will not alias: +fmax = 1/2∆t. TESS observations’ sampling intervals +are 2 minutes and 30 minutes, that means the maximum +possible frequencies should be less than 360 and 24 +d−1 (i.e. the Nyquist frequency), respectively. However, +besides the regular sampling intervals, there are additional +time gaps in a data set which was finally used in frequency +analysis. We know the single-site ground photometry +suffered from daily aliases, an effect of daytime without +observing and almost started observation 24 hours late at +the same time the next night. There are additional time +gaps in TESS data between two consecutive orbits in a +sector as well as between two successive sectors. +In the case of ι Boo, current available TESS data +are from sectors 22,23,49 and 50. This set of data +spanned 792.19 days, starting on BJD 2458899.3216408 +to 2459691.51163725 (i.e. between 2020.02.19 19:43 and +2022.04.22 00:16), the gap between two orbits in each +sector are 0.96665 days in sector 23, 0.90692 days in sec- +tor 50, and 1.33205 days while the gaps between the two +successive sectors are 1.6125 between 22–23, 0.956937 +between 49–50, which act as two arbitrary TESS sectors. +This would produce aliases around 1.0/(13.7+1.6126), +1.0/(13.7+0.96665) or 1.0/(13.7+1.33205), i.e. 0.065306, +0.066524 d−1 aliases, similar to the ground observations’ +daily aliases. As a composite effect, the spectral window +reflects the aliasing structure: in the current case, Figure 7 +depicts the aliases: two strong aliases at fa2=0.001384 and +fa1=0.025752 d−1. Nyquist sampling theorem indicates +that a frequency peak will be accompanied by the +combination side peaks at f ± n ∗ fa1 where each is +further slightly aliased to be at fa1 ± m ∗ fa2 (n,m are +integers). That is, f + 0.13788, 0.21092, 0.28292, 0.35526, +0.42622, 0.49787, and 0.56712 are the strongest among +others. Then, f + 0.002705, 0.004069, etc. +In addition, the frequency resolution is restricted by +the timebase length (T) of a dataset and is estimated +by ∆f = 1/T, which in current case T=792.19 days, +which means ∆f= 0.001262 d−1. However, actually, four +sectors, eight orbits, T ∼13.7*8 days) — effective time +span with data is only 13.8% of that observing dates +interval, i.e. 1/(8 ∗ 13.7) = 0.009124 d−1. +7 RESULTS +We report in this section our results for each analyzed +star in tables of frequency solution and in graphs of light +curves, periodograms and prewhitened residuals. Detailed +works on several of the stars are underway and will be +reported elsewhere. +7.1 HD 52788 = TIC 279361762 +TESS observed HD 52788 in total of 24 sectors across +2 to 39 over 1035 days from 2018 August 22 through +2021 May 26. Light curves are available as a source of +TESS-SPOC, HLSP-QLP, and HLSP-SPOC. The TESS- +SPOC 2-minute cadence SAP flux data were used. +We see the MIT HLSP-QLP team attempted resolving +the data with multiple planets fittings, but no final +positive results. The light curves exhibit a much more +complex structure visually. Frequency analysis shows that +the residual spectrum even with 130+ frequencies pre- +whitened is still full of peaks in the range of 5–15 d−1. +With successive prewhitenings and multiple frequencies +fittings, the residuals are improved slightly (see Table 5). +With a big number of parameters when simultaneous +optimization for frequencies, amplitudes, and phases, non- +linear least-squares fitting of PERIOD04 encountered a +calculation issue of ’matrix cannot be inverted’, which +failed to improve the three parameters of the fitting. So +we have to stop prewhitening procedure. We computed +noise levels based on both residuals and original data +for comparison of significant peaks. Current preliminary + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +11 +Fig. 2 Newly discovered pulsating variable stars of δ Sct and other types plotted contrast with the known 59350 +DSCT. Red squares refer to classical δ Sct domain: Teff in [6300, 9100], log Teff in [3.80, 3.96], log g in [3.2, 4.3], +log L/L⊙ in [0.65, 1.90] consistent with the H-R diagram of pulsators given by Handler (2009); Jeffery (2008); +Breger (2000); magenta dashed squares refer to an extended domain: Teff in [3800, 10000], log Teff in [3.58, 4.00], +log g in [3.05, 5.0], log L/L⊙ in [-1.15, 1.90], MV in [9.9, -0.95]. δ Sct domain is observationally much enlarged. +results are reported in Figure 9 and Table 7. Further study +is underway. +Table 5 Fitting Residuals with Successive Prewhitening. +Number of Frequencies +Fitting Residuals +Imporvement +12 +0.00423889 +– +32 +0.003242 +23.5% +79 +0.0030626 +5.5% +108 +0.0030232 +1.29% +120 +0.0029725 +1.68% +158 +0.0028472 +4.22% +169 +0.0028147 +1.14% +191 +0.0027301 +3.00% +7.2 C1=HD 53166 = TIC 279431011 +C1= HD 53166 (=TIC 279431011, V =8.m17, A1V, +RA=07:00:34.60 Dec=-58:23:36.27 J2000.0), its TESS +light curves are available in sectors 2–13 in 2-minute +cadence, and 24 sectors (2–13, 27–39) as HLSP- +QLP products in cadence of 30-minute and 10-minute, +respectively. With TESS data we reveal the star to be a new +δ Scuti-type pulsating variable star. Pulsation frequency +contents are given in Table 6. See Figure 10. +7.3 C2=HD 53349 = TIC 279476396 +C2= HD 53349 (=HR 2662 = TIC 279476396, V =6.m0, +A8III, RA=07:01:05.11 Dec=-58:56:23.77) is a High +Proper Motion Star. It was listed as NSV 3349 with +F0V spectral type at the International Variable Star Index +(VSX) of AAVSO but without identified variability type +23 . +Analyses based on all available TESS data during 24 +sectors 1 through 39 lead to multiple pulsational variation +of light with two dominant frequencies at f1=1.888139 +and f2=1.809503 d−1. See Figure 11 and Table 8. +23 https://www.aavso.org/vsx/index.php?view=search.top + +loas +4'S +4'0 +a.E +3'4 +4'S +4'0 +8.E +8.E +3'e +3'4 +0.E +3 +ИGM D2C+bn* +2 +2.E +-S +10.↓ +loarir +0 +p4 +0 +p +0.2 +2'2 +S +ИGM D2C+bN* +2a340 KU0MU D2C +0.a +sT +loaiGut +J2000J5200 +1200 +2000 +5200 +4'S +4'0 +8.E +a.E +3'4 +J200 +J0000 +V6M b* +ИGM D2CI+b* +0.21 +12'0 +ИGM D2CI +2a340 KU0MU Q 2Cf 2f92 +2ae340 ku0mu Q 2cf 2f912 +JS'2 +JS'2 +0.0r +JO'O +12 +1'2 +0.2 +0.己 +52 +5'2 +0.0 +0.0 +5'212 +A.-Y. Zhou +Fig. 3 Newly discovered δ Sct stars and other variables plotted contrast with the known 59350 DSCT. +Fig. 4 +Newly discovered δ Sct stars and other variables +plotted contrast with the known 59350 DSCT in Gaia color- +magnitude diagram. +Fig. 5 +New discoveries of δ Sct and other types compared +with Gaia stars. + +omIm +omIm +0.0 +0°2 +1'O +1'2 +s'O +5'2 +0.E +2.0 +1O +1'2 +s'O +5'2 +0.E +0.0 +3'0 +0 +3'2 +I - +4'0. +B +42 +0 +3 - +0.2 +4- +2'2 +И6M D2CI+bn* +2a340 KU0MU D2Cl +2 ++ +0.a +omIm +omIm +0.0 +2.0 +1' +1'2 +s'0 +s'2 +0.E +0.0 +2.0 +JO +1'2 +0.S +s'2 +0.E +0 +4000 +20 - +eooo +JO0 - +8000 +120 +J0000 +500 - +JS000 +520 - +J4000 +Jeooo +300 +ИGM D2CI+bn* +ИGM D2CI+bn* +2a340 KU0MU D2Cl +2a340 ku0mu Q 2Cf 2f912 +J8000 +320Bb - Bb (wga) +S +1S'2 +9 +6 +JO'O +1 9ulo2d6 +12 +0.2 + gbutinp6m +s2 +0.0 +(p6m) +S'2 +ИGM D2CI+bn* +0.2 +2a340 KU0M D2Cl() +0 +S +3 +4 +1↓'2 +J2'0 +JS'2 +9tulo2dA +JO'O +D +2.1 +0.2 +5'2 +0.0 +ИGM D2CI +5424aa C919 27912Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +13 +Fig. 6 59350 δ Sct stars compared with Gaia stars. +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Frequency (d−1) +0 +200 +400 +600 +800 +1000 +Normalized Amplitude (mmag) +fa2=0.025752 d−1 +fa2 + 2fa1 +fa1=0.001384 d−1 +4fa1 +Spec(ra Window +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +0 +200 +400 +600 +800 +1000 +Spec(ra window +A iasing s(r)c()re for TESS sec(ors 22,23,49,50 combina(ion +Fig. 7 Aliasing structure for the combination data of TESS sectors 22,23,49 and 50. + +(Bb - Bb) +(Bb - Bb) +3 +s +0 +丁 +0 +4 +e +J12 +J2 +J2'0 +Julo2dA +Julo2dA +JS'2 +JO +JO'O +gauifnqe +12 +2 +6 +0.2 +(wga) +(wg +0 +p +s'2 +0.0 +-2 +И6M D2CI +2a320 D2Cl +S424aa C919 2f912 +S424aa C919 2f91214 +A.-Y. Zhou +Fig. 8 Finder chart of HD 52788 and comparisons used in +Kurtz(1979,1981). +Table 6 +Frequency Solution of HD 53166 (=TIC +279431011) Based on 24 TESS Sectors 2–13 and 27–39. +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 0.859989 +0.000326 +0.264577 +19.8 +f1= 2.488853 +0.000307 +0.804333 +44.27 +f2= 4.977580 +0.000301 +0.296669 +78.67 +f2 =2f1± 0.000127 +0.000301 +0.296669 +78.67 +f3= 0.856983 +0.000120 +0.250005 +7.30 +f4= 0.140282 +0.000093 +0.700786 +4.70 +f5 =2f0± 0.003007 +0.000049 +0.921085 +5.84 +f5= 1.722984 +0.000049 +0.921085 +5.8 +f6 =3f1± 0.000204 +0.000028 +0.752082 +11.32 +f6= 7.466356 +0.000028 +0.752082 +11.32 +f7 =2f2± 0.000092 +0.000023 +0.138647 +10.41 +f7= 9.955068 +0.000023 +0.138647 +10.41 +f7 =f1 + f2± 0.000077 +0.000028 +0.752082 +11.3 +f8= 12.444062 +0.000018 +0.916512 +7.80 +f8 =f1 + 2*f2 ± 0.000049 +0.000018 +0.916512 +7.8 +f9 =3f2± 0.000216 +0.000016 +0.509646 +6.84 +f9= 14.932523 +0.000016 +0.509646 +6.84 +Zeropoint: -4.43230403e-06 mag +Residuals: 0.000590436668 mag +Table 7 +Frequency Solution of HD 52788 (=TIC +279361762) Based on All Available TESS Data Collected +in 24 Sectors. +Frequency (d−1) Amplitude (mag) Phase (0-1) +SNR +f0= 10.760711 +0.003757 +0.653190 +299.1 +f1= 8.912941 +0.003336 +0.487635 +208.7 +f2= 6.718369 +0.003145 +0.554220 +233.8 +f3= 8.322361 +0.002863 +0.089991 +193.2 +f4= 7.361251 +0.002730 +0.563008 +191.2 +f5= 9.667860 +0.002711 +0.603559 +188.3 +f6= 7.625737 +0.001647 +0.408994 +109.2 +f7= 9.827841 +0.001635 +0.075182 +113.6 +f8= 5.891104 +0.001634 +0.916897 +142.2 +f9= 9.940035 +0.001520 +0.568099 +105.6 +f10= 5.692326 +0.001507 +0.393682 +131.2 +f11= 8.912464 +0.001483 +0.539822 +92.7 +f12= 8.139382 +0.001330 +0.098920 +89.8 +f13= 9.197316 +0.001324 +0.772519 +84.2 +f14= 6.072851 +0.001311 +0.032528 +105.1 +f15= 6.816985 +0.001264 +0.183063 +94.0 +f16= 10.301894 +0.001257 +0.019626 +89.7 +f17= 6.828059 +0.001200 +0.408745 +89.2 +f18= 9.667937 +0.001152 +0.247872 +80.0 +f19= 9.939957 +0.001120 +0.260189 +77.8 +f20= 5.434878 +0.001012 +0.131651 +94.3 +f21= 8.546385 +0.000980 +0.952377 +61.3 +f22= 10.761122 +0.000973 +0.142553 +77.5 +f23= 7.260373 +0.000968 +0.754004 +67.8 +f24= 7.260386 +0.000946 +0.225250 +66.2 +f25= 7.059142 +0.000897 +0.838202 +62.8 +f26= 7.626010 +0.000894 +0.685751 +59.3 +f27= 9.512210 +0.000799 +0.576298 +55.5 +f28= 7.772967 +0.000754 +0.671621 +50.0 +f29= 9.827725 +0.000693 +0.613502 +48.2 +f30= 7.575876 +0.000693 +0.429183 +45.9 +f31= 4.954131 +0.000687 +0.406177 +67.8 +f32= 8.032471 +0.000686 +0.778436 +46.3 +f33= 9.598536 +0.000678 +0.601792 +47.1 +f34= 8.425755 +0.000609 +0.244856 +41.1 +f35= 11.284097 +0.000544 +0.413103 +44.9 +f36= 6.786902 +0.000540 +0.368252 +40.1 +f37= 7.834032 +0.000494 +0.267889 +32.7 +f38= 9.269924 +0.000479 +0.945852 +30.4 +f39= 9.515054 +0.000466 +0.266285 +32.4 +f40= 9.598494 +0.000448 +0.166360 +31.1 +· · · · · · +The full 135 frequencies are provided in electronic version online. +Zeropoint: 7.81185449 mag +Residuals: 0.00273008 mag + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +15 +1630 +1635 +1640 +1645 +1650 +Time (TBJD) +97000 +98000 +99000 +100000 +101000 +102000 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 279361762-S0012 +5 +10 +15 +20 +25 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Amplitude (mmag) +Four peaks over 3 mmag +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +SNR=4.0, noise based original data +5 +10 +15 +20 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +191 significant peaks prewhitened +TIC279361762_S0002--S0039 +Fig. 9 TESS light curves and amplitude spectrum of HD 52788=TIC 279361762 (sectors 2–10, 12,13, 27–39). + +16 +A.-Y. Zhou +1630 +1635 +1640 +1645 +1650 +Time (TBJD) +88600 +88700 +88800 +88900 +89000 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 279431011-S0012 +0 +10 +20 +30 +40 +50 +Frequenc2 (d31) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +2 +4 +6 +8 +10 +12 +14 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +Six significant peaks at 0.8566,2.4889,4.9776,0.8548,1.7193,7.4664 d31 pre0hi−ened +TIC 279431011 = HD 53166 in TESS ,ec−or, 2 and 12 +5 +10 +15 +20 +25 +Frequenc2 ( −1) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Amplitude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2 +4 +6 +8 +10 +12 +14 +16 +0.00 +0.05 +0.10 +0.15 +0.20 +10 significant peaks prewhitened +TIC279431011_S2739-HLSP_QLP_llc (S0027--S0039, 10-minute cadence) +Fig. 10 Amplitude spectrum of TIC 279431011: second panel to bottom: 2-minute cadence: sector 2, sectors +2 and 12, sectors 2–13; 30-minute cadence: sectors 27–39. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +17 +Table 8 +Frequency Solution of HD 53349 (=TIC +279476396) Based on 24 TESS Sectors Data. +Frequency (d−1) Amplitude (mmag) Phase (0-1) +SNR +f0= 1.888139 +0.039 +0.476718 +24.90 +f1= 1.809503 +0.035 +0.229343 +22.35 +f2= 1.538036∗ +0.018 +0.000155 +11.58 +f3= 7.287296 +0.009 +0.773619 +9.57 +f4= 6.148190 +0.008 +0.131089 +8.34 +f5 =2f3± 0.017 +0.008 +0.458507 +6.43 +f6= 3.074502 +0.008 +0.458507 +6.43 +f7= 1.251712 +0.008 +0.215842 +4.76 +Zeropoint: 4.43898411e-07 mag +Residuals: 0.000186609086 mag +* Accompanied with an equal amplitude peak at f2 – 0.000794 +within frequency resolution of 0.0009656 d−1 +7.4 NGC 6871 +During the course of observing a known δ Sct star V1821 +Cyg in the open cluster NGC 6871, one of the field star +GSC 2683-3076 (=V2238 Cyg) was discovered to be a +new δ Scuti star (Zhou et al. 2001b). Now we are going +to revisit this field. TESS observed this field in sectors +14, 15, and 41. Light curves are available in 2-minute +cadence SPOC (S0041) and in 30-minute cadence HLSP- +QLP (S0014 and S0015) products. A check led to the +discoveries of four additional new variable stars nearby +(Table 9). +7.4.1 NGC 6871: V1821 Cyg = TIC 90350726 +V1821 Cyg (=HD 227695 = TIC 90350726 = 2MASS +J20063348+3552420, B=10.m56,V =10.m22, A5p) is a +known δ Sct star. We provide the newly detected 39 +pulsational contents based on TESS data in Table 10 +and Figure 12 as updates to the previous two pulsation +modes (Zhou et al. 2001a). +7.4.2 NGC 6871: V2238 Cyg = GSC 2683-3076 = TIC +41195917 +V2238 Cyg (= GSC 2683-3076 = TIC 41195917) was +observed by TESS in sectors 14, 15 and 41. SPOC light +curves are available with cadence of 2 minutes only for +sector 41, 30-minute cadence HLSP-QLP data for sectors +14 and 15. With TESS data, we resolved 38 pulsational +frequency contents, see Figure 13 and Table 11. +7.4.3 NGC 6871: TIC 89757305 +TIC +89757305 +(= +HD +227658 += +2MASS +J20061465+3551028,B=11.m18, +V =11.m09, +B2) +was +observed by TESS in sectors 14, 15 and 41. SPOC light +curves are available with cadence of 2 minutes only for +sector 41, 30-minute cadence HLSP-QLP data for sectors +14 and 15. With TESS data, we identified the star to be +a pulsating star of possibly β Cephei-type and resolved +8 pulsational frequency contents, see Figure 14 and +Table 12. +7.4.4 NGC 6871: TIC 41195818 +TIC +41195818 +(= +2MASS +J20061931+3555462, +V =14.m37, +Tmag=12.m6093) +is +a +plain +star +with +only +one +identifier, +[MJD95] +J200619.32+355546.4 +and +2 +references +linked +in +Simbad/CDS +(http://simbad.cds.unistra.fr/simbad/sim- +ref?bibcode=1995ApJ...454..151M). +TESS +observed +the star in sectors 14, 15 and 41. Light curves in sectors +14 and 15 are in 30-minute cadence without removal +of systematic affects and detrending. Light curves in +sector 41 are in 2-minute cadence, which show evident +periodic variations. Based on sector 41 light curves, it +is now identified to be a new W UMa-type eclipsing +binary system with an orbit period of 1.370563 days +(f0=0.729626 d−1). See Figure 15 and Tables 13 and 14. +However, the Zwicky Transient Facility (ZTF +24 ) +discovered this star (20 06 19.31 +35 55 45.8, =ZTF +J200619.30+355545.8=[MJD95] +J200619.32+355546.4, +spectral type K8) to be a BY Draconis-type variables +with g-band period= 1.3702414 d (Chen et al. 2020), +which are emission-line dwarfs of dKe-dMe spectral type +showing quasi-periodic light changes with periods from +a fraction of a day to 120 days and amplitudes from +several hundredths to 0.5 mag. in V . The light variability +is caused by axial rotation of a star with a variable degree +of non-uniformity of the surface brightness (spots) and +chromospheric activity. Some of these stars also show +flares similar to those of UV Ceti stars, and in those cases +they also belong to the latter type and are simultaneously +considered eruptive variables. In the above point of view, +light curves in sectors 14 and 15 did show some kind +of emission features. We do not understand the periodic +variability disappeared in the duration of sectors 14 and +15. +7.4.5 NGC 6871: TIC 41195988 +TIC +41195988 +(=GSC +02683-03318 +=2MASS +J20062044+3553079, B=12.m32, V =11.m99) is plain +star with fewer identifiers and references in Simbad/CDS, +where TESS ID (TIC) is not cross-matched. TESS Light +curves are available in HLSP-PATHOS and HLSP-QLP +two products in sectors 14 and 15. No pulsation can be +identified, but some kind of flare-like features are seen in +sector 15 PATHOS light curves, see Figure16. +7.4.6 NGC 6871 22: TIC 41196013 +TIC 41196013 = NGC 6871 22 (V =11.m65, A5) was +observed in TESS sector 41, the TESS PDCSAP light +24 https://www.ztf.caltech.edu/ + +18 +A.-Y. Zhou +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Frequency (d−1) +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Amplitude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2 +4 +6 +8 +10 +12 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +8 significant peaks prewhitened +TIC279476396-S0139-LC (S0001--S0039, 2-minute cadence) +Fig. 11 Amplitude spectrum of TIC 279476396 (=HD 53349) +Table 9 A Group of 6 Stars in NGC 6871 Observed by TESS in Sectors 14, 15 and 41. +Stars Identifiers +V , Sp. +RA, Dec(J2000.0)/Notes +V1821 Cyg = TIC 90350726 +8.m37,Fm dD +20:06:33.388 +35:52:42.88 +V2238 Cyg = TIC 41195917 +10.m53, A6 +20:06:24.311 +35:54:16.08 +V1=HD 227682 = TIC 41195891 +10.m4, F8 +20:06:29.467 +35:54:41.88 +V2=HD 227658 = TIC 89757305 +11.m09, B2 +20:06:14.547 +35:51:01.10 +V3=TIC 41195818 +15.m69, +20:06:19.276 +35:55:46.78 +V4=TIC 41195988 +11.m99 +20:06:20.439 +35:53:08.07 +curves demonstrated that the star is constant within the +time scale of several weeks at least down to the level of +0.125 mmag. Fourier spectrum basically exhibits white +noise except that a peak at 6.576045 d−1 with amplitude +of 2.82 mmag (Figure 17), a little bit over significant +level. This peak also appeared when only the data of +the second orbit in this sector were used. Direct use +of ’to periodogram’ of LIGHTKURVE to the light curves +(*llc.fits) did show this peak too. The value is close to half +of the ∼13.7 days orbital period. PDCSAP’s detrending +should have removed most instrumental variations. A +further visual inspection on the other two TESS sectors +14 and 15 at 30-minute cadence as HLSP-QLP products +did not support a positive light variability. Thus, the +star is non-variable at time scales less than days. The +results support that it is a photometric standard star, +and the color data listed in the uvby98 photoelectric +photometric catalogue (Hauck & Mermilliod 1998) can +still be referenced. +7.4.7 NGC 6871: HD 227682 = TIC 41195891 +HD 227682 = TIC 41195891 (V =10.m4, F8) was observed +in TESS sectors 14 and 15. The SPOC and QLP light +curves data are available in 30-minute cadence as HLSP +mission products. A frequency analysis applied to the two +sectors data shows no variability over 0.05 mmag level. +Amplitude spectrum is just noise below 0.035 mmag. +7.4.8 NGC 6871: TIC 90350490 +TIC +90350490 +(= +2MASS +J20064266+3550552 +, +V =14.m35, G=14.m238817, Tmag=13.m8359) is a plain +star with fewer references (less than 5) in Simbad/CDS +database. No periodic variability was identified. +7.4.9 NGC 6871: TIC 90350607 +TIC +90350607 +(=2MASS +J20064698+3551472, +B=14.m15, V =13.m86) is a plain star with a couple +of identifiers and one linked reference back to year +1995 in Simbad/CDS. TESS observed the star in sectors +14, 15 and 41. Light curves are available in 30-minute +cadence for sectors 14 and 15. The uncorrected SAP +light curves produced by HLSP-QLP from sectors 14 +and 15 did not show periodic light variability. We tried +4–6 order polynomial fittings to remove trends in each +orbit but failed to detect variability. No TPF images to +download, FFI images could be the last chance to check +any variability. +7.5 HD 191025 = TIC 41189624 +HD 191025 (=TIC 41189624 = 2MASS J20062426+ +3643560, V =8.m75, A5V) is a plain star in Simbad/CDS, +it is now identified to be a new δ Sct-type pulsating +variable star with TESS data in sector 41. See Figure 18 +and Table 15. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +19 +2420 +2425 +2430 +2435 +2440 +2445 +Time (TBJD) +20400 +20600 +20800 +21000 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 90350726-S0041 +10 +20 +30 +40 +50 +Frequency (d−1) +0 +1 +2 +3 +4 +5 +Amplitude (1000 ppm) +aliases +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +5 +10 +15 +20 +25 +30 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +39 significant peaks prewhitened +V1821Cyg-S141541 (TESS sectors 14, 15 and 41) +Fig. 12 TESS light curves and amplitude spectrum of V1821 Cyg +7.6 TYC 2671-577-1 = TIC 90869850 +TYC 2671-577-1 (=TIC 90869850 = 2MASS J20074805+ +3148345, V =11.m44), a plain star in Simbad/CDS, is now +identified to be a new δ Scuti pulsating variable star +based on TESS light curves in sector 41 of 2-minute +cadence. 30-minute cadence light curves in sector 14 +without detrendings were not analyzed together. Results +are reported in Figure 19 and Table 16. According to color +magnitudes and indices in existing catalogs, B=11.m93, +V =11.m44 (B − V =0.49 means about 6300 K according +to the basic empirical formula (B − V ) = −0.865 + +8540/Teff); J=11.m061, H=11.m017, K=10.m975, (J − +H) means K3, (H − K) means A–F, and LAMOST +spectrum we derived a spectral type in A0–F9 for the +star, along with a consideration of effective temperature of +9058±423.817 K and gravity log g = 4.41791 ± 0.07722 +in TIC v8.2. +7.7 TIC 40831024=HD 227647 +HD 227647 (=TIC 40831024, V =10.m29, A2), was +observed in TESS sector 41, it is identified to be a new +δ Sct-type pulsating variable star. Figure 20 and Table 17 +report the pulsation contents. +7.8 V2455 Cyg = HD 204615 = TIC 266794067 +V2455 Cyg (= HD 204615 =TIC 266794067, V =8.m84, +F2), is a known HADS with f0=10.61 d−1 (Ostadnezhad +et al. 2020) which was observed by TESS in sectors 15 and +16 with cadence of 2 minutes. +For this star, because of its high amplitude and short +period, we used SAP fluxes other than PDCSAP, which +removed long-term trends. Our goal is to detect non- +radial and long-term pulsation contents. Now besides the +harmonics of the fundamental radial frequency, non-radial +contents are revealed. See Figure 21 and Table 18. + +20 +A.-Y. Zhou +2420 +2425 +2430 +2435 +2440 +2445 +Time (TBJD) +11100 +11200 +11300 +11400 +11500 +11600 +11700 +PDCSAP Flux (e-/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 41195917-S0041 +0 +10 +20 +30 +40 +50 +Frequenc1 (d21) +0 +1 +2 +3 +4 +5 +6 +PDCSAP (mmag) +Signi ican− level a− SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +10 +20 +30 +40 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +40 significant peaks in 5--23 d−1 prewhi−ened +TIC 41195917 = V2238 Cyg in TESS sector 41 +Fig. 13 TESS light curves and amplitude spectrum of V2238 Cyg +7.9 NGC 6910: V2245 Cyg = HD 229196 = TIC +13876370 +V2245 Cyg (=HD 229196 = TIC 13876370, V =8.m59, +O6II), a member of the young open cluster NGC 6910, +is classified in Simbad/CDS as a pulsating variable. +Its variability exhibiting irregular light variations was +confirmed by Karitskaya et al. (2000). It is known to be +a double-line spectroscopic binary early (Sanford 1949). +This Galactic O-type blue giant star (Ma´ız-Apell´aniz et al. +2004) at a distance of about 1.3 kpc, was previously +known only from some peculiarities in the ultraviolet +spectrum (Massa et al. 1983; Crawford et al. 1977). +Gaia updated the parallaxes to be 0.5535±0.0324 mas +corresponding to a distance of 1.81±0.11 kpc. Hα line in +emission reported by Kolaczkowski et al. (2004) was not +verified in the spectroscopic studies of Kub´at et al. (2007) +who instead found variable Hα line profile. +It was observed by TESS in sectors 15, 16 and 41. +SPOC light curves are available with a cadence of 2 +minutes only for sector 41. Both SAP and PDCSAP +fluxes light curves show clear W UMa-type (a kind of +close contact binary) featured with strong pulsation over +the orbit period. So the primary component is pulsating. +Fourier analysis exclusively showed just two significant +frequencies at 0.52276988 and 0.2613849 d−1. Clearly, +the second term is half of the former. The orbital period +is then should be 1.9128875 days. See Figure 22. +As pointed out by Kub´at et al. (2007) that these +changes and the presence of the extended blue wing in +the Hα profile, together with the rarely seen photometric +variability in pulsating stars, make this star an interesting +object for further study. +7.10 AL Tri = GSC 2293-1021 +AL Tri = GSC 2293-1021 = TIC 61236485 (V =13.m98) is +a W UMa-type eclipsing binary discovered by our group +(Liu et al. 2000b), where they folded the light curves in +an orbit period of 0.262 days. W UMa-type is a kind of +contact binaries. It is listed in the most recent catalogs +of 9380 samples (Marsh et al. 2017), the earlier catalogs +of 7216 stars in Avvakumova et al. (2013) and 6330 stars +in Malkov et al. (2006). +AL Tri was observed in TESS sector 17 and the data +are available to download in 30-minute cadence as the +TESS High Level Science Products (HLSP mission) which +are produced by the MIT Quick-Look Pipeline (QLP) +team. QLP light curves come in two forms as *llc.fits and +*llc.txt files. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +21 +2420 +2425 +2430 +2435 +2440 +2445 +Time (TBJD) +9150 +9200 +9250 +9300 +9350 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Cu ve - TIC 89757305-S0041 +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +10 +20 +30 +40 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +7 significant peaks in 0.125--6.528 d−1 prewhitened +TIC89757305-S0041 in TESS sector 41 +Fig. 14 TESS light curves (dots) with fitted solid lines (upper) and amplitude spectrum of TIC 89757305 +Besides a strong frequency peak at f0=7.502196 d−1 +that would correspond to orbit period of Porb= 2/f0= +0.2665886 days, there are several significant frequen- +cies presented after removing the aliased peaks (fi ± +0.057814): 0.5f0, 2f0, 0.5(f0 + f1) and f1 + f4. See +Table 19 and Figure 23. The presence of period-doubling +(0.5f0 and 0.5f1) can be referred to those detections in +RR Lyrae stars (Plachy & Szab´o 2021; Szab´o et al. 2014) +and other cases (Kemp et al. 1979). This period doubling +bifurcation means a chaotic pulsation behaviour of the star, +being worthwhile to further investigate. +7.11 IT Dra = SAO 16394 +IT Dra (= SAO 16394 =HD 127411 = TIC 166177270) +is a δ Scuti star first discovered by our group (Liu et al. +1998, 2000a) to be pulsating with two detected frequencies +(f1=16.8493, f2=23.0613 d−1), which are confirmed in +TESS data with more additional pulsation contents. TESS +observed IT Dra in 7 sectors (15, 16, 22, 23, 48, 49 and 50) +in 2-minute cadence. The data are summarized in Table 20, +pulsational frequency analysis results are presented in +Table 21 and Figures 24 and 25. +7.12 AD Ari +AD Ari (= HD 14147 = HIP 10701 = TIC 246938869, +F0, V =7.m43) was misclassified and collected in the δ +Scuti star catalog (Rodr´ıguez et al. 2000), Zhou (2002) +observed the star on four nights with a result of 991 +photometric points through two photoelectric 3-channel +and 4-channel photometers. Due to actual longer periodic +light variations, the author only saw a portion of light +variation at different phases each night. With the data +from STEREO TRansiting Exoplanet and Stellar Survey +(STRESS), Sangaralingam & Stevens (2011) showed full +periodic light curves with two distinct amplitudes and +suspected this star could be a binary system rather than a δ +Sct star. Ziaali et al. (2019) derived a period–luminosity +pair values of log P += +−0.57, MV += +2.26 using +Gaia DR2 parallaxes, they quoted the star as being an +ellipsoid variable following Handler & Shobbrook (2002). +No updates on the star’s variability until TESS. TESS +observed AD Ari in two sectors 42 and 43 in 2-minute +cadence. Uninterrupted monitors during four successive +∼13.7-day orbits, TESS collected perfect phase-coverage +light curves which show distinct binary nature and the +QLP team paid close attention to the light curves’ fitting +for identifying an exoplanet. As shown now in Figure 26, +it is ultimately an eclipsing binary system. + +22 +A.-Y. Zhou +Table 10 +Frequency Solution of V1821 Cyg (=TIC +90350726) Based on TESS Sectors 14, 15 and 41. +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 11.081879 +0.005575 +0.296366 +149.5 +f1= 8.817484 +0.004505 +0.989323 +131.7 +f2= 8.242338 +0.003427 +0.694973 +99.4 +f3= 7.773465 +0.002468 +0.391910 +71.6 +f4= 9.032012 +0.002067 +0.389771 +60.5 +f5= 10.523965 +0.001996 +0.361299 +58.7 +f6= 8.240781 +0.001981 +0.992853 +57.4 +f7= 8.822225 +0.001654 +0.918925 +48.4 +f8= 7.424433 +0.001392 +0.028127 +40.5 +f9= 6.529672 +0.001343 +0.412738 +45.8 +f10= 6.068134 +0.001339 +0.692078 +51.0 +f11= 10.380492 +0.001329 +0.876536 +39.1 +f12= 8.699882 +0.001314 +0.148762 +35.8 +f13= 11.230837 +0.001253 +0.281064 +33.6 +f14= 7.788070 +0.001237 +0.332489 +35.9 +f15= 6.603249 +0.001155 +0.653380 +39.4 +f16= 6.790211 +0.000984 +0.894335 +28.6 +f16 =3f0 − 3f1± 0.002974 +0.000984 +0.894335 +28.6 +f17= 11.333399 +0.000956 +0.114926 +30.9 +f18= 7.753926 +0.000677 +0.201504 +19.6 +f19= 9.748771 +0.000633 +0.101019 +17.3 +f20= 10.917850 +0.000553 +0.270140 +14.8 +f21= 9.582702 +0.000513 +0.791551 +14.0 +f22= 8.868577 +0.000510 +0.131971 +14.9 +f23= 11.037073 +0.000509 +0.317912 +13.6 +f24= 9.795599 +0.000501 +0.724841 +14.3 +f25= 7.807337 +0.000440 +0.905796 +12.8 +f26= 9.871035 +0.000410 +0.170139 +11.7 +f27= 6.632588 +0.000348 +0.763343 +11.9 +f28= 8.790085 +0.000315 +0.136601 +9.2 +f29= 9.012559 +0.000265 +0.702011 +7.8 +f30= 1.794221 +0.000221 +0.903107 +5.3 +f31= 11.552661 +0.000211 +0.363582 +6.8 +f32= 11.001767 +0.000209 +0.049064 +5.6 +f33= 5.354732 +0.000206 +0.437978 +8.9 +f34= 13.845783 +0.000184 +0.476908 +9.6 +f35= 11.774323 +0.000158 +0.213792 +5.2 +f36= 18.767678 +0.000134 +0.086756 +6.7 +f37= 59.082709 +0.000091 +0.932831 +6.0 +f38= 36.913990 +0.000087 +0.324406 +5.9 +Zeropoint: 7.16854912e-05 mag +Residuals: 0.00138046064 mag +7.13 NGC 2632 +NGC 2632 or Praesepe is well known for its dense +population of 14 δ Sct stars in a small region. It is a +young nearby open cluster (0.8 Gyr, 172±47 pc). Here we +checked a few of them. We will address the detection +of pulsation frequencies of these stars and leave mode +characterization in future works. +7.13.1 NGC 2632: BR Cnc +BR Cnc (= HD 73175 = TIC 307678320, F0Vn, V =8.m25, +B=8.m48) is one of the members of stellar cluster Praesepe +(NGC 2632), a known δ Sct star (Zhou et al. 2001c; +Zhou 2002) . TESS observed BR Cnc in sectors 44 +and 46 at the short cadence of 2 minutes. The Data +Validation Report Summary produced in the TESS Science +Processing Operations Center Pipeline (SPOC) presented +a transit analysis with a period of 4.18766±0.00109 days. +Table 11 +Frequency Solution of V2238 Cyg (=TIC +41195917) Based on TESS Sectors 14, 15 and 41. +Frequency (d−1) Amplitude (mag) Phase (0-1) SNR +f0= 9.505837 +0.005955 +0.474658 +56.0 +f1= 13.695551 +0.004294 +0.497070 +41.9 +f2= 13.956938 +0.003410 +0.971645 +33.3 +f3= 8.050000 +0.003323 +0.636755 +40.6 +f4= 15.707666 +0.003184 +0.808053 +32.1 +f5= 12.375452 +0.003055 +0.874708 +28.3 +f6= 13.005414 +0.002577 +0.678555 +24.9 +f7= 7.992049 +0.002576 +0.633647 +33.1 +f8= 14.844525 +0.002256 +0.416459 +22.7 +f9= 8.061000 +0.002019 +0.383568 +24.7 +f10= 14.780000 +0.001801 +0.752388 +18.1 +f11= 7.390000 +0.001730 +0.270568 +23.7 +f12= 13.214147 +0.001623 +0.339179 +15.7 +f13= 16.661000 +0.001520 +0.106067 +16.9 +f14= 13.220000 +0.001443 +0.422392 +13.9 +f15= 8.920000 +0.001290 +0.221862 +14.1 +f16= 13.260000 +0.001190 +0.413340 +11.5 +f17= 15.077705 +0.000940 +0.942938 +9.3 +f18= 8.930000 +0.000871 +0.398617 +9.5 +f19= 15.090000 +0.000822 +0.409740 +8.2 +f20= 14.399000 +0.000789 +0.185057 +7.8 +f21= 15.360000 +0.000721 +0.902924 +7.2 +f22= 7.403000 +0.000716 +0.689828 +9.8 +f23= 15.891000 +0.000709 +0.610698 +7.1 +f24= 14.849000 +0.000702 +0.411147 +7.1 +f25= 11.708000 +0.000640 +0.861762 +5.9 +f26= 12.940000 +0.000636 +0.819852 +6.0 +f27= 13.292000 +0.000620 +0.655752 +6.0 +f28= 7.260000 +0.000567 +0.978260 +7.8 +f29= 10.460000 +0.000562 +0.892636 +5.4 +f30= 12.676329 +0.000558 +0.908948 +5.2 +f31= 8.310000 +0.000499 +0.387129 +6.1 +f32= 7.521000 +0.000478 +0.735857 +6.1 +f33= 10.120000 +0.000469 +0.767173 +4.5 +f34= 15.499000 +0.000428 +0.579135 +4.2 +f35= 22.190000 +0.000376 +0.855627 +7.4 +f36= 21.890000 +0.000341 +0.233820 +6.1 +f37= 5.560000 +0.000307 +0.337988 +4.9 +Zeropoint: 10.8558103 mag +Residuals: 0.00247465132 mag +Table 12 +Frequency Solution of HD 227658 (=TIC +89757305) Based on TESS Sectors 14, 15 and 41. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 0.470121 +0.001449 +0.826953 +10.8 +f1= 0.163602 +0.001092 +0.054359 +8.1 +f2= 0.125992 +0.000779 +0.118758 +5.8 +f3= 0.514000 +0.000659 +0.508592 +5.5 +f4= 2.286000 +0.000590 +0.505777 +6.4 +f5= 0.808000 +0.000577 +0.651683 +4.8 +f6= 1.955200 +0.000447 +0.034838 +4.2 +f7= 6.528000 +0.000280 +0.322140 +6.3 +Zeropoint: 11.0829341 mag +Residuals: 0.00254047713 mag +It was not a TESS Objects of Interest (TOI,Guerrero +et al. 2021), but SPOC checked it as a potential transiting +planets by searching for periodic flux decreases, known +as threshold-crossing events (TCEs). Suspicion of hosting +an exoplanet was not confirmed either on the NASA + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +23 +Table 13 Astronomic Parameters of TIC 41195818 from +TIC v8.2. +Parameter +TIC 41195818 +B +15.535±0.176 +V +14.136±0.08 +J +11.283 pm0.028 +H +10.641±0.031 +K +10.465±0.026 +G +13.6009±0.000972 +Tmag +12.6093±0.007496 +Teff +3814±157 +log g +4.66061±0.0107662 +R/R⊙ +0.588903±0.0177 +M/M⊙ +0.57894±0.0.0203863 +Table 14 Frequency Solution of TIC 41195818 Based on +TESS Sector 41 . +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 0.729627 +0.030967 +0.323611 +50.3 +f1= 0.498328 +0.005566 +0.330653 +7.9 +f2= 2.190000 +0.001967 +0.192861 +5.0 +f2 =3f0± 0.001119 +0.001967 +0.192861 +5.0 +f3= 4.020000 +0.001346 +0.924510 +7.1 +f4= 9.504000 +0.000834 +0.143969 +4.9 +f4 =2f0 + 2f2± 0.004746 +0.000834 +0.143969 +4.9 +Zeropoint: 13.4511123 mag +Residuals: 0.0134016886 mag +Table 15 +Frequency Solution of HD 191025 (=TIC +41189624) Based on TESS Sector 41. +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 24.145378 +0.000452 +0.483759 +40.1 +f1= 21.153532 +0.000208 +0.448056 +16.8 +f2= 27.043500 +0.000153 +0.353013 +13.8 +f3= 32.116700 +0.000136 +0.294803 +10.7 +f4= 20.611954 +0.000132 +0.634321 +10.6 +f5= 12.877480 +0.000118 +0.488746 +9.0 +f6= 6.661400 +0.000113 +0.649834 +8.1 +f7= 37.477900 +0.000112 +0.909331 +8.0 +f8= 47.621060 +0.000104 +0.968834 +9.7 +f8 =f2 + f4± 0.034394 +0.000104 +0.968834 +9.7 +f9= 34.946900 +0.000104 +0.177139 +8.0 +f10= 20.000800 +0.000102 +0.654042 +8.2 +f11= 2.867700 +0.000099 +0.608690 +6.2 +f11 =f2 − f0± 0.030422 +0.000099 +0.608690 +6.2 +f12= 17.722100 +0.000095 +0.009082 +6.7 +f13= 47.692300 +0.000090 +0.667469 +8.4 +f14= 20.801000 +0.000083 +0.445203 +6.7 +f15= 50.305560 +0.000083 +0.409245 +7.6 +f16= 33.644500 +0.000081 +0.167615 +6.4 +f17= 44.338500 +0.000079 +0.661790 +6.7 +f18= 61.840990 +0.000078 +0.383978 +8.6 +f18 =3f4± 0.005129 +0.000078 +0.383978 +8.6 +f19= 40.888300 +0.000071 +0.765283 +5.2 +f20= 59.502000 +0.000065 +0.980044 +6.2 +f21= 75.891680 +0.000062 +0.339682 +6.9 +f22= 54.332300 +0.000062 +0.259201 +5.7 +f23= 15.146000 +0.000059 +0.563436 +4.2 +Zeropoint: 8.5517624 mag +Residuals: 0.00074233798 mag +2420 +2425 +2430 +2435 +2440 +2445 +Time (TBJD) +980 +1000 +1020 +1040 +1060 +1080 +1100 +1120 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Cu ve - TIC 41195818-S0041 +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0 +5 +10 +15 +20 +25 +30 +PDCSAP (mmag) +S gn f cant level at SNR=4.0 +P(e−h tened Re) dual) +Ampl tude )pect(um +0 +10 +20 +30 +40 +0 +1 +2 +3 +4 +5 ) gn f cant peak) n 0.1--15 d−1 pre−h tened +TIC41195818-S0041 in TESS sector 41 +Fig. 15 TESS light curves and amplitude spectrum of TIC +41195818. +Exoplanet Catalog +25 +or on the NASA Exoplanet Archive +(Exoplanet and Candidate Statistics) at Caltech +26 . Results +25 https://exoplanets.nasa.gov/discovery/exoplanet-catalog/ +26 https://exoplanetarchive.ipac.caltech.edu/docs/counts detail.html + +24 +A.-Y. Zhou +1685 +1690 +1695 +1700 +1705 +1710 +BJD 2457000+ +0.98 +1.00 +1.02 +1.04 +Normalized Flux +HLSP_QLP_S0014_41195988*_llc.fits +TIC 41195988_S0014 Normalized SAP_flux +Fig. 16 TESS light curves and amplitude spectrum of TIC +41195988. +Table 16 Frequency Solution of TYC 2671-577-1 (=TIC +90869850) Based on TESS Sector 41. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 39.930228 +0.000796 +0.055957 +20.5 +f1= 39.403692 +0.000260 +0.133480 +6.7 +f2= 38.687227 +0.000237 +0.317018 +6.1 +f3= 35.953001 +0.000190 +0.348816 +4.5 +f4= 41.092368 +0.000184 +0.907349 +4.8 +Zeropoint: 11.4195204 mag +Residuals: 0.00294914535 mag +of an attentive pulsational frequency analysis are presented +in Table 22 and Figure 27. +7.13.2 NGC 2632: EX Cnc +EX Cnc (= TIC 437039231, B9V, V =10.m92, B=11.m17) +is one of the members of NGC 2632, a known δ Sct +star (Zhou 2002). TESS observed EX Cnc in four sectors +42, 44, 45, and 46 at 2-minute cadence. Results of an +attentive pulsational frequency analysis are presented in +Table 23 and Figure 28. +7.13.3 NGC 2632: BU Cnc = EPIC 211936696 +BU Cnc (= HD 73576 = EPIC 211936696 = TIC +175240124 (V =7.m65, A7Vn) is a known δ Sct star +in the Praesepe cluster which was once observed +at five international observatories during a multi-site +photoelectric photometry campaign during 1989 February +2 through 26 and resulted in five frequencies of pulsation +with millimag amplitudes (19.76, 17.36, 16.69, 18.62 +Table 17 +Frequency Solution of HD 227647 (= TIC +40831024) Based on TESS Sector 41 . +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 51.624848 +0.002395 +0.947638 +136.7 +f1= 61.912960 +0.001782 +0.207245 +107.3 +f2= 44.992391 +0.000682 +0.724975 +36.9 +f3= 48.396062 +0.000600 +0.611759 +33.7 +f4= 48.258787 +0.000444 +0.063170 +25.0 +f5= 41.615047 +0.000242 +0.865100 +13.7 +f6= 36.678784 +0.000232 +0.269233 +13.7 +f7= 45.332758 +0.000219 +0.464750 +11.9 +f8= 42.352195 +0.000159 +0.128891 +9.0 +f9= 41.259636 +0.000158 +0.981180 +8.9 +f10= 48.766517 +0.000127 +0.800103 +6.9 +f11= 51.598521 +0.000112 +0.322200 +6.4 +f12= 43.134476 +0.000103 +0.142620 +5.5 +f13= 48.715744 +0.000103 +0.251687 +5.6 +f14= 38.720986 +0.000097 +0.780971 +5.7 +f15= 38.647648 +0.000096 +0.808365 +5.7 +f16= 38.557385 +0.000085 +0.087259 +5.0 +f17= 51.453724 +0.000083 +0.957313 +4.6 +Zeropoint: 9.95198413 mag +Residuals: 0.00126494106 mag +Table 18 +Frequency Solution of V2455 Cyg (=TIC +266794067) Based on TESS Sectors 15 and 16. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 10.615119 +0.129642 +0.710469 +3106.4 +f1= 21.230239 +0.037485 +0.849221 +1175.2 +f1 =2f0± 0.000000 +0.037485 +0.849221 +1175.2 +f2= 31.845358 +0.013105 +0.881155 +419.1 +f2 =3f0± 0.000000 +0.013105 +0.881155 +419.1 +f3= 42.460478 +0.005583 +0.957322 +287.9 +f3 =4f0± 0.000000 +0.005583 +0.957322 +287.9 +f4= 10.599000 +0.001178 +0.351336 +28.2 +f5= 19.455000 +0.000904 +0.342271 +33.3 +Zeropoint: 0.00293803817 mag +Residuals: 0.00419628259 mag +Table 19 Frequency Solution of AL Tri (= TIC 61236485) +Based on TESS Sector 17. +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 7.501947 +0.095829 +0.457005 +74.1 +f1= 3.415383 +0.025524 +0.603479 +13.0 +f2= 3.751931 +0.014389 +0.232549 +7.9 +f2 =0.5f0± 0.000958 +0.014389 +0.232549 +7.9 +f3= 1.709439 +0.013531 +0.537993 +5.6 +f3 =0.5f1± 0.001748 +0.013531 +0.537993 +5.6 +f4= 5.126617 +0.012306 +0.201879 +7.8 +f4 =3f3± 0.001701 +0.012306 +0.201879 +7.8 +f5= 6.834864 +0.011506 +0.136301 +8.0 +f5 =2f1± 0.000602 +0.011506 +0.136301 +8.0 +f6= 3.361258 +0.010165 +0.055429 +5.2 +f7= 3.361258 +0.010165 +0.055429 +5.2 +f8= 15.002650 +0.008016 +0.343246 +7.0 +f8 =2f0± 0.001244 +0.008016 +0.343246 +7.0 +f1 + f2 =8.547700 +0.006265 +0.862245 +5.4 +Zeropoint: 0.996309697 mag +Residuals: 0.0292512619 mag + +LIWG - S42I000 LBLID d9A2 +J2 +J52 +T130 +J32 +3450 +3440 +tInx +34e0 +cOL +2 +3200 +3250 +IIC JJa2a88-2002-- b2 inx col +LICJJa2a88-20012 Hr2b bV1H02* 1C'!f2Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +25 +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Amplit(de (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +TIC 41196013 - Sector S0041 +Fig. 17 TESS light curves and amplitude spectrum of TIC 41196013. +20 +40 +60 +80 +Frequency (d−1) +0.0 +0.1 +0.2 +0.3 +0.4 +Amplitude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +25 significant peaks prewhitened +TIC 41189624 - S0041 +Fig. 18 TESS light curves and amplitude spectrum of HD 191025 (=TIC 41189624). +25 +30 +35 +40 +45 +50 +55 +60 +Frequency (d−1) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Amplitude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.00 +0.05 +0.10 +0.15 +5 significant peaks prewhitened +TIC 90869850 - S0041 +Fig. 19 TESS light curves and amplitude spectrum of TYC 2671-577-1 (=TIC 90869850) +and 19.87 d−1, Breger et al. 1993). This is one of the +highest cited works in the δ Sct field. In addition, the +two δ Scuties in the Praesepe cluster, namely BN Cnc +and BU Cnc, were observed during the 1992 STEPHI IV +campaign (lasting 3 weeks). Five and six frequency peaks +were detected, respectively (Perez Hernandez et al. 1995). +A re-analysis of the published photoelectric photometry +showed that using statistical weights results in a dramatic + +26 +A.-Y. Zhou +30 +40 +50 +60 +70 +80 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +Amplitude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.00 +0.02 +0.04 +0.06 +0.08 +18 significant peak prewhitened +TIC 40831024 - Sector S0041 +Fig. 20 TESS light curves and amplitude spectrum of HD 227647 (=TIC 40831024). +Table 20 +TESS Data of IT Dra (=SAO 16394 = TIC +166177270). +Term +Notes +Sectors +15, 16, 22, 23, 48, 49, 50 +fa1 +0.001203 (primary alias +fa2 +0.026936 (secondary alias) +Time span +BJD 2458711.3629–2459691.51 +(2019.08.15–2022.04.22) +Data length +23523.5 hours, 127968 points +Days +13.7*7*2= 191.8 days +reduction in the noise level along with the detection of +seven frequencies in BU Cnc and eight in BN Cnc, with +three of the latter being previously unknown(Arentoft et al. +1998). +K2 observed the Praesepe cluster (=NGC 2632) area +during 2015.04.27–2018.07.02 in Campaigns c05, c16 and +c18. A total of 9450 data points were collected over +1162.75 days time span with actual 208 observing days. +Frequency or spectral resolution is down to 0.00086 d−1, +which ensures closely spaced frequencies can be resolved. +After removing outliers, the rest 9244 points were +analyzed. Table 24 lists the main stellar parameters for +four of the Praesepe members. +Now with TESS and K2 data, the pulsation frequencies +are updated to 28 and 10 for BU Cnc and BN Cnc, +respectively. That additional frequency at 23.91 d−1 +suspected by Breger et al. (1993) was not confirmed. +The current frequency solution of pulsation fits the +observations to 5.4 mmag, even worse than that of ±2.6 +mmag in the first campaign work. Figure 29 and Table 25 +summarized the results. +7.13.4 NGC 2632: BN Cnc and BV Cnc +BN Cnc (= EPIC 211933524) and BV Cnc (= EPIC +211931309) are two δ Sct stars in NGC 2632. No +observational study on them in the past 20 years. +Using Kepler K2 data we resolved 10 and 27 pulsation +frequencies for each of them, respectively. See Tables 26 +and 27, demo light curves and Fourier analyses are given +in Figures 30 and 31. +7.13.5 NGC 2632: HD 73712 = EPIC 211941583 = TIC +175261925 +HD 73712 (= TIC 175261925, V =6.m78, A9V) is a +spectroscopic binary system with the primary component +being a pulsating star of δ Sct type. TESS observed the star +in sectors 44 and 46 at 2-minute cadence. K2 observed it in +campaigns c05, 16 and 18. We applied frequency analyses +separately to TESS and K2 data, and their combination. +We resolved 30 significant pulsational frequencies. Final +results are reported in Tables 28 and 29 and Figure33. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +27 +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0 +25 +50 +75 +100 +125 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +10 +20 +30 +40 +0 +1 +2 +3 +f0=10.615119 c/d and its Harmonics prewhitened +V2455 Cyg=TIC 266794067 - TESS sectors 15+16 +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PDCSAP (mmag) +Significant l . l at SNR=4.0 +f0 2 5f0 Pr whit n d R siduals +V2455 C0g=TIC 266794067 - TESS sectors 15+16 +Fig. 21 TESS light curves (dots) with fitted solid lines (upper) and amplitude spectrum of TIC 266794067 (=V2455 Cyg). +7.13.6 NGC 2632: TIC 175261942 +TIC 175261942 is an eruptive variable in NGC 2632 +(=HSHJ 295, M3.4). Simbad/CDS lists V =18.m02 and +B=20.m63, which are quite faint for TESS if it was staying +at that brightness. TESS observed the star in sectors 44 +and 46 at 2-minute cadence, and the 5-minute binned +SAP light curves show clear stable unique mono-periodic +variation with a period of 2.62299 days (f=0.3812437 +d−1). See Figure 34. Eruptive variable stars show irregular +or semi-regular brightness variations caused by material +being lost from the star, or in some cases being accreted +to it. They may vary in brightness because of the violent +progresses such as flares that occur on the surface of +the star. The changes in luminosity coincide with shell +events or mass outflow in the form of stellar wind, or +interaction with the outside interstellar medium. This star +may be R Coronae Borealis sub-type (RCB) which are +high luminosity, simultaneously pulsating and eruptive +variables. Slow non-periodic fading or eruptive changes +up to several hundred days are superposed on cyclic +pulsations, with periods in the range of 30-100 days. RCB +stars light-curves show variation in the luminosity when it +erupts or pulsates (Bruch 2021) +27 . +7.14 BL Cam = TIC 392774261 +BL Cam (= GD 428 = TIC 392774261, B=13.m10, +G=13.m015898) is a typical SX Phe-type pulsating +variable with fruitful photometric investigations (Fauvaud +et al. 2006; Zhou et al. 1999; Hintz et al. 1997). +It was observed in TESS sector 19. Light curves are +available for the public in three formats including HLSP +QLP, HLSP SPOC, and the general 2-minute cadence +results. After a comparison among them, we selected +the SAP Flux (*-0164-s lc.fits) which have no visible +instrumental systematic variations. A frequency analysis +led to the disclosure of more new pulsation contents in the +range of 25–60 d−1. See Figure 35 and Table 30. +7.15 TIC 309661089 and TIC 356473060 +TYC 3413-228-1 (= TIC 309661089, V =9.m82, G1V) +and TYC 3413-187-1 (= TIC 356473060, V =12.m06) are +two high proper motion stars. The variability of TYC +27 https://www.astro.keele.ac.uk/workx/superwasp-variable- +stars/Eruptive.html + +28 +A.-Y. Zhou +2426 +2427 +2428 +2429 +2430 +Time (TBJD) +166000 +167000 +168000 +169000 +170000 +171000 +172000 +173000 +SAP F (x (e−/s) +SAP_F (xes +SAP Light C(r)e - TIC 13876370-S0041 - a portion zoomed +Fig. 22 TESS light curves (dots) with fitted solid lines (mid-panel) of TIC 13876370 (=V2245 Cyg). +3413-228-1 (=TIC 309661089) is clear from the PDCSAP +light curves of TESS sector 47 and the second orbit of +TESS sector 20. We resolved a primary variation period +of 13.1235 d (f0=0.076199 d−1), which is much close to +the ∼13.7 days TESS orbit period. The profile of light +variations in middle panel of Figure 36 and G1V spectral +type (with Teff=5814) would suggest a rotating variables. +See Table 31. +For TYC 3413-187-1, variations are fundamentally +at a period of about 9.80346 days (f0=0.1020048 d−1). +The dip seen in the 30-minute binned light curves around +BJD 2451848 deserves further investigation. Regarding +its effective temperature of 4851 (spectral type K2), it +probably is a Mira-type variable. See Figure 37. +7.16 HD 48270 = TIC 307618601 +HD 48270 (= TIC 307618601, V =6.m64, G5) is a bright +giant. G and K giants have often been used in variable +star research as photometric comparison stars because they +are bright, relatively numerous, and not expected to be +intrinsically variable. But Henry et al. (2000) found low- +amplitude photometric variability on timescales of days to +weeks for 81 of 187 selected giants, where HD 48270 was +listed as K2 III star without detected light variations. It +was observed in TESS sector 20. Two kinds of light curves +data available: 2-minute cadence and 30-minute cadence +(as HLSP-QLP light curves). Frequency analysis showed +irregular or quasi-periodic light variations exceeding a day +timescale. A dominant period would be around 1.8 days +(f1=0.553853 d−1). See Figure 38 and Table 32. +All stars, including red giant stars, with convective +outer layers, can exhibit solar-like oscillations, which +are standing waves excited by the turbulent motion of +convective envelopes. These waves are not coherent, +instead, these waves are stochastic and damped, which +means that the lifetimes and amplitudes of the waves are +limited and variable. So what we got here is a normal case. +7.17 TIC 309661100 +TIC 309661100 (V =13.m09, M4.0Ve, =PM J07472+5020) +is a high proper motion star in Simbad/CDS. It is an +M dwarf (L´epine & Gaidos 2011) which was selected +to be a planet-hosting Candidate by both CARMENES +exoplanet survey (D´ıez Alonso et al. 2019) and TESS. +The stellar parameters had been recently updated: Teff = +3284 ± 109 K, R = 0.307 ± 0.067R⊙, log L/L⊙ = +−2.006 ± 0.224 (Khata et al. 2021). ASAS-SN Catalogs +of Variable Stars V and X identified the star to be ROT +(spotted stars showing rotational variability, Jayasinghe +et al. 2020; Christy et al. 2022). +This star was observed in TESS sectors 14 and 20 +at the cadence of 2 minutes, and late in sector 47 at +20 seconds cadence. The 20-second data unexpectedly +do not show clear variability patterns, but the low-rate +cadence data exhibit evident periodic variations. Then + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +29 +1779 +1780 +1781 +1782 +1783 +1784 +1785 +Time - 2457000 [BTJD days] +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +sap_flux +TIC 61236485 -- Normalized SAP Flux +0 +5 +10 +15 +20 +25 +Frequency (d−1) +0 +20 +40 +60 +80 +100 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +5 +10 +15 +20 +25 +0 +2 +4 +6 +8 +10 +12 +14 +7 significant peaks in 0.125--6.528 d−1 prewhitened +AL Tri = TIC 61236485 - TESS sectors 17 +Fig. 23 TESS light curves (upper) and amplitude spectrum of TIC 61236485 (=AL Tri). +0 +10 +20 +30 +40 +50 +60 +70 +80 +Frequency (d−1) +0 +1 +2 +3 +4 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.00 +0.05 +0.10 +0.15 +0.20 +18 significant peaks prewhitened +IT Dra = TIC 166177270 - TESS sectors 15 +Fig. 24 TESS light curves (upper) and amplitude spectrum of TIC 166177270 (=IT Dra = SAO 16394). +we used binned time series data to resolve the light +variations. A frequency analysis based on 5-minute and +10-minute binning fluxes consistently showed that it is +likely a pulsating variable star with merely a dominant +fundamental period of 1.31280 days and the second and +fourth harmonics: f0=0.761729 d−1, 2f0=1.517933 d−1, +and 4f0. The odd-order harmonic of 3f0 was not present +in a Fourier power spectrum. See Figure 39 and Table 33. +7.18 ι Boo = HR 5350=TIC 310381204 +ι Boo is a bright δ Sct star (V =4.m75, A7V, =21 Boo +=HR 5350 =HD 125161=TIC 310381204). Liakos & +Niarchos (2017) listed it as a suspected pulsator in + +30 +A.-Y. Zhou +0 +10 +20 +30 +40 +50 +60 +70 +Frequency (d−1) +0 +1 +2 +3 +4 +SAP (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +35 Significant peaks prewhitened +TIC 166177270 - Sectors 15+16+22+23+48+49+50 +Fig. 25 TESS light curves (upper) and amplitude spectrum of TIC 166177270 (=IT Dra). +2481.0 +2481.5 +2482.0 +2482.5 +2483.0 +2483.5 +Time (TBJD) +207500 +210000 +212500 +215000 +217500 +220000 +222500 +SAP Flux (e./() +SAP_Fluxe( +SAP Lig ) Curve - TIC 246938869-S0043 - a portion zoomed +0 +10 +20 +30 +40 +50 +Frequency (d−1) +0 +5 +10 +15 +20 +25 +30 +PDCSAP (mmag) +Significant level at SNR=4.0 +Prewhitened Residuals +Amplitude spectrum +0 +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Significant peaks prewhitened +AD Ari = TIC 246938869-S42+S43 - TESS sectors 42+43 +Fig. 26 TESS light curves (upper) and amplitude spectrum of TIC 246938869 (=AD Ari). + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +31 +2501.5 +2502.0 +2502.5 +2503.0 +2503.5 +Time (TBJD) +94000 +94500 +95000 +95500 +96000 +SAP Flux (e./() +SAP_Fluxe( +SAP Lig ) Curve - TIC 307678320-S0044 - a portion zoomed +0 +10 +20 +30 +40 +50 +60 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +PDCSAP (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +0.0 +0.1 +0.2 +0.3 +0.4 +Significant peaks prewhitened +TIC 307678320 - Sectors S44+S46 +Fig. 27 TESS light curves (upper) and amplitude spectrum of TIC 307678320 (=BR Cnc). +0 +10 +20 +30 +40 +50 +60 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +PDCSAP (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +0.0 +0.1 +0.2 +0.3 +0.4 +Significant peaks prewhitened +TIC 437039231 - Sectors S44+45+46 +Fig. 28 TESS light curves (upper) and amplitude spectrum of TIC 437039231 (=EX Cnc). +eclipsing binary system and adopted a dominant frequency +of 37.750±0.001 d−1 according to the result of 1995– +1998 photoelectric observations by Kiss et al. (1999) and +The Washington Double Star Catalog (Mason et al. 2001). +The star is too bright to be a good target for telescopes +sized over 50 cm with CCD photometry due to saturation +issues concerning a reasonable CCD exposure time and +restricted readout rates. TESS observed the star in four +sectors 22, 23, 49, and 50 at 2-minute cadence. The TESS +stacked 2-minute cadence observations are good for a +pulsation analysis. For this star, SAP fluxes are lined up +and look like little systematic variations, thus SAP fluxes +are used. +As seen now, it is absolutely true that a one- +component sinewave function is not enough for account- +ing for the light variations (see Fig. 40). We update the +pulsation contents with 65 frequencies in Table 34. +7.19 CD-58 1608 =TIC 279476440 +CD-58 1608 (=TIC 279476440, B=10.m97, V = 9.m36 +G= 8.m763089) is a plain star with which there is +no reference linked in Simbad/CDS database. TESS + +32 +A.-Y. Zhou +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Amplitude (1000 ppm) +Amplitude )pect(um +P(e− itened Re)idual) +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.1 +0.2 +0.3 +0.4 +28 )ignificant peak) p(e− itened +EPIC211936696 (K2 Campaign) 05, 16 and 18) +Fig. 29 Kepler K2 mission light curves (upper) and amplitude spectrum of BU Cnc (=EPIC 211936696). +14 +16 +18 +20 +22 +Frequency (d−1) +0.0 +0.1 +0.2 +0.3 +0.4 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.1 +0.2 +0.3 +0.4 +3+ significant peaks prewhitened +EPIC211933524-c051618r (K2 Campaigns 05, 16 and 18) +Fig. 30 Kepler K2 Mission light curves (upper) and amplitude spectrum of BN Cnc (=EPIC 211933524). +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.00 +0.05 +0.10 +0.15 +0.20 +20+ significant peaks prewhitened +EPIC211931309-c051618r (K2 Campaigns 05, 16 and 18) +Fig. 31 Kepler K2 Mission light curves (upper) and amplitude spectrum of BV Cnc (=EPIC 211931309). +light curves (HLSP-QLP products) in 30-minute cadence +(sectors 1–13) and in 10-minute cadence (sectors 27– +39) exhibit clear light variations. But the combination +of all 24 sectors’ data led to confusing results due to + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +33 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Frequency (d−1) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +30+ significant peaks prewhitened +EPIC211941583-c051618r (K2 Campaigns 05, 16 and 18) +Fig. 32 Kepler K2 Mission light curves (upper) and amplitude spectrum of HD 73712 (=EPIC 211941583) +2555 +2560 +2565 +2570 +2575 +2580 +Time (TBJD) +409000 +409500 +410000 +410500 +411000 +411500 +412000 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Cu ve - TIC 175261925-S0046 +0 +5 +10 +15 +20 +25 +30 +Frequency (d−1) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Observed (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +30 Significant peaks prewhitened +TIC 175261925 - Sectors S4446 +Fig. 33 TESS light curves (upper) and amplitude spectrum of TIC 175261925 (=HD 73172). +undetrended SAP light curves without the removal of +spacecraft’s systematic effects and variation from sector +to sector were not cleanly removed. With only sector 39 +we derived a 7 low frequencies solution with the goodness +of fitting residual less than 0.00037526. Then we selected +the data that suffered from less scatter and abnormal +variations from sectors 1,2,7,36,37 and 39 as a subset. We +summarized our final results from sector 39, the 6-sector +subset, and all 24-sector full datasets in Figure 41. The star +is a longer multi-periodic pulsating star of type mostly γ +Doradus, which is pulsating in non-radial gravity modes. +7.20 TYC 8549-1255-1=TIC 279476423 +TYC +8549-1255-1 +(=TIC +279476423 +=2MASS +J07010497-5854230, +B= +11.m71, +V = +10.m78, +G= +10.m552827) is a plain star with a null reference linked in +Simbad/CDS. Frequency analysis showed two twin peaks +at +(47.941631,47.940385) +and +(48.07122,48.072548) +d−1. They should be aliases and prewhitening can +not eliminate the closely spaced peaks (see Fig. 42). +Overall, concerning the long-cadence sampling, these +high frequencies are wrongly introduced, mostly by the +undetrended SAP light curves. No intrinsic pulsational + +34 +A.-Y. Zhou +2555 +2560 +2565 +2570 +2575 +Time - 2457000 [BTJD days] +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +Normalized pdcsap_flux +PDCSAP: 10-min binned +0 +5 +10 +15 +20 +25 +30 +Frequency (d−1) +0 +500 +1000 +1500 +2000 +Normali2ed PDCSAP_Fl−0 (e −s−1) +Am(lit−de s(ectr−m +Prewhitened Resid−als +Si nificant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +One significant peak prewhitened +TIC 175261942 - Sectors S4446 +Fig. 34 TESS light curves (upper) and amplitude spectrum of TIC 175261942. +20 +25 +30 +35 +40 +45 +50 +55 +60 +Frequency (d−1) +0 +20 +40 +60 +80 +Ampli)ude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +20 +40 +60 +80 +100 +0.0 +0.5 +1.0 +1.5 +2.0 +50 significant peaks prewhitened +20 +30 +40 +50 +60 +70 +80 +90 +0 +1 +2 +3 +4 +5 +f0 and 2f0 prewhitened +TIC 392774261 - S0019 +Fig. 35 TESS light curves (upper) and amplitude spectrum of BL Cam (=TIC 392774261). +frequency was detected. This star seemed to vary in +brightness irregularly over a time scale of days to weeks. +7.21 IRAS 20069+3648 = TIC 42247329 +IRAS 20069+3648 (= TYC 2683-2241-1 = TIC 42247329, +V =10.m43, B=12.m02) is a plain star with none reference + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +35 +0 +1 +2 +3 +4 +5 +Frequency (d−1) +0 +1 +2 +3 +4 +Normalized PDCSAP_Flux (e −s−1) +Amplit−de spectr−m +Prewhitened Resid−als +Si nificant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Significant peaks prewhitened +TIC 309661089 - Sector S47 +Fig. 36 TESS light curves (upper) and amplitude spectrum of TYC 3413-228-1 (=TIC 309661089). +in Simbad/CDS database. TESS sectors 14, 15, and 41 +observed the star. Variability on a time scale of days is +evident, but it seems irregular or non-periodic variations. +A frequency based on these three sectors’ data failed to +resolve a good fit for the light variations. See Figure 43. +7.22 HD 227681 = TIC 41194739 +HD 227681 (=TIC 41194739, V =9.m807, B=9.m798) is +a plain star with just 3 references where the latest +was back to 1993 according to CDS/Simabd. ASAS +archived its light curves (as AP13958649.csv) on HJD +2457069.16843 – 2458327.89257 with 184 data points at +a mean photometric error of 0.m02 but no variability was +visible. TESS observed the star in sectors 14, 15, and 41. +It is now surely identified to be an eclipsing binary. See +Table 35 and Figure 44. +7.23 TYC 2682-229-1 = TIC 89755468 +TYC 2682-229-1 (= TIC 89755468, V =9.m77, B=11.m04) +is a plain star with merely two references linked in +Simbad/CDS database. It is a binary star. TESS observed +the star in sectors 14, 15, and 41. A preliminary analysis +based merely on the data in sector 41 shows strong +variability. Detailed analyses will be given elsewhere +when combining the FFI images’ light curves on sectors +14 and 15. See Table 36 and Figure 45. +7.24 21 Com = TIC 393819751 +21 Com (= TIC 393819751, B=5.m496, V =5.m436, A2pv) +is a classical magnetic chemically peculiar (Ap/CP2) +star showing increased abundances. 21 Com has been +extensively studied in the past, with widely differing +and sometimes contradictory results, concerning the +occurrence of short-term variability (Paunzen et al. 2019). +The star exhibits rotational light variability with a period +of 2.05219(2) d, with no significant frequencies were +found beyond 5 d−1. Their radial velocity data also do +not indicate short-term variability. Pulsational models +assuming different metallicities and ages, which do not +predict the occurrence of unstable modes. The star 18 +Com, employed as a comparison star for 21 Com in +the past, had been identified as a periodic variable (P = +1.41645 d). This is similar to the case of HD 52788, that is +previous differential photometry of both 21 Com and HD +52788 had used a variable as comparison star. +We checked the TESS 2-minute cadence data (S0022 +and S0049) for this star. The 30-minute cadence data +provided by HLSP-QLP and HLSP-SPOC looks nice – a +few outliers are gone. We found that the rotational period + +36 +A.-Y. Zhou +1845 +1850 +1855 +1860 +1865 +1870 +Time - 2457000 [BTJD days] +0.998 +0.999 +1.000 +1.001 +1.002 +1.003 +Normalized pdcsap_flux +PDCSAP: 30-min binned +1845 +1850 +1855 +1860 +1865 +1870 +Time - 2457000 [BTJD days] +0.998 +0.999 +1.000 +1.001 +1.002 +1.003 +Normalized pdcsap_flux +PDCSAP: 30-min binned +0 +1 +2 +3 +4 +5 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Normalized PDCSAP_Flux ( −s−1) +Amplitud −p ctrum +Pr 1hit ( d R −idual− +Sig(ifica(t l v l at SNR=4.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.2 +0.4 +0.6 +One significant peak prewhitened +TIC 356473060 - Sector S20-HLSP-QLP +Fig. 37 TESS light curves (upper) and amplitude spectrum of TYC 3413-187-1 (=TIC 356473060). +becomes to be 2.046107(=1.0/0.488733) from 2.05219 +(=1.0/0.487284). Light curves can be well-fitted with 10 +low frequencies. See Figure 46 and Table 37. No trace of +short-term variability. +7.25 TIC 99180739 +TYC +6672-772-1 +(= +TIC +99180739, +B=11.m66, +V =10.m91) is a plain star. TESS light curves from +sectors 10 and 37 at the cadence of 2-minute and +10-minute are available, respectively. We obtained a +6-frequency solution to fit the light variations. Regarding +to the B − V =0.m743 value (could roughly correspond an +effective temperature of 5310 K) and other astronomical +parameters from TIC catalog v8.2: Teff= 5644.0 K, log g= +4.53865, M/M⊙= 1.001, and R/R⊙= 0.891099, this +star is probably a solar-like oscillator. Check results in +Table 38 and Figure 7.25. +7.26 CD-54 7154=TIC 173503902 +CD-54 +7154 +(=TIC +173503902 +=Gaia +DR3 +5923586600813106816, +B=10.m65,V =10.m26) +is +a +known δ Scuti star with GCVS designation of V0952 +Ara (Kazarovets et al. 2015). We list its parameters +by combining Gaia DR3 parallax and TIC v8.2. The +star’s light curves are wonderful for its conspicuous +long-time periodic light variations at a period of about +4.81 days, superimposed by shorter periodic variations. +Such dumbbell-shaped profile (see Fig. 49) is strongly +reminiscent of amplitude modulation presented in RR +Lyr-type pulsators. It also conjured up the demonstration +theoretically expected in overcontact eclipsing binary +systems, where both stellar components have overfilled +their Roche lobes, resulting in a dumbbell-shaped shared +envelope. +As we know that a significant number of RRab +stars (up to 50% according to Jurcsik et al. (2009)) +exhibit long-term modulations of the amplitudes and + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +37 +1845 +1850 +1855 +1860 +1865 +1870 +Time - 2457000 [BTJD days] +0.9985 +0.9990 +0.9995 +1.0000 +1.0005 +1.0010 +Normalized pdcsap_flux +PDCSAP: 10-min binned +0 +5 +10 +15 +20 +Frequency (d−1) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Observed (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +0.00 +0.05 +0.10 +0.15 +0.20 +6 Significant peaks prewhitened +TIC 307618601 - Sectors S0020-LC +Fig. 38 TESS light curves (upper) and amplitude spectrum of TIC 307618601 (=HD 48270) +phases of their light curves – a phenomenon first +discovered by Sergey Blazhko in 1907 and the origin +of this effect remains a mystery to the present day. +The Blazhko modulation of light curves may be strictly +periodic (with periods ranging from days to years), +multi-periodic, or irregular. Demo examples given in +OGLE project +28 , for instances OGLE discovered: RRab +stars (just fundamental-mode pulsators) OGLE-BLG- +RRLYR-09193 (17:58:21.32 −27:37:18.2), OGLE-BLG- +RRLYR-11419 (18:02:12.27 −28:47:59.9), OGLE-BLG- +RRLYR-11992 (18:03:18.50 −29:10:48.6); RRc star +(pulsate in the first-overtone mode) OGLE-BLG-RRLYR- +12135 (18:03:37.67 −29:07:34.8); RRd star (with fun- +damental and first overtone double-mode pulsations) +OGLE-BLG-RRLYR-05762 (17:52:54.04 −29:33:34.7, +Pf=0.4663027,P1O=0.3472630 days). They all exhibit +strong Blazhko modulation over 100 days period. The +Blazhko effect may shed light on the cause of the +light variation profile of TIC 173503902, i.e. periodic +amplitude modulation. Furthermore, this is also similar to +the observed pulsation amplitude changes with the orbital +phase in OGLE-SMC-T2CEP-28, an eclipsing binary with +a pulsating primary component. From the light curve +being subtracted the eclipsing modulation, the residual +pure pulsations, the amplitudes of pulsations change with +28 http://ogle.astrouw.edu.pl/atlas/RR Lyr.html +half the orbital period, which reflects complex oscillations +of a star that is distorted by tidal interactions from its +companion. The observed pulsation amplitudes change +depending on the angle at which we observe the star +29 . +On the other hand, The longer periodic variation of +brightness may be due to its rotation, so might be a +rotational brightness variation with a period of ∼4.81 +days. +Last, resonance occurs when an external oscillation is +exerted on the star, with a frequency in the neighborhood +of a certain resonance frequency. Resonance describes +the phenomenon of increased amplitude that occurs when +the frequency of an applied periodic force (or a Fourier +component of it) is equal or close to the natural frequency +of the system on which it acts. When an oscillating +force is applied at a resonant frequency of a dynamic +system, the system will oscillate at a higher amplitude +than when the same force is applied at other, non-resonant +frequencies The two strongest peaks at f0=9.162285 and +f1=9.36902 d−1 probably can interact. The star deserves +further studies. Check Figure 49 and Table 39. +29 http://ogle.astrouw.edu.pl/atlas/W Vir.html + +38 +A.-Y. Zhou +0 +10 +20 +30 +40 +50 +60 +Frequency (d−1) +0 +5000 +10000 +15000 +20000 +25000 +SAP_Flux (e 1s11) +Amplitude spect)um +P)ew itened Residuals +Significant le−el at SNR=4.0 +0 +10 +20 +30 +40 +50 +0 +2000 +4000 +6000 +8000 +10000 +3 Significant peaks prewhitened +TIC 309661100_S20 - Sectors 5m-bin +Fig. 39 TESS light curves (upper: with the fitted line) and amplitude spectrum of TIC 309661100. +7.27 GSC 04040-01606 = UCAC4 771-012013 +GSC 04040-01606 (= UCAC4 771-012013 = TIC +372724683 = Gaia DR2 512143690071260800) is a +known δ Sct star which variability was first claimed +by Handler & Meingast (2011) during a search for new +β Cephei stars in the young open cluster NGC 637. Due +to fewer observations, it was suspected to be multiple- +frequency pulsation with apparent multiple frequencies +between 6 and 8 d−1 with amplitudes around 12 mmag. +No more research since discovery. TESS observed the +star in Sectors 18, 24, and 25 at the 30-minute cadence. +Light curves are available as HLSP-QLP products. We +resolved 15 significant pulsation frequencies, see Table 40 +and Figure 48. +7.28 Six Photometric Standards +We +checked +a +few +Landolt +photometric +standard +stars (Landolt 2013, 1992) as an inauguration of another +project. Since CoRoT, with unprecedented photometry +precision, variability is becoming basically universal for +a huge population of celestial objects down to the sub- +milli-magnitude level. In turn, it becomes hard to confirm +a star to be non-variable or constant. So we go to check +a group of photometry standards for variability with data +from space telescopes (see Table 41). +7.28.1 SA 41-660 = UCAC4 678-116341 = TIC +455553286 +As +Landolt +photometric +standard +star, +SA +41-660 +(V =12.m804) is a high proper motion star. TESS observed + +L!WG (IBID) +18e2 +J810 +J82 +J820 +822 +J8e0 +asoo +as20 +a320 +a400 + a420 +a200J +bDC2Vb_LInx62 +bDc2vb r!aμf Cnla6 - lIC 30aeeJJ00-20050Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +39 +1901.5 +1902.0 +1902.5 +1903.0 +1903.5 +Time (TBJD) +2.630 +2.635 +2.640 +2.645 +2.650 +2.655 +2.660 +SAP Fl)x (e−/s) +1e6 +SAP_Fl)xes +SAP L gh( C)rve - TIC 310381204-S0022 - a portion zoomed +0 +10 +20 +30 +40 +50 +60 +70 +80 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +SAP (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.00 +0.02 +0.04 +0.06 +0.08 +64 Significant peaks prewhitened +TIC 310381204 - Sectors 22+23+49+50 +Fig. 40 TESS light curves (middle: red dots with the fitted line) and amplitude spectrum of TIC 310381204 (=ι Boo). +it in sectors 15 and 16, light curves are extracted as +HLSP-QLP results in fits and txt formats, where the latter +has deleted unusual light variations caused by external +sources. Instead, we used the fits file and did custom +detrending, that is, divide each sector into two portions, +which were separately removed from a 6-order polynomial +fitting. Then we analyzed the residual light curves. The +result of a frequency analysis makes clear the star’s longer +periodic variability, see Figure 50 and Table 42. +7.28.2 SA 26-93 = IRAS 06415+4434 = TIC 307650624 +Simbad/CDS classifies SA 26-93 (= IRAS 06415+4434 = +TIC 307650624) to be a long periodic variable candidate +(also refers to Gaia DR2: Mowlavi et al. 2018). ASAS-SN +Catalogs of Variable Stars V and X identified the star to be +SR (semi-regular variable, Jayasinghe et al. 2020; Christy +et al. 2022). +TESS observed the star in sector 20, light curves are +available as HLSP-QLP products in 30-minute cadence. +This sector data suffered from severe instrumental issues +(see top panel of Fig. 51), After detrending for each orbit +– removing a third-order polynomial fitting from the data, +the residual light curves confirmed clear variability, being +consistent with Simbad/CDS classification. A Fourier +analysis shows a couple of peaks lower than 0.5 d−1 with +the highest peak at 0.21317 d−1, corresponding to a period +of about 4.69 days. Significant frequencies are not a good + +40 +A.-Y. Zhou +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Amplitude (1000 ppm) +No ) gn f cant peak) la(ge( than 1 d−1 +Ampl tude )pect(um +P(e−h tened Re) dual) +S gn f cant level at SNR=4.0 +1 +2 +3 +4 +5 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +7 ) gn f cant peak) p(e−h tened +TIC279476440_S0039 (S0039 at 30-minute cadence) +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Freq.enc1 (d21) +0.0 +0.2 +0.4 +0.6 +0.8 +Am)li−.de (mmag) +N( signi ican− )eaks larger −han 1 d21 +Am)li−.de s)ec−r.m +Prewhi−ened Resid.als +Signi ican− level a− SNR=4.0 +1 +2 +3 +4 +5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +4 signi ican− )eaks )rewhi−ened +TIC279476440_S0139 (S0001--0013, 10-minute; S0027--0039, 30-minute cadence) +Fig. 41 TESS light curves and amplitude spectrum of TIC 279476440 (sectors 1–13, 27–39). + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +41 +10 +20 +30 +40 +50 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +Ampli.ud (mmag) +Ampli.ud −p c.rum +Pr 1hi. ( d R −idual− +Sig(ifica(. l 0 l a. SNR=4.0 +SNR=4.0, noise based original data +47.0 +47.5 +48.0 +48.5 +49.0 +49.5 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +4 significant peaks prewhitened +TIC279476423_S0139 (S0001--0013, 10-minute; S0027--0039, 30-minute cadence) +Fig. 42 TESS light curves and amplitude spectrum of TIC 279476423 (sectors 1–13, 27–39). +1685 +1690 +1695 +1700 +1705 +1710 +Time (TBJD) +88000 +88200 +88400 +88600 +88800 +89000 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 42247329-S0014 +1715 +1720 +1725 +1730 +1735 +Time - 2457000 [BTJD days] +0.997 +0.998 +0.999 +1.000 +1.001 +1.002 +1.003 +Normalized pdcsap_flux +PDCSAP: 10-min binned +Fig. 43 TESS light curves of TIC 42247329 in TESS sectors 14, 15 and 41. +fit for the light variations. Higher precision photometry +over a longer time is needed to disclose the true nature +of the variability. +7.28.3 SA 41-654 = BD+44 3978 =TIC 455517315 +BD+44 3978 (V =10.m01, F8) is a Landolt photometric +standard star without light variability reported before. A +frequency analysis of the TESS data in sectors 15 and 16 +showed that the star is constant down to 78 ppm. Long +time-scale variations may be involved in systematic issues, +detrending was not applied to derive any long periodic +variability. +7.28.4 SA 26-96 = UCAC4 673-048808 = TIC +307650596 +This Landolt photometric standard star is confirmed +constant in TESS photometry down to 0.00108 in flux. +Though an enforced frequency analysis was applied and +three significant peaks were presented in the low domain, +we are hardly certain light variations in a time scale over +days or weeks. See Figure 52. + +42 +A.-Y. Zhou +2420 +2425 +2430 +2435 +2440 +2445 +Time - 2457000 [BTJD days] +0.992 +0.994 +0.996 +0.998 +1.000 +1.002 +1.004 +1.006 +1.008 +Normalized pdcsap_flux +PDCSAP: 10-min binned +1685 +1690 +1695 +1700 +1705 +1710 +Time - 2457000 [BTJD days] +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +1.020 +sap_flux +TIC 41194739_S0014-- Normalized SAP_flux +1715 +1720 +1725 +1730 +1735 +Time - 2457000 [BTJD days] +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +1.020 +sap_flux +TIC 41194739_S0015-- Normalized SAP_flux +10 +20 +30 +40 +50 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Amplitude (mmag) +Aliasing +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0 +20 +40 +60 +80 +0.0 +0.2 +0.4 +0.6 +0.8 +5 significant peaks prewhitened +TIC 41194739 - S141541 +Fig. 44 TESS light curves and amplitude spectrum of TIC 41194739. +7.28.5 SA 26-95 = UCAC4 673-048807 = TIC +307650637 +This Landolt photometric standard star is confirmed +constant in TESS photometry down to 4.188 mmag, +the standard deviation of the residuals after removing +polynomial fittings individually to the SAP fluxes which +suffered from instrumental variations during each orbit in +TESS sector 20. See Figure 53. +8 SUMMARY AND ENDING REMARKS +With the uninterrupted high-precision photometry from +TESS and Kepler K2, we are able to insight the real +nature of those stars which would have been confused +or misunderstood due to short-term poor-quality ground +observations. The author has explored, further processed, +and analyzed the TESS data for a small sample of 50 +stars. This paper reports our preliminary TESS discoveries + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +43 +1715 +1720 +1725 +1730 +1735 +Time (TBJD) +81200 +81400 +81600 +81800 +82000 +82200 +82400 +PDCSAP Flux (e−/s) +PDCSAP_Fluxes +PDCSAP Light Curve - TIC 89755468-S0015 +1 +2 +3 +4 +5 +Frequency (d−1) +0 +50000 +100000 +150000 +200000 +250000 +Ampli)ude (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0 +2000 +4000 +6000 +8000 +10000 +12000 +5 significant peaks prewhitened +TIC 89755468 - S0015 +Fig. 45 TESS light curves and amplitude spectrum of TIC 89755468. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.1 +0.2 +0.3 +9 significant peaks prewhitened +TIC393819751-S2249 (TESS Sectors 22 and 49 ) +Fig. 46 TESS light curves and amplitude spectrum of TIC 393819751. + +44 +A.-Y. Zhou +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Frequency (d−1) +0 +1 +2 +3 +4 +5 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +5 +10 +15 +20 +25 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +6 significant peaks prewhitened +TIC99180739-S1037 (TESS Sectors 10 and 37 ) +Fig. 47 TESS light curves and amplitude spectrum of TIC 99180739. +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Frequency (d−1) +0 +1 +2 +3 +4 +5 +6 +7 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +12 significant peaks prewhitened +TIC372724683_S182425 (TESS Sectors 18, 24 and 25 ) +Fig. 48 TESS light curves and amplitude spectrum of TIC 372724683. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +45 +10 +20 +30 +40 +50 +Frequency (d−1) +0 +5 +10 +15 +20 +25 +Amplitude (1000 ppm) +aliases +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +SNR=4.0, noise based original data +10 +20 +30 +40 +50 +0 +1 +2 +3 +30+ significant peaks prewhitened +TIC173503902-S1239 (TESS Sectors 12 and 39) +Fig. 49 TESS light curves and amplitude spectrum of TIC 173503902. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Frequency (d−1) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Amplitude (1000 ppm) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.5 +1.0 +1.5 +3 significant peaks prewhitened +TIC455553286 (TESS Sectors 15 and 16 ) +0 +1 +2 +3 +4 +5 +Frequency (d−1) +0 +2 +4 +6 +8 +Normali2ed PDCSAP_Fl−0 (e −s−1) +Am(lit−de s(ectr−m +Prewhitened Resid−als +Si nificant level at SNR=4.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0 +1 +2 +3 +4 +5 +6 +One significant peak prewhitened +TIC 455553286 - Sectors S1516-llc +Fig. 50 TESS light curves (upper) and amplitude spectrum of TIC 455553286 (UCAC4 678-116341). +resulting from frequency analyses. We unveiled the +variability types and pulsation contents down to an +unprecedented precision for the selected targets. We +discovered a sum of 20 new pulsating variable stars + +46 +A.-Y. Zhou +Table 21 Frequency Solution of IT Dra (= SAO 16394 = +TIC 166177270) Based on TESS Sectors 15, 16, 22, 23, +48, 49 and 50. +Frequency(d−1) +Amplitude (mag) +Phase (0–1) +SNR +f0= 16.850786 +0.004126 +0.681244 +340.3 +f1= 23.062767 +0.002091 +0.401832 +192.1 +f2= 11.785623 +0.001015 +0.273699 +106.0 +f3= 18.141353 +0.000701 +0.473206 +59.1 +f4= 14.656449 +0.000654 +0.217638 +64.0 +f5= 23.061051 +0.000612 +0.855977 +56.2 +f6= 31.213879 +0.000561 +0.744394 +63.5 +f7= 17.507855 +0.000515 +0.373155 +43.2 +f8= 12.004583 +0.000490 +0.530261 +51.2 +f9= 15.442096 +0.000436 +0.963979 +41.4 +f10= 16.852875 +0.000407 +0.312884 +33.6 +f11= 15.208178 +0.000381 +0.262093 +36.1 +f12= 25.060636 +0.000302 +0.547621 +27.9 +f13= 20.825305 +0.000292 +0.397240 +25.7 +f14= 18.078680 +0.000283 +0.587844 +23.8 +f15= 0.036672 +0.000266 +0.057030 +11.6 +f16= 33.289025 +0.000215 +0.142259 +25.3 +f17= 21.232822 +0.000198 +0.928254 +16.9 +f18= 6.656792 +0.000189 +0.580865 +14.7 +f19= 15.049642 +0.000188 +0.177053 +17.8 +f20= 19.008615 +0.000175 +0.157325 +14.1 +f21= 20.331594 +0.000173 +0.445997 +14.8 +f22= 32.187801 +0.000172 +0.633056 +18.3 +f23= 23.574049 +0.000168 +0.467395 +14.8 +f23 =2f2± 0.002803 +0.000168 +0.467395 +14.8 +f24= 19.425886 +0.000167 +0.015433 +13.4 +f25= 14.564082 +0.000163 +0.766823 +16.0 +f26= 34.398346 +0.000161 +0.454109 +18.9 +f27= 37.454100 +0.000147 +0.043645 +21.8 +f28= 42.082750 +0.000135 +0.022363 +24.4 +f29= 8.834044 +0.000106 +0.827261 +10.4 +f30= 55.084270 +0.000073 +0.662737 +20.3 +f31= 50.749873 +0.000061 +0.636523 +14.9 +f32= 39.934971 +0.000060 +0.858786 +10.3 +f33= 48.768254 +0.000056 +0.569036 +13.0 +Zeropoint: 7.97601139 mag +Residuals: 0.00109769352 mag +Table 22 +Frequency Solution of BR Cnc (=TIC +307678320) Based on TESS Sectors 44 and 46. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 24.978248 +0.003077 +0.830430 +68.8 +f1= 11.464797 +0.001870 +0.025719 +59.1 +f2= 8.202908 +0.001238 +0.576726 +41.9 +f3= 12.784615 +0.000797 +0.215056 +28.2 +f4= 10.364524 +0.000761 +0.255704 +26.0 +f5= 10.971461 +0.000724 +0.075013 +25.2 +f6= 24.185591 +0.000675 +0.212619 +17.9 +f7= 11.744333 +0.000494 +0.123568 +19.5 +f8= 15.790582 +0.000426 +0.223130 +11.5 +Zeropoint: 8.52563563 +Residuals: 0.0015644164 +and eclipsing binaries in the sample, including 5 δ Sct +stars; 4 EB/EW; 1 BCEP; 1 gDor; 1 solar-like oscillator; +2 pulsators in SB+EB/EW systems and 6 unclassified +variable stars. Table 43 outlines the results of the selected +targets. Among them, the two comparison stars HD 53349 +and HD 53166 used in the differential photometry of +Table 23 +Frequency Solution of EX Cnc (= TIC +437039231) Based on TESS Sectors 42, 44, 45 and 46. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 5.536208 +0.002487 +0.706622 +56.0 +f1= 5.557908 +0.002413 +0.425665 +54.3 +f2= 19.604583 +0.002321 +0.150933 +34.1 +f3= 20.639830 +0.001998 +0.094616 +29.9 +f4= 17.758755 +0.001479 +0.126502 +26.3 +f5= 5.573865 +0.001386 +0.072498 +31.2 +f6= 20.759821 +0.001288 +0.289023 +19.3 +f7= 0.478052 +0.001210 +0.134550 +26.2 +f8= 0.717397 +0.001079 +0.123829 +24.4 +f9= 5.520251 +0.000990 +0.046869 +22.3 +f10= 19.757764 +0.000977 +0.347460 +14.3 +f11= 16.756059 +0.000802 +0.061488 +15.2 +f12= 5.475574 +0.000674 +0.956620 +14.8 +f13= 19.563735 +0.000636 +0.023164 +9.3 +f14= 5.496636 +0.000622 +0.375365 +13.7 +f15= 5.496636 +0.000622 +0.375365 +13.7 +f16= 17.553237 +0.000589 +0.880286 +10.5 +f17= 18.302547 +0.000574 +0.823215 +9.6 +f18= 19.006540 +0.000570 +0.188721 +8.8 +f19= 5.608968 +0.000549 +0.140147 +12.3 +f20= 5.466000 +0.000526 +0.743352 +11.6 +f21= 5.593650 +0.000506 +0.657455 +11.4 +f22= 0.955465 +0.000494 +0.694499 +11.2 +f23= 16.163760 +0.000484 +0.041466 +9.8 +f24= 0.237430 +0.000482 +0.555291 +10.4 +f25= 18.603164 +0.000475 +0.166906 +7.5 +f26= 17.013275 +0.000467 +0.932997 +8.5 +Zeropoint: 11.2592026 +Residuals: 0.00276130809 +HD 52788 are identified to be δ Sct stars. AD Ari, +formerly known as δ Sct due to fewer data, actually is +now identified to be an eclipsing binary. The pulsation +frequency spectrum of HD 52788 over 130 frequencies is +an outstanding record and unique among the δ Sct stars. +In addition, we compiled a comprehensive catalogue +consisting of more than 59,350 individual δ Sct stars based +on existing catalogs, data releases of various surveys as +well as recent publications. This is the largest collection +for this class of pulsating stars by far. With TIC and Gaia +DR3 cross-identifiers and stellar parameters extracted +from the TESS Input Catalog and Gaia DR3, the catalog +is much more useful. The H-R diagrams using this biggest +amount of DSCT show a much extended DSCT domain +than that demonstrated in earlier literature. It is interesting +and necessary to further examine those members outside +of the extended domain for a confirmed observational +region of δ Sct pulsators, because real observational +borders will strictly constrain and impact on the theories +of stellar evolution and pulsation. In special, the HTML +version of the catalog provides hyperlinks to MAST +Portal, CDS portal, and Gaia data archive, which play the +role of observation planning and follow-up investigation +portal for researchers. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +47 +Table 24 Cataloged Parameters for BU, BV, BN Cnc and HD 73712 in NGC 2632 (Praesepe). +ID & Parameters +V* BU Cnc +V* BV Cnc +V* BN Cnc +HD 73712 +TIC +TIC 175240124 +TIC 175264756 +TIC 175264749 +TIC 175261925 +HD +HD 73576 +HD 73746 +HD 73763 +HD 73712 +2MASS +J08394466+1916308 +J08403296+1911395 +J08403924+1913418 +J08402013+1920564 +EPIC +EPIC 211936696 +EPIC 211931309 +EPIC 211933524 +EPIC 211941583 +RA +08 39 44.6645 +08 40 32.9592 +08 40 39.2340 +08 40 20.1451 +RA(deg) +129.936103332169 +130.137330060834 +130.163484048856 +130.08393919638 +Dec ++19 16 30.772 ++19 11 39.569 ++19 13 41.848 ++19 20 56.315 +Dec(deg) +19.2752147347028 +19.1943252992796 +19.2282823919751 +19.3489787893529 +B= +7.823 +8.933 +7.986 +7.019 +V = +7.66791 +8.66168 +7.81068 +6.78136 +Tmag +7.4828 +8.38683 +7.60499 +6.4987 +Teff +7890.0 +7333.36 +7744.0 +7408.0 +log g +3.85145 +4.11231 +3.81905 +3.30218 +M/M⊙ +1.89 +1.663 +1.827 +1.692 +R/R⊙ +2.70108 +1.8764 +2.75664 +4.81 +Table 25 +Frequency Solution of BU Cnc (=EPIC +211936696) Based on Kepler K2 Mission Campaigns 05, +16 and 18. +Frequency (d−1) +µHz Amplitude (106 ppm) Phase (0-1) SNR +f0= 19.767731 +*228.79 +0.001199 +0.424305 +46.3 +f1= 19.766586 +228.78 +0.001017 +0.545122 +39.3 +f2= 9.799938 +*113.43 +0.000885 +0.145044 +36.0 +f3= 7.893159 +91.36 +0.000613 +0.007861 +22.3 +f4= 9.797775 +113.40 +0.000602 +0.651391 +24.5 +f5= 18.619838 +*215.51 +0.000579 +0.146193 +24.8 +f6= 16.706031 +*193.36 +0.000566 +0.071571 +23.8 +f7= 16.859086 +*195.13 +0.000514 +0.303013 +21.6 +f8= 17.054589 +197.39 +0.000485 +0.636388 +20.3 +f9= 14.584224 +168.80 +0.000480 +0.793880 +23.6 +f10= 17.053340 +197.38 +0.000422 +0.013330 +17.7 +f11= 7.890973 +91.33 +0.000399 +0.595773 +14.5 +f12= 18.616678 +215.47 +0.000362 +0.984162 +15.5 +f13= 16.703890 +193.33 +0.000359 +0.524754 +15.1 +f14= 14.582041 +168.77 +0.000340 +0.353052 +16.7 +f15= 11.913144 +137.88 +0.000254 +0.049845 +10.8 +f16= 19.858675 +229.85 +0.000211 +0.149757 +8.2 +f17= 16.849761 +195.02 +0.000208 +0.423314 +8.7 +f18= 19.828865 +229.50 +0.000201 +0.471820 +7.8 +f19= 9.915434 +114.76 +0.000174 +0.107316 +7.1 +f20= 10.752580 +124.45 +0.000168 +0.127492 +7.2 +f21= 14.357945 +166.18 +0.000156 +0.561674 +7.6 +f22= 6.874254 +79.56 +0.000154 +0.010981 +6.1 +f23= 17.359629 +200.92 +0.000148 +0.855211 +6.3 +f24= 16.874621 +195.31 +0.000131 +0.761835 +5.5 +f25= 11.681082 +135.20 +0.000129 +0.762046 +5.5 +f26= 7.474528 +86.51 +0.000122 +0.497753 +4.5 +f27= 18.219348 +210.87 +0.000122 +0.021513 +5.2 +Zeropoint: 2.26827204e-05 +Residuals: 0.00151853162 +Table 26 +Frequency Solution of BN Cnc (=EPIC +211933524 ) Based on Kepler K2 Mission Campaigns 05, +16 and 18. +Frequency (d−1) +µHz Amplitude (106ppm) Phase (0-1) SNR +f0= 20.968746 +242.69 +0.000656 +0.081217 +35.3 +f1= 20.971849 +242.73 +0.000539 +0.347899 +29.0 +f2= 20.670825 +239.25 +0.000214 +0.544785 +11.9 +f3= 20.982369 +242.85 +0.000213 +0.123163 +11.5 +f4= 20.673376 +239.28 +0.000202 +0.613332 +11.1 +f5= 19.323688 +223.65 +0.000165 +0.433353 +9.1 +f6= 20.986503 +242.90 +0.000141 +0.956876 +7.6 +f7= 19.327869 +223.70 +0.000130 +0.117184 +7.2 +f8= 20.663785 +239.16 +0.000129 +0.281142 +7.2 +f9= 20.958855 +242.58 +0.000119 +0.693097 +6.4 +Zeropoint: -1.24703794e-06 +Residuals: 0.00125937867 + +48 +A.-Y. Zhou +Table 27 +Frequency Solution of BV Cnc (=EPIC +211931309 ) Based on Kepler K2 Mission Campaigns 05, +16 and 18. +Frequency (d−1) +µHz Amplitude (106ppm) Phase (0-1) SNR +f0= 16.454138 +190.44 +0.001119 +0.844489 +75.1 +f1= 15.708030 +181.81 +0.000542 +0.148140 +33.8 +f2= 20.254236 +234.42 +0.000455 +0.355272 +32.7 +f3= 11.236416 +130.05 +0.000401 +0.292173 +21.1 +f4= 19.281025 +223.16 +0.000380 +0.515107 +26.6 +f5= 11.079418 +128.23 +0.000328 +0.703124 +17.2 +f6= 16.027988 +185.51 +0.000325 +0.654100 +20.6 +f7= 16.437150 +190.24 +0.000323 +0.464989 +21.7 +f8= 10.523808 +121.80 +0.000260 +0.833700 +13.5 +f9= 17.745867 +205.39 +0.000246 +0.092627 +18.4 +f10= 17.745996 +205.39 +0.000240 +0.348701 +18.0 +f11= 13.818642 +159.94 +0.000234 +0.633451 +16.6 +f12= 18.454742 +213.60 +0.000231 +0.210441 +16.8 +f13= 9.387727 +108.65 +0.000215 +0.270959 +12.2 +f14= 1.143246 +13.23 +0.000163 +0.374624 +5.6 +f15= 10.093334 +116.82 +0.000158 +0.217808 +9.2 +f16= 16.946492 +196.14 +0.000157 +0.538174 +11.0 +f17= 10.437652 +120.81 +0.000141 +0.353718 +7.3 +f18= 16.388637 +189.68 +0.000141 +0.488430 +9.5 +f19= 9.936352 +115.00 +0.000139 +0.571865 +8.1 +f20= 18.421911 +213.22 +0.000138 +0.974453 +10.1 +f21= 20.967817 +242.68 +0.000135 +0.224543 +9.8 +f22= 16.037943 +185.62 +0.000126 +0.359109 +8.0 +f23= 16.016966 +185.38 +0.000122 +0.064544 +7.7 +f24= 18.333466 +212.19 +0.000112 +0.323422 +8.1 +f25= 16.787254 +194.30 +0.000112 +0.519380 +7.8 +f26= 21.760402 +251.86 +0.000097 +0.072052 +7.0 +Zeropoint: -2.19461936e-06 +Residuals: 0.000994116206 +Table 28 +Frequency Solution of HD 73712 (=EPIC +211941583 ) Based on K2 Mission Campaigns 05, 16 and +18. +Frequency (d−1) +µHz +Amplitude (mag) Phase (0-1) SNR +f0= 7.076171 +81.90 +0.000356 +0.209552 +20.4 +f1= 7.169653 +82.98 +0.000359 +0.974926 +20.6 +f2= 8.485967 +98.22 +0.000274 +0.233071 +19.1 +f3= 3.142627 +36.37 +0.001106 +0.649304 +47.7 +f4= 3.142682 +36.37 +0.000990 +0.986130 +42.7 +f5= 7.082444 +81.97 +0.000473 +0.589538 +27.1 +f6= 2.073726 +24.00 +0.000285 +0.639527 +10.9 +f7= 0.415086 +4.80 +0.000259 +0.864588 +9.0 +f8= 2.458471 +28.45 +0.000253 +0.297834 +10.2 +f9= 1.430804 +16.56 +0.000239 +0.449906 +8.5 +f10= 0.394456 +4.57 +0.000203 +0.529552 +7.0 +f11= 9.544075 +110.46 +0.000186 +0.389608 +14.9 +f12= 2.093481 +24.23 +0.000183 +0.126856 +7.0 +f13= 8.667229 +100.32 +0.000174 +0.155538 +12.2 +f14= 6.154719 +71.24 +0.000159 +0.098029 +8.7 +f15= 14.912859 +172.60 +0.000146 +0.268945 +18.5 +f16= 9.671240 +111.94 +0.000140 +0.393567 +11.2 +f17= 5.993164 +69.37 +0.000114 +0.895376 +6.3 +f18= 7.088776 +82.05 +0.000111 +0.271384 +6.4 +f19= 7.487203 +86.66 +0.000108 +0.061886 +6.6 +f20= 6.118509 +70.82 +0.000108 +0.023652 +5.9 +f21= 4.212949 +48.76 +0.000102 +0.894734 +5.4 +f22= 13.570806 +157.07 +0.000093 +0.586826 +10.0 +f23= 8.474514 +98.08 +0.000091 +0.059191 +6.3 +f24= 9.713114 +112.42 +0.000089 +0.006138 +7.1 +f25= 12.731512 +147.36 +0.000075 +0.535333 +8.5 +f26= 16.617048 +192.33 +0.000062 +0.019669 +8.4 +f27= 14.741815 +170.62 +0.000061 +0.385736 +9.1 +f28= 14.748357 +170.70 +0.000058 +0.811897 +7.0 +f29= 11.900878 +137.74 +0.000050 +0.246791 +6.7 +Zeropoint: 7.29679206e-06 mag +Residuals: 0.000806757647 mag + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +49 +Table 29 +Frequency Solution of HD 73712 (=TIC +175261925) Based on TESS Sectors 44 and 46. +Frequency (d−1) +Amplitude (mag) Phase (0-1) SNR +f0= 7.079656 +0.000672 +0.630162 +24.9 +f1= 7.166453 +0.000527 +0.164607 +19.5 +f2= 0.723091 +0.000427 +0.638129 +9.9 +f3= 2.002063 +0.000342 +0.029550 +9.0 +f4= 0.411645 +0.000341 +0.290673 +8.0 +f5= 2.460935 +0.000319 +0.458195 +8.4 +f6= 1.434695 +0.000291 +0.340839 +7.0 +f6 =2f2± 0.011488 +0.000291 +0.340839 +7.0 +f7= 2.078010 +0.000285 +0.146605 +7.5 +f8= 6.157444 +0.000270 +0.953221 +9.2 +f9= 9.541868 +0.000263 +0.176722 +13.8 +f10= 9.668234 +0.000250 +0.865863 +13.1 +f11= 8.482500 +0.000224 +0.813607 +9.3 +f12= 14.746600 +0.000198 +0.700070 +18.3 +f13= 11.895700 +0.000144 +0.317200 +9.9 +f14= 5.985360 +0.000138 +0.127113 +4.6 +f15= 13.788900 +0.000134 +0.309149 +11.1 +f16= 16.619800 +0.000121 +0.426514 +17.2 +f17= 12.730170 +0.000120 +0.602488 +9.0 +f18= 12.577700 +0.000080 +0.012942 +6.7 +f19= 14.159080 +0.000078 +0.144756 +6.0 +f19 =2f0± 0.000233 +0.000078 +0.144756 +6.0 +f20= 14.915990 +0.000073 +0.838670 +6.9 +f21= 16.031650 +0.000064 +0.480007 +6.8 +f22= 18.491500 +0.000045 +0.615876 +8.1 +f23= 27.713280 +0.000043 +0.408965 +7.6 +f24= 21.414600 +0.000029 +0.981252 +10.8 +f24 =f0 + 2f1± 0.002038 +0.000029 +0.981252 +10.8 +Zeropoint: 6.88636843 mag +Residuals: 0.00068237 mag +Table 30 +Frequency Solution of BL Cam (=TIC +392774261) Based on TESS Sector 19. +Frequency (d−1) +Amplitude (mag) Phase (0-1) +SNR +f0= 25.577327 +0.084966 +0.734851 +448.7 +f1= 51.154524 +0.019832 +0.042085 +171.2 +f1 =2f0± 0.000129 +0.019832 +0.042085 +171.2 +f2= 25.253052 +0.004748 +0.617959 +25.1 +f3= 0.040388 +0.004443 +0.167166 +9.2 +f4= 76.733486 +0.003587 +0.197802 +38.7 +f4 =3f0± 0.001506 +0.003587 +0.197802 +38.7 +f5= 46.392190 +0.002723 +0.608231 +23.0 +f6= 50.830059 +0.002195 +0.306296 +18.3 +f6 =f0 + f2± 0.000321 +0.002195 +0.306296 +18.3 +f7= 1.581920 +0.001988 +0.939041 +4.3 +f8= 51.395083 +0.001818 +0.101890 +15.7 +f9= 53.860289 +0.001652 +0.217159 +13.9 +f10= 51.115324 +0.001535 +0.262394 +15.1 +f10 =f1 − f3± 0.001188 +0.001535 +0.262394 +15.1 +f11= 44.129976 +0.001297 +0.142844 +13.3 +f12= 45.198400 +0.001073 +0.794374 +14.4 +f13= 33.670584 +0.001049 +0.538667 +14.7 +f14= 71.972096 +0.001018 +0.987347 +12.4 +f15= 31.259214 +0.000956 +0.314487 +10.2 +f16= 50.690170 +0.000928 +0.439177 +11.3 +f17= 41.657884 +0.000895 +0.582047 +10.6 +f18= 73.461830 +0.000841 +0.757529 +9.8 +f19= 30.321565 +0.000829 +0.056768 +7.7 +f20= 50.261369 +0.000818 +0.786735 +12.6 +f21= 31.684624 +0.000815 +0.848857 +8.7 +f22= 24.000882 +0.000802 +0.586480 +8.1 +f23= 76.413228 +0.000780 +0.688821 +6.8 +f23 =f1 + f2± 0.005651 +0.000780 +0.688821 +6.8 +f24= 47.545388 +0.000725 +0.950933 +8.4 +f25= 52.555813 +0.000708 +0.829014 +4.3 +f26= 79.435438 +0.000693 +0.330379 +4.9 +f27= 24.104497 +0.000652 +0.520928 +8.4 +f28= 23.906069 +0.000646 +0.628600 +6.1 +f29= 76.967925 +0.000628 +0.416855 +5.9 +f30= 53.447445 +0.000620 +0.279101 +7.7 +f31= 40.627385 +0.000514 +0.490527 +4.0 +f32= 32.758887 +0.000501 +0.904324 +4.0 +f33= 58.126402 +0.000474 +0.830386 +6.8 +f34= 56.833269 +0.000462 +0.386790 +5.7 +f35= 20.810244 +0.000457 +0.869261 +5.8 +f36= 71.647385 +0.000457 +0.276940 +5.0 +f37= 33.275041 +0.000436 +0.192975 +6.8 +f38= 102.310527 +0.000418 +0.953923 +5.3 +f38 =f0 + f3± 0.000286 +0.000418 +0.953923 +5.3 +f39= 78.129373 +0.000406 +0.841000 +4.6 +Zeropoint: 12.6150921 mag +Residuals: 0.00696801437 mag +Table 31 Frequency Solution of TYC 3413-228-1 (=TIC +309661089) Based on TESS Sector 47. +Frequency +Amplitude +Phase +f0 = 0.076199 +0.004246 +0.546173 +f1 = 0.145024 +0.001386 +0.266956 +f2 = 0.172062 +0.000716 +0.734760 +f3 =0.5f0± 0.001229 +0.000435 +0.306296 +f4 =f2 + f3± 0.002458 +0.000632 +0.899568 +f5 =3f0 + f3± 0.002458 +0.000334 +0.292986 +f6 =3f1 +0.000148 +0.912223 +Zeropoint: -0.000165870177 mag +Residuals: 0.000774585274 mag + +50 +A.-Y. Zhou +Table 32 +Frequency Solution of HD 48270=TIC +307618601 Based on TESS Sector 20. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +SNR +f0= 0.161778 +0.000451 +0.861076 +7.6 +f1= 0.982090 +0.000290 +0.896451 +5.5 +f2= 0.553853 +0.000270 +0.674589 +5.1 +f3= 1.052511 +0.000263 +0.249170 +5.7 +f4= 0.430140 +0.000253 +0.713577 +4.2 +f4 =f1 − f2± 0.001903 +0.000253 +0.713577 +4.2 +f5= 0.810795 +0.000235 +0.306757 +4.5 +f5 =5f0 ± 0.001903 +0.000235 +0.306757 +4.5 +Zeropoint: 5.9495583 mag +Residuals: 0.00037776471 mag +Table 33 Frequency Solution of TIC 309661100 Based +on TESS Sectors 20 and 47. +Label +Frequency (d−1) +Amplitude (ppt) +Phase +f0 +0.761729 +25.961933 +0.614266 +2f0 +1.517933 +9.926124 +0.992423 +4f0 +3.040717 +3.764533 +0.617396 +Zeropoint: 8307.4597 +Residuals: 15.4959 +Table 34 Frequency Solution of ι Boo (= TIC 310381204) +Based on TESS Data in Sectors 22,23,49 and 50. +Label +Frequency +Amplitude +Phase +f0 +36.752011 +0.002364 +0.804260 +f1 +32.632110 +0.000934 +0.138520 +f2 +29.694026 +0.000678 +0.305748 +f3 +25.142709 +0.000463 +0.195451 +f4 +20.680305 +0.000437 +0.987688 +f5 +29.950102 +0.000353 +0.279074 +f6 +12.425428 +0.000349 +0.214981 +f7 +12.577021 +0.000327 +0.401567 +f8 +25.200827 +0.000321 +0.806041 +f9 +0.029640 +0.000477 +0.498484 +f10 +23.724616 +0.000295 +0.757859 +f11 +16.697468 +0.000267 +0.904054 +f12 +0.038078 +0.000314 +0.808798 +f13 +22.456986 +0.000222 +0.383094 +f14 +0.127140 +0.000239 +0.073000 +f15 +11.482119 +0.000209 +0.606351 +f16 +27.568840 +0.000198 +0.349852 +f17 +48.018360 +0.000190 +0.042267 +f18 +0.262600 +0.000185 +0.736287 +f19 +50.137099 +0.000182 +0.158605 +f20 +8.950240 +0.000179 +0.131638 +f21 +27.679861 +0.000169 +0.100654 +f22 +45.536400 +0.000167 +0.623741 +f23 +29.617119 +0.000168 +0.633017 +f24 +0.019760 +0.000190 +0.583419 +f25 +20.450560 +0.000158 +0.167599 +f26 +35.879740 +0.000142 +0.183426 +f27 +11.656840 +0.000141 +0.391520 +f28 +37.090820 +0.000132 +0.458191 +f29 +0.148720 +0.000137 +0.501795 +f30 +43.350000 +0.000124 +0.898948 +f31 +39.157760 +0.000117 +0.607877 +f32 +35.896580 +0.000113 +0.345798 +f33 +23.929040 +0.000106 +0.947335 +f34 +25.129980 +0.000102 +0.108191 +f35 +36.767840 +0.000102 +0.304765 +f36 +15.487620 +0.000097 +0.863676 +f37 +43.722060 +0.000097 +0.102456 +f38 +39.105500 +0.000093 +0.338326 +f39 +11.979180 +0.000089 +0.799689 +f40 +6.720160 +0.000089 +0.524168 +· · · · · · +Full 65 frequencies are provided in online version +Zeropoint: 4.95133591 +Residuals: 0.00040283 +Table 35 +Frequency Solution of HD 227681 (=TIC +41194739) Based on TESS Sectors 14, 15 and 41. +Frequency (d−1) +Amplitude (mag) +Phase (0-1) +f0= 0.332810 +0.004006 +0.956256 +f1= 0.542065 +0.000688 +0.382507 +f2 =f1 + f4± 0.006669 +0.000992 +0.583921 +f3= 1.244771 +0.000656 +0.306526 +f4= 11.495900 +0.000139 +0.580036 +Zeropoint: -4.02857291e-05 mag +Residuals: 0.00249460618 mag + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +51 +Table 36 Frequency Solution of TYC 2682-229-1 (=TIC +89755468) Based on TESS Sector 15. +Frequency (d−1 +Amplitude (mag) +Phase (0–1) +f0= 0.159915 +298150.869000 +0.633925 +f1 =0.5f0± 0.008776 +260.034291 +0.011480 +f2 =0.5f1± 0.008335 +187.564444 +0.772442 +f3 =3f1± 0.009316 +182.704475 +0.163471 +f3 =f0 + f1± 0.008236 +182.704475 +0.163471 +f4 =f1 + 2f2± 0.000904 +298101.124000 +0.169587 +f5 =f0 + 2f2± 0.000076 +105.265641 +0.566421 +f6 =f1 + f3± 0.003702 +52.322193 +0.607845 +f7 =2f0 + f3± 0.003504 +19.676675 +0.608003 +f8 =2f0 + 2f2± 0.002804 +26.807164 +0.328779 +Zeropoint: 78719.1848 +Residuals: 52.8136565 +Table 37 +Frequency Solution of 21 Com (=TIC +393819751) Based on TESS Sectors 22 and 49. +Frequency (d−1) +µHz Amplitude (mag) Phase (0-1) +SNR +f0= 0.974758 +11.28 +0.003109 +0.863137 +110.3 +f1= 0.488733 +5.66 +0.001760 +0.017190 +51.8 +f2= 1.462129 +16.92 +0.000407 +0.112578 +28.0 +f3= 2.438256 +28.22 +0.000081 +0.714194 +8.2 +f4= 2.438256 +28.22 +0.000081 +0.714194 +8.2 +f5= 3.410295 +39.47 +0.000072 +0.119697 +8.1 +f6= 1.941371 +22.47 +0.000068 +0.362410 +6.0 +f7= 2.924262 +33.85 +0.000066 +0.461166 +6.9 +f8= 3.900367 +45.14 +0.000050 +0.950478 +5.9 +f9= 4.879241 +56.47 +0.000048 +0.411910 +6.4 +f10= 4.390272 +50.81 +0.000033 +0.451993 +4.1 +Zeropoint: -7.38265987e-06 mag +Residuals: 0.000354434559 mag +Table 38 Frequency Solution of TYC 6672-772-1 (=TIC +99180739) Based on TESS Sectors 10 and 37. +Frequency (d−1) µHz Amplitude (mag) Phase (0-1) SNR +f0= 0.132213 +1.53 +0.005182 +0.123044 +38.1 +f1= 0.280924 +3.25 +0.002417 +0.215510 +22.4 +f2= 0.159831 +1.85 +0.001221 +0.682503 +9.0 +f3= 0.052176 +0.60 +0.001117 +0.331570 +8.2 +f4= 0.291053 +3.37 +0.001051 +0.214937 +9.8 +f5= 0.341499 +3.95 +0.000876 +0.619822 +8.1 +f6= 0.212683 +2.46 +0.000727 +0.591576 +5.3 +Zeropoint: 3.62211924e-05 mag +Residuals: 0.00168076396 mag +Table 39 +Frequency Solution of CD-54 7154 (=TIC +173503902) Based on TESS Sectors . +Frequency (d−1) +µHz Amplitude (ppt) Phase (0-1) +SNR +f0= 9.162285 +106.04 +0.024388 +0.032680 +461.0 +f1= 9.369020 +108.44 +0.015415 +0.784659 +291.4 +f2= 15.404979 +178.30 +0.002431 +0.997096 +62.3 +f3= 9.136657 +105.75 +0.002213 +0.693960 +41.8 +f4= 0.025979 +0.30 +0.001865 +0.229600 +12.8 +f5= 18.531188 +214.48 +0.001747 +0.362383 +36.6 +f6= 16.246765 +188.04 +0.001589 +0.387674 +34.7 +f7= 38.816643 +449.27 +0.001492 +0.974287 +52.7 +f8= 38.818696 +449.29 +0.001491 +0.599908 +52.7 +f9= 9.879882 +114.35 +0.001317 +0.638641 +27.3 +f10= 9.128695 +105.66 +0.001211 +0.108822 +22.9 +f11= 18.327258 +212.12 +0.001005 +0.123664 +21.1 +f12= 9.188212 +106.35 +0.000881 +0.364610 +16.7 +f13= 18.528015 +214.44 +0.000719 +0.153648 +15.1 +f14= 18.740728 +216.91 +0.000599 +0.551185 +12.6 +f15= 16.459082 +190.50 +0.000568 +0.611640 +12.4 +f16= 20.629477 +238.77 +0.000566 +0.550562 +14.4 +f17= 10.258152 +118.73 +0.000514 +0.340838 +10.6 +f18= 27.265133 +315.57 +0.000501 +0.225157 +12.4 +f19= 15.427366 +178.56 +0.000452 +0.971293 +11.6 +f20= 16.213636 +187.66 +0.000430 +0.879982 +9.4 +f21= 19.408859 +224.64 +0.000415 +0.419254 +9.5 +f22= 9.769503 +113.07 +0.000389 +0.399423 +8.1 +f23= 17.375864 +201.11 +0.000385 +0.785259 +7.8 +f24= 28.278836 +327.30 +0.000364 +0.726266 +10.1 +f25= 10.380897 +120.15 +0.000353 +0.300692 +7.3 +f26= 26.763543 +309.76 +0.000327 +0.803219 +8.1 +f27= 24.569955 +284.37 +0.000322 +0.961122 +7.7 +f28= 14.695983 +170.09 +0.000304 +0.724394 +9.5 +f29= 27.774338 +321.46 +0.000276 +0.617392 +7.7 +f30= 34.370361 +397.81 +0.000271 +0.707302 +8.9 +Zeropoint: 0.000301970746 +Residuals: 0.00212318609 +Table 40 Frequency Solution of GSC 04040-01606 (=TIC +372724683) Based on TESS Sectors 18, 24 and 25. +Frequency (d−1) +µHz +Amplitude (ppt) +Phase (0-1) +SNR +f0= 7.543132 +87.30 +0.007421 +0.094978 +45.6 +f1= 7.168275 +82.97 +0.006057 +0.579494 +36.4 +f2= 6.515472 +75.41 +0.002124 +0.868479 +13.2 +f3= 11.854332 +137.20 +0.001177 +0.756068 +8.4 +f4= 6.754491 +78.18 +0.001052 +0.915680 +6.3 +f5= 14.217330 +164.55 +0.001037 +0.574169 +6.7 +f6= 11.977436 +138.63 +0.000950 +0.933337 +6.8 +f7= 6.190357 +71.65 +0.000880 +0.662468 +5.8 +f8= 12.324753 +142.65 +0.000823 +0.847252 +6.0 +f9= 13.612377 +157.55 +0.000810 +0.430857 +5.5 +f10= 6.500103 +75.23 +0.000747 +0.543673 +4.7 +f11= 7.948547 +92.00 +0.000727 +0.034663 +4.6 +f12= 14.442319 +167.16 +0.000705 +0.560596 +4.5 +f13= 9.109439 +105.43 +0.000643 +0.394886 +4.7 +f14= 3.252631 +37.65 +0.000612 +0.442822 +4.2 +Zeropoint: 0.998299271 +Residuals: 0.00589591286 + +52 +A.-Y. Zhou +Table 41 Discoveries and Updates of 5 Landolt Photometric Standard Stars in Present Work. +Star Identification +TIC +Status +1. SA 41-660= UCAC4 678-116341, 12.m804 +TIC 455553286 +New DSCT +2. SA 26-93 = IRAS 06415+4434, 1107 +TIC 307650624 +New variable +3. SA 26-95 = UCAC4 673-048807, 11.m98 +TIC 307650637 +non-variable +4. SA 26-96 = UCAC4 673-048808, 10.m75 +TIC 307650596 +non-variable +5. SA 41-654= BD+44 3978, V =10.m03 +TIC 455517315 +non variable +Table 42 +Frequency Solution of UCAC4 678-116341 +(=TIC 455553286) Based on TESS Sectors 15 and 16. +Frequency +Amplitude(ppt) +Phase +S/N +f0 = 0.26964824 +0.00291296 +0.087694 +7.4 +f1 = 0.54242835 +0.00206943 +0.812136 +5.3 +f2 = 0.23642194 +0.00189728 +0.615010 +4.6 +Zeropoint: 0.000154424229 +Residuals: 0.00599494917 +Table 43 Discoveries and Updates Presented in Current +Work on a Sample of 50 Stars. +SN TIC ID +Simbad Identification +Variability Status +01 TIC 279431011 HD 53166 +New DSCT +02 TIC 279476396 HD 53349 +New DSCT +03 TIC 90869850 +TYC 2671-577-1 +New DSCT +04 TIC 40831024 +HD 227647 +New DSCT +05 TIC 41189624 +NGC 6871: HD 191025 +New DSCT +06 TIC 89757305 +NGC 6871: HD 227658 +New BCEP +07 TIC 279476440 CD-58 1608 +New gDor +08 TIC 99180739 +TYC 6672-772-1 +New solar-like oscillator +09 TIC 356473060 TYC 3413-187-1 +New Pu* +10 TIC 309661089 TYC 3413-228-1 +New Pu* +11 TIC 309661100 PM J07472+5020 +New Pu* +12 TIC 307618601 HD 48270 +New Pu*: Irregular +13 TIC 279510617 TYC 8549-1773-1 +New Var. +14 TIC 89755468 +TYC 2682-229-1 +New EB +15 TIC 246938869 AD Ari +New EB +16 TIC 41194739 +NGC 6871: HD 227681 +New EB +17 TIC 41195818 +NGC 6871: UCAC4 630-092212 New EW +18 TIC 42247329 +IRAS 20069+3648 +New SR +19 TIC 175261925 NGC 2632: HD 73712 +SB+EB+dS* +20 TIC 13876370 +NGC 6910: V2245 Cyg +SB+EW+Pu* +21 TIC 41195917 +NGC 6871: V2238 Cyg +dS* +22 TIC 90350726 +NGC 6871: V1821 Cyg +dS* +23 TIC 27936176 +V383 Car = HD 52788 +dS* +24 TIC 166177270 IT Dra = SAO 16394 +dS* +25 TIC 307678320 NGC 2632: BR Cnc +dS* +26 TIC 175240124 NGC 2632: BU Cnc +dS* +27 TIC 175264749 NGC 2632: BN Cnc +dS* +28 TIC 175264756 NGC 2632: BV Cnc +dS* +29 TIC 437039231 NGC 2632: EX Cnc +dS* +30 TIC 173503902 CD-54 7154 +dS* +31 TIC 372724683 GSC 04040-01606 +dS* +32 TIC 310381204 ι Boo = HR 5350 +dS* +33 TIC 266794067 V2455 Cyg = HD 204615 +HADS +34 TIC 392774261 BL Cam +SX* +35 TIC 175261942 2MASS 08400416+1924502 +ER, mono-periodic +33 TIC 307650624 IRAS 06415+4434 +SR/LPV +36 TIC 393819751 21 Com +Ap/CP2 +37 TIC 61236485 +AL Tri = GSC 2293-1021 +EW +38 TIC 279476423 TYC 8593-1255-1 +uncertain +39 TIC 41195988 +GSC 02683-03318 +? constant +40 TIC 455553286 UCAC4 678-116341 +constant +41 TIC 307650596 UCAC4 673-048808 +constant +42 TIC 307650637 UCAC4 673-048807 +constant +43 TIC 455517315 BD+44 3978 +constant +44 TIC 41195891 +NGC 6871: HD 227682 +constant +45 TIC 41196013 +NGC 6871: UCAC4 630-092237 constant +46 TIC 90350490 +2MASS J20064266+3550552 +constant +47 TIC 90350607 +2MASS J20064698+3551472 +constant +48 +TIC 279361762 HD 52788 +dS* +49 +TIC 402338608 IP Vir +no TESS LC +50 +TIC 416302408 V577 Oph +no TESS LC + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +53 +1842 +1844 +1846 +1848 +1850 +1852 +1854 +TBJD +0.98 +0.99 +1.00 +1.01 +1.02 +1.03 +1.04 +Amplitude (mmag) +Orbit-1 of TESS Sector 20 +Observed +3th-order polyfit +1844 +1846 +1848 +1850 +1852 +1854 +TBJD +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Amplitude (mmag) +Orbit-1 of TESS Sector 20 +Residuals +1858 +1860 +1862 +1864 +1866 +1868 +TBJD +0.994 +0.996 +0.998 +1.000 +1.002 +Amplitude (mag) +Orbit-2 of TESS Sector 20 +Observed +3th-order polyfit +1858 +1860 +1862 +1864 +1866 +1868 +TBJD +−1.25 +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +Amplitude (mmag) +Orbit-1 of TESS Sector 20 +Residuals +0 +1 +2 +3 +4 +5 +Frequency (d−1) +0 +50 +100 +150 +200 +Observed (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0 +10 +20 +30 +40 +50 +60 +4 Significant peaks prewhitened +TIC 307650624 - Sectors S20-Res-LC +Fig. 51 TESS light curves (upper) and amplitude spectrum of TIC 307650624 (IRAS 06415+4434). + +54 +A.-Y. Zhou +1845 +1850 +1855 +1860 +1865 +TBJD +0.996 +0.997 +0.998 +0.999 +1.000 +1.001 +1.002 +1.003 +Normalized flux +TIC 307650596 in TESS Sector 20 +Observed +0 +1 +2 +3 +4 +5 +Frequency (d−1) +0.0 +0.2 +0.4 +0.6 +0.8 +Observed (mmag) +Amplitude spectrum +Prewhitened Residuals +Significant level at SNR=4.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +0.1 +0.2 +0.3 +3 Significant peaks prewhitened +TIC 307650596 - Sectors S20 +Fig. 52 TESS light curves (upper) and amplitude spectrum of TIC 307650596. + +Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS +55 +1845 +1850 +1855 +1860 +1865 +1870 +Time - 2457000 [BTJD days] +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +sap_flux +TIC 307650637_S0020-- Normalized SAP_flux +1842 +1843 +1844 +1845 +1846 +1847 +1848 +TBJD +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +Amplitude (mag) +1st-Orbit of TESS Sector 20 +Observed +3th-order polyfit +1848 +1849 +1850 +1851 +1852 +1853 +1854 +1855 +TBJD +0.975 +1.000 +1.025 +1.050 +1.075 +1.100 +1.125 +1.150 +Amplitude (mag) +1st-Orbit of TESS Sector 20 +Observed +3th-order polyfit +1858 +1860 +1862 +1864 +1866 +1868 +TBJD +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +1.01 +Amplitude (mag) +2nd-Orbit of TESS Sector 20 +Observed +4th-order polyfit +1845 +1850 +1855 +1860 +1865 +TBJD +−15 +−10 +−5 +0 +5 +10 +15 +Amplitude (mmag) +TIC 307650637 in TESS Sect r 20 +Observed +Fig. 53 TESS light curves (upper) and amplitude spectrum of TIC 307650637. + +56 +A.-Y. Zhou +Acknowledgements This paper includes data collected +with the TESS mission and by the Kepler mission and +obtained from the MAST data archive at the Space +Telescope Science Institute (STScI). Funding for the +TESS mission is provided by the NASA Explorer +Program. Funding for the Kepler mission is provided +by the NASA Science Mission Directorate. STScI is +operated by the Association of Universities for Research +in Astronomy, Inc., under NASA contract NAS 5–26555. +We acknowledge the use of TESS data, which are +derived from pipelines at the TESS Science Processing +Operations Center (SPOC). TESS High Level Science +Products (HLSP) produced by the Quick-Look Pipeline +(QLP) at the TESS Science Office at MIT, which +are publicly available from the Mikulski Archive for +Space Telescopes (MAST). This research has made use +of the SIMBAD/VizieR databases, operated at CDS, +Strasbourg, France. This research has made use of +the International Variable Star Index (VSX) database, +operated at AAVSO, Cambridge, Massachusetts, USA. +Special acknowledgements go to all the projects and +surveys mentioned in this work including but not +limited to: OGLE-V, ASAS-3, ASAS-SN, GCVS, ZTF, +SuperWASP, 2MASS, ROTSE, CRTS/CSS, LAMOST, +and other surveys and websites from where catalogs +of δ Sct stars and relevant data were downloaded and +compiled into the new catalog in this paper. Some of +them are footnoted in text. This work has made use of +data from the European Space Agency (ESA) mission +Gaia +( +h +t +t +p +s +: +/ +/ +w +w +w +. +c +o +s +m +o +s +. +e +s +a +. +i +n +t +/ +g +a +i +a +), +processed by the Gaia Data Processing and Analysis +Consortium (DPAC, +h +t +t +p +s +: +/ +/ +w +w +w +. +c +o +s +m +o +s +. +e +s +a +. +i +n +t +/ +w +e +b +/ +g +a +i +a +/ +d +p +a +c +/ +c +o +n +s +o +r +t +i +u +m +). 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X., et al. 2019, AJ, 157, +178 + diff --git a/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/load_file.txt b/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f226ffa7e26870641a33ad11816ca65a7c796fa --- /dev/null +++ b/jdE_T4oBgHgl3EQf4xzd/content/tmp_files/load_file.txt @@ -0,0 +1,6134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf,len=6133 +page_content='RAA Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22 (2022) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9, 000–000 © 2022 National Astronomical Observatories, CAS and IOP Publishing Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='raa-journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org http://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org/raa Research in Astronomy and Astrophysics A Catalog of 59 Thousand δ Scuti Stars and A Dozen Discoveries of New Variables with TESS Data A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' aiying@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='cn Abstract We present discoveries of stellar pulsation, variability, binarity and multiperiodicity among a sample of 50 stars including types of DSCT, HADS, SX Phe, EB, and photometric standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We initially aimed at checking the known δ Scuti star HD 52788 and its field stars with TESS data and found that the previously reported complex light variations with uncertain frequency solutions were partly caused by the two comparison stars, which turn out to be pulsating variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 52788 exhibits 135 pulsation frequencies in a small domain in 4–12 d−1 based on the non-differential Pre-search Data Conditioning Simple Aperture Photometry results of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The record high rich frequency solution turns HD 52788 into a distinctive and very interesting object among δ Sct stars for testing current stellar evolution and pulsation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Inspired by the discoveries around HD 52788, we extended our exploration to a small group of interested stars and resulted in discovery of 20 new variables including 5 δ Sct stars, 4 eclipsing binaries, and other kinds of pulsating variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In addition, based on existing sources, we have compiled a new comprehensive catalog of 59350 δ Sct stars, which is by far the largest collection of DSCT with TESS Input Catalog and Gaia DR3 cross-identifiers and a number of astronomical parameters extracted from TIC and Gaia archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With the new catalog covering almost a hundred times the earlier list, the δ Sct domain on the pulsating H-R diagram is largely extended, which would impact the theoretical borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Key words: stars: oscillation (pulsation) — stars: binaries: eclipsing: — stars: variables: δ Scuti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' γ Doradus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HADS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SX Phe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' β Cephei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Solar-like — stars: variables: general — methods: statistical — techniques: photometric — catalogues — stars: individual: HD 52788,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 53166,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 53349,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' AD Ari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 191025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 227658,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 227647,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 227681,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' CD-58 1608,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TIC 41195818,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TIC 309661100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TYC 2671-577-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' AL Tri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' IT Dra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' BR Cnc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' BU Cnc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' BV Cnc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' BN Cnc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' EX Cnc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HD 73712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V1821 Cyg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V2238 Cyg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V2245 Cyg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V2455 Cyg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' BL Cam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ι Boo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TYC 6672-772-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 21 Com 1 INTRODUCTION HD 52788 (=V383 Carinae = TIC 279361762 = SAO 234839 = HIP 33616 = ASASSN-V J065904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08- 583054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1, V = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m37, α2000 = 06h59m04s [104◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7667], δ2000 = −58◦30′54′′ [−58◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5147] ) was announced to be a δ Scuti star by Kurtz (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Based on a total of 104 hours of differential photometric observations obtained in 1978–1980, both Kurtz (1981) and Zhou (2004) failed to establish a consistent frequency solution for the pulsational behaviour of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The pulsation frequency spectrum of HD 52788 changed yearly with somewhat complex light variability, makes it a much interesting δ Sct target than others, referring to reviews of δ Sct variables by Handler (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Rodr´ıguez & Breger (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Breger (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' What makes the author interested is the cause of the unstable pulsation frequency spectrum of HD 52788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The poor frequency resolution could result from inadequate 0 Submitted to RAA on 2022 October 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This work is dedicated to my wife Jingyun Zhang who has been supporting my research all along.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' time series data, intrinsic rapid changing, and data issues using wrong stars as comparison among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Which is the true source?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Potential comparison stars in the target field were often discovered to be new variables, for example, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Jurcsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Due to the star in the southern sky, while the interested observers in the northern hemisphere, the question is could we have a chance of obtaining adequate data to derive a unique set of frequencies that represents the light variations of this star?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fortunately, the Transiting Exoplanet Survey Satellite (TESS, Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2015) data provide us with the best opportunity to check any star’s light variations more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We are delighted with our unexpected good fortune when checking TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We do find rich intrinsic pulsation of HD 52788 based on the perfect data sets, and serendipitously we reveal that the two comparison stars used in Kurtz’s differential photometry turned out to be pulsating variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This fact of using the wrong comparison would have added uncertainties in resolving the target’s pulsation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Inspired by the stellar variability detection results of HD 52788 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08355v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='SR] 19 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou and its comparisons, we decided to extend our studies to a group of interested stars in a couple of sky fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Now we report all our discoveries and preliminary results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Meanwhile, we describe the data reduction with newly developed Python-based tools for the undergoing project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Furthermore, we will report a series of results showing that TESS full-sky photometry survey has a huge space for serendipity and discovery even only examining a few selected known pulsating stars’ fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS is an MIT-led NASA mission dedicated to discovering transiting exoplanets orbiting nearby bright stars by an all-sky photometry survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS rotates every ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days per cycle along a unique highly elliptical lunar-resonant orbit around the Earth (about 600 km from Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS is equipped with four identical refractive cameras with a combined field-of-view (FOV) of 24x96 degrees (a segment of sky, known as an observing sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The lens assembly in each camera has a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 cm diameter entrance pupil (aperture) and a focal ratio f/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each of the four cameras has four 2048x2048 CCD detectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' total 16 CCDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each detector pixel sized 15x15 micron corresponds to 21 arcseconds in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS collects light in 600–1000 nm centered on the traditional Cousins I-band (IC, central wavelength = 786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Please refer to the details to TESS Science Data Products Description Document 1 and ”Characteristics of the TESS space telescope” web page 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS filter basically matches with Gaia Red Photometer with bandpass spanning in 630–1050 nm (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS cameras actually expose at a cadence of 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' That is, the CCDs take images and read them out continuously at 2-second intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, the 2- second frames are used for spacecraft guiding, they were not downloaded to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The images are processed on the spacecraft by the data handling unit (DHU, a Space Micro Image Processing Computer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The DHU stacks 2-second images in groups of 60 to produce 2-minute cadence images (in cut Target Pixel File, TPF) or 30- minute cadence for general observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' High cadence is needed for the detection of exoplanets, so exposures of planet search targets and other stars of particular interest (cataloged 200,000 primary stars) are obtained every 2 minutes while the Full-Frame Images (FFIs) of the entire field of view are returned every 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Finally, pixels in postage stamps around TESS mission target stars will be downloaded at a 2-minute cadence, while FFIs will be downloaded at the 30-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' These two sets of data will allow general variability studies for the vast majority of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The data on spacecraft are transmitted to Earth when the spacecraft reaches orbital perigee every ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each sector of the sky will be observed twice 1 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/missions-and-data/tess;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/files/live/sites/mast/files/home/missions-and-data/active- missions/tess/ documents/EXP-TESS-ARC-ICD-TM-0014-Rev-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='pdf 2 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gov/docs/tess/the-tess-space- telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html or https://tess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/science/ with 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 days observing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' So the TESS observation is most sensitive to exoplanets with periods of less than 13 days so that at least two transits are used for discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In the meantime, TESS high-precision uninterrupted photometry and time resolution (2-minute and 30- minute cadence) are perfect for stellar pulsation and asteroseismology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' the works by Lund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Furthermore, TESS produces additional pixels in little postage stamps surrounding a few (1000) bright asteroseismology targets and downloads at 20-second cadence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' the works by Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' After TESS first two-year prime mission, started in July 2020, TESS was revisiting the sky in an extended ongoing mission that records full-frame images at a fast ten-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With the advantage of TESS along with Kepler data, we are about to re-visit those special targets, including the selected High-Amplitude δ Scuti stars (HADS), SX Phe- type stars, pulsating subdwarf B stars (sdBV), pulsating white dwarfs, eclipsing binary systems with pulsating components, exoplanet-host pulsating stars, photometric standard stars, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We are going to characterize their true variability and complete pulsation contents, binarity, multiperiodicity, and whether or not they are hosts of exoplanets in a high precision level of less than a few parts per million (ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We check typical stars over the pulsational H-R diagram and even photometric standard stars over the full spectral series from O to M types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We will first pay close attention to those highly cited stars and poorly studied stars in the literature that either suffered from unresolved puzzles in light variations due to poor ground observations or were neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The aim is to refine our current understanding of variable stars by resolving high-precision light variations data from space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2 KEPLER K2 DATA The Kepler second mission (K2, Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2014) had released its data for open access 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each K2 Campaign has a duration of approximately 80 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curve data are usually provided in a 30-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Long-time coverage improves the frequency resolution of variability detection and is suitable for both stable and unstable periodic and non-periodic variations of a wide range of variable star studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' There are three kinds of light curve data available to the public: (1) K2 Systematics Correction (K2SC 4 ): The K2SC-detrended light curves are especially suited for studying variable stars in K2 photometry (by allowing us to remove the position-dependent systematics while 3 K2: Extending Kepler’s Power to the Ecliptic, The Ecliptic Plane Input Catalog (EPIC) for Kepler’s K2 mission, see http://keplerscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gov/K2/ 4 K2SC (K2 Systematics Correction) is a K2 light curve detrending tool that uses Gaussian processes to robustly model the systematics due to the Kepler telescope pointing jitter together with the astrophysical variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 3 keeping time-dependent variability intact), and searches for transiting planets (by allowing us to remove both the systematics and astrophysical variability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2) K2 Extracted Lightcurves (K2SFF): The lightcurves from K2 contain larger systematics than the original Kepler mission, due to the reduction in pointing precision as a result of having to rely on only two reaction wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Vanderburg & Johnson have created a technique to correct for the pointing-dependent nature of the pixel- level fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This correction improves the photometric precision by typical factors of 2-5, and results in the median photometric performance of K2 targets to within a factor of two of the original, 4-wheeled mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Such extracted lightcurves (using a variety of photometric apertures), as well as diagnostic plots, for each target, have been released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The FITS lightcurves include all data points (with flags to indicate thruster firing), as well as multiple versions (each FITS extension is an extracted lightcurve for a different aperture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' There are a total of twenty apertures provided: ten circular and ten based on the pixel response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The final extension contains the summed image from all the postage stamp frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (3) EVEREST (EPIC Variability Extraction and Removal for Exoplanet Science Targets) light curves, these are products from an open-source pipeline for removing instrumental systematics from K2 light curves, using a combination of pixel-level decorrelations to remove spacecraft pointing error and Gaussian processes to capture astrophysical variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves from campaigns 0 through 8, 102, 111, 112, 12, and 13 are currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Either one of K2SFF, K2SC, and EVEREST will be used depending upon whichever looks better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For our current work, we used K2SFF light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, the K2SFF data is still not flat, we have to further remove a profile by the polynomial fitting of orders between 4 and 24 to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each campaign data set might be divided into 2 or 3 portions for better fitting and detrending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then we put the residuals together for pulsation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 3 TESS DATA We first checked the TESS archive at the MAST Portal (Mikulski Archive for Space Telescopes 5 ) for available data of the interested target star HD 52788 (=V383 Carinae =TIC 279361762 = ASASSN-V J065904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08- 583054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1, α2000=06:59:04, δ2000=−58:30:53 [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7667◦, −58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5147◦], V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m40) and the two comparison stars used in literature C1=HD 53166 (=TIC 279431011, V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m1) and C2=HD 53349 (=TIC 279476396, V =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We downloaded the NASA’s Science Processing Operations Centre (SPOC) generated files, including both the extracted light curves (*-s lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits) and Target Pixel files (*-s tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The light curve files *-s lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits will be read directly with ASTROPY and LIGHTKURVE packages (The 5 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/portal/Mashup/Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' or https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/tess/ A s t r o p y C o l l a b o r a t i o n e t a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2 0 1 8 a , b ) i n a P y t h o n s c r i p t f o l l o w i n g t h e TESS Archive Manual 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The TESS data information for these three stars is listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' There are two kinds of SPOC-extracted light curves: one is Simple Aperture Photometry flux (SAP flux),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' which is the flux after summing the calibrated pixels within the TESS optimal photometric aperture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' and the other is Pre-search Data Conditioned Simple Aperture Photometry (PDCSAP flux),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' it is the SAP flux from which long-term trends have been removed using so-called Co- trending Basis Vectors (CBVs) and nominally corrected for instrumental variations and excessive scattered light removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' PDCSAP flux is usually cleaner data than the SAP flux and will have fewer systematic trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Thus PDCSAP flux is widely used for final analysis without further processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Dumusque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Astudillo- Defru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Demory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Battley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, the PDCSAP flux might suffer from loss or wrong-deletion of long-term and transient burst variability intrinsic to stars, for instance, as that pointed out by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Littlefield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With these concerns, SAP flux is used by some authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='von Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Steindl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Hon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Prˇsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Southworth & Van Reeth 2022) accompanied with additional custom processing such as detrendings depending on science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In the current work, we first did graphic screenings on each sector’s light curves for choosing SAP or PDCSAP flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A close visual inspection of both light curves plotted up and down and overlapping helped the author compare and judge which one is better for undergoing pulsation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' If the SAP flux looks no evident difference with PDCSAP flux, then use SAP flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' If the PDCSAP flux data lost too many points while the SAP flux data are flat, then we choose SAP flux, otherwise, PCDSAP is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In case of severe systematic trends (slope and steep brightness down or up in most cases), PDCSAP flux is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In any SAP or PDCSAP flux, all data points flagged as ”bad quality” were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' During a course of surveying a large amount of TESS light curves the author found that both PDCSAP and SAP fluxes could be sometimes suffered from a systematic shift in brightness between two orbits of a TESS sector, which would result in abnormal discontinuity and actual displacement of two orbits’ light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This is quite frequently occurred in HLSP-QLP (uncorrected SAP) light curves for long-period variables, for instance, Mira variables TIC 329891910, TIC 457021714, and TIC 359401246 on sector 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Displacements of a segment light curve also occurred within individual orbits, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' RRLyr star TIC 127088233 on sector 38 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Thus, a custom processing of correction to the displacement is 6 The Beginner Tutorial Notebooks: https://outerspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/ display/TESS/TESS+Archive+Manual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='com/spacetelescope/ notebooks/blob/master/notebooks/MAST/TESS/beginner how to use lc/ beginner how to use lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ipynb 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou applied whenever needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With the above criteria, these final used fluxes of PDCSAP and SAP mixture are ensured the TESS best estimate of the intrinsic variability of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' One thing was kept in mind that low-frequency instrumental systematics, which is either not removed by the TESS science pipeline, or introduced by the TESS reduction pipeline (Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Caution will be taken in picking up a frequency when strong low- frequency instrumental noise is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The final derived periods have ejected all possible aliases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We then checked the two comparison stars in General Catalog of Variable Stars (GCVS database, Version 2022 Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1, Samus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2017) 7 and Revised Version of the New Catalogue of Suspected Variable Stars (NSV Release 2, 1982, 1998, Kazarovets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022) 8 , and found that C2=HD 53349 =HR 2661 =NSV 3349 is a dwarf (DM, spectral type F0V) and a new suspected variable star but without further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We further looked for TESS Input Catalog (TIC) v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 and CTL v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01 catalog (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Paegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021), and adopted the basic astronomical parameters in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No variability type is provided in TIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 1 TESS observations (at two-minute cadence) of HD 52788, HD 53166 and HD 53349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Star Identifiers V Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V =HD 52788 = TIC 279361762 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m37 Fm dD C1=HD 53166 = TIC 279431011 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m1 A1V C2=HD 53349 = TIC 279476396 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m0 A8III Orbits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Sector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (Camera,CCD) 11–25, 61–85;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2–10,12,13, 27–39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='(4,4),(4,1),(4,3),(4,2) S0036 started on 2021 Mar 1 Time span BJD 2458354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='107–2459389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='716 (2018 Aug 22 23:50 – 2021 May 26 02:43 UT, 1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 days) Data length 24 sectors * 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days * 2 = 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='599 days Frequency resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0009656 d−1 Number of data points analyzed 419,915 (outliers and unusual data points excluded) Total number of data 422,046 Last, we proceed to process the TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In a few cases, we used fluxes counts directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, for comparison with literature data, we prefer to use magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Using pixels counts, the fluxes (SAP Flux and PDCSAP Flux), corresponding magnitudes (SAP mag and PDCSAP mag) are calculated using the following for- mula PDCSAP mag = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 log(PDCSAP Flux) + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The magnitude zero point of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4436 mag was selected for a resulting magnitude matching with the value in existing catalogs (Simbad/CDS) and especially with Tmag of TESS CTL v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01 or TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' By arbitrarily selecting a star, TIC 41196013 (Tmag=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m4745) which was observed in TESS sector 41 as non-variables in a time scale of days for calibrating the magnitude zero point, 7 online URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='sai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='su/gcvs/cgi-bin/search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='htm#cat 8 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gov/W3Browse/all/gcvsnsvars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html Table 2 Astronomical Parameters of HD 52788, HD 53166 and HD 53349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V=HD 52788 C1=HD 53166 C2=HD 53349 B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='788±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='028 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='473±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='029 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='006±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='245±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='016 Tmag 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0396 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='722 Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fm dD A1V A8III Teff 6838±134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='87 9562±186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='18 7118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='48±111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='54 log g 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='319±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='092 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='039±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='079 R/R⊙ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='412±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='216 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='026±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='053 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='988±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='069 M/M⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 and using the average magnitude in a sector or an orbit, we found 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4436 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2531 mag for converting SAP and PDCSAP fluxes into magnitudes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The data in a TESS sector which consists of two successive orbits observations (each ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days) have sometimes slightly different mean values or zero-point offsets, so the average of each orbit’s magnitudes or fluxes was subtracted individually to obtain a flattened final set of data with unique zero mean or zero-offset in order for Fourier analysis and least-squares fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Such flattening at a uniform level by subtracting individual means of each part was applied whenever necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A similar aligning- up also applied to the uneven cases from sector to sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Therefore light curves between two orbits and among sectors are always lined up before doing Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This adjustment of magnitude/flux zero points ensures the elimination of externally-entailed additional noise in the low-frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Concerning no atmospheric effect on the TESS photometry, it is not necessary to establish differential magnitude as in traditional ground astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Thus the above-calculated magnitudes were used in our analyses, this way also allows us to diagnose each star without effects brought in by comparison stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 4 SAMPLE OF TARGETS Besides the initial target HD 52788 and its two comparison stars, an extended small sample of 50 stars were selected in following categories (refer to Table 43): – Poorly studied,neglected known pulsating variable stars and eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For those with a few identifiers and linked references in Simbad/CDS, we annotate them ’plain stars’ — stars mostly with object types of ’* and IR’ in Simbad/CDS, the author’s notation hereafter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – Stars well studied and highly cited, but still not completely clear for its pulsation or light variations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – The stars that the author personally observed but not well characterized together with their neighborhoods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – Photometric standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 5 5 CATALOGS FOR CHECKING NEW VARIABLES For a given star with one of its identifiers, we first check it in popular existing catalogs for astronomical parameters and variability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Upon a finding of light variations, one needs to know whether the result is already reported in a published article, on the web pages of any known sky surveys or not ever, and make sure that the discovery of variability is not yet listed in existing catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, the situation of checking whether a suspected variable star was already reported in a catalog or not, it is just a recovered variable, becomes more challenging and more time-consuming than years ago due to the vast outputs of various surveys in the past two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As various-purpose sky surveys continue to evolve, the amount of newly discovered variable stars including eclipsing binary stars databases are increasing rapidly, some still need to be sifted through to identify a class of variabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The author attempted to find a com- plementary way for checking a star whose variability type is not classified in Simbad/VizieR/CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' After a survey of various sources (catalogs and websites) providing online databases of variables, especially those with dynamic updating, we established that for identifying variability types and classifications of newly discovered variables, a safe checking procedure should be first conducted against those well-known primary databases and catalogs of Simbad/VizieR at CDS, GCVC, the International Variable Stars Index (VSX) database of AAVSO (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022, 2006) 9 , key surveys including ASAS-SN Catalog of Variable Stars X (Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022) 10 , OGLE- IV (Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Pietrukowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020) 11 , ZTF (Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Ofek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020) 12 , SuperWASP (Pollacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' McMaster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021) 13 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=', see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' First, we scan all variable stars which have been reported or cataloged in literature and websites, then we focus on δ Sct stars mainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' How many δ Sct stars now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' What is the number of pulsators in binary systems reported so far?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' These are two tough questions in the author’s mind all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Finding answers to these questions may be beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In the earlier works (Zhou 2010, 2015) the author initiated a statistical exploration on pulsating binaries and hybrid pulsators, one of the results is a web-version catalog consisting of 4697 δ Sct stars extracted from Simbad/CDS as of the time 2014 October 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As an update, the author has made an effort to combine the known δ Sct stars based on the existing catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 9 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='aavso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org/vsx/ 10 Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=', 2022, MNRAS, in press :https://asas- sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/variables 11 https://ogledb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='pl/ ogle/OCVS/ 12 Zwicky Transient Facility: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ztf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/ 13 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='superwasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org/vespa/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org/ projects/ajnorton/superwasp-variable-stars/about/research Meanwhile, several surveys operated in the past two decades served as DSCT secondary sources were also checked: (1) ASAS-3 (Pojmanski 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Pojmanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2005) 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2) SuperWASP did not classify it discovered pulsators into different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' By plotting them in the Hertzsprung-Russell diagrams using the parameters extracted from Gaia DR3 and TIC, luminosity, effective temperature and surface gravity, along with data from LAMOST, compared with those known DSCT, we would sift a good number of δ Sct candidates from the SuperWASP catalog of 24667 pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (3) Pan- STARRS (Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2010) 15 surveys all the sky north of declination 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5◦, about three-quarters of the entire sky on a short duty cycle of days, down to 23 magnitude, it had discovered numerous Cepheid variables and eclipsing binary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' An extremely large number of variable stars are expected from Pan-STARRS DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (4) ROTSE 16 , (5) WISE/AllWISE (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2010) 17 , (6) NSVS (Wo´zniak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2004) 18 , (7) CRTS/CSS 19 , (8) WFCAM 20 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' On the other hand, the catalog of Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013) was carefully checked and found that multiple stars cross-matching failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No stars within 2′ radius, at 23:51:24 −25:45:00, the nearest star is BY Scl (SX Phe- type) apart from 120′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star [KPA2010] 36 (22:01:08 +24:44:33, B=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m42, V=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m01) and HD 5076 (00:52:40 +06:39:55, [KPA2010] 1) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2010) compiled in the table of Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013) without discovery source ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No source ID for the star at 23:43:00 −29:52:00 (V=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m51), and within 67′′ radius no star in Simbad, in 24′′ no star in TIC, can be identified, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' At 16:27:51 −49:07:36 in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='39′′ identified to be Gaia DR3 5941411883325639296 (G=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' At 19:38:06 +30:51:54, nearest star is [MPC98] V798 8, but as a DSCT it should be identified to be V2116 Cyg, 17′′ apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' At 19:53:46 +18:46:42 is PSR J1953+1846A, but it should be identified to NGC 6838 1084, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21′′ apart, multiple stars share the same coordinats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Some duplicate coordinates with no source identifiers for variables in stellar clusters such as NGCA 288, 2099, 3201, 4590, 5139, 6809, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' MACHO and OGLE ID were not provided either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In a few cases, we can not simply take the nearest star 14 1275+2263 rows/DSCT from The All Sky Automated Survey (ASAS): h t t p : / / w w w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' a s t r o u w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' e d u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' p l / a s a s / ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' p a g e = a s a s 3 15 The Panoramic Survey Telescope And Rapid Response System: https://panstarrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ifa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/pswww/ 16 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='rotse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='net/transients/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The Robotic Optical Transient Search Experiment (ROTSE) is a multi-telescope experiment designed to observe the optical afterglow of gamma-ray bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Most ROTSE- discovered variable stars have been reported in VSX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 17 The Wide-field Infrared Survey Explorer mission: https://wise2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/docs/release/allwise/ 18 Northern Sky Variability Survey (NSVS): http://skydot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gov/nsvs/nsvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='php 19 The Catalina Real-Time Transient Survey, Catalina Sky Survey (CSS): http://crts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/ 20 The Wide Field Camera (WFCAM) on UKIRT – The United Kingdom Infrared Telescope (a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8m IR-dedicated telescope, Mauna Kea, Hawaii): http://wsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='roe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='uk/ 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Table 3 Primary Catalogs for Checking Existence of a New Variable Star†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Project/Catalog VarType Number of Stars Notes GSC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 * 3,485,671,481 2020 pan-STARAS * 1,919,106,885 2016 Gaia DR3 * 1,811,709,771 2022 TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 * 1,727,987,580 2021 Gaia DR2 * 1,692,919,135 2018 USNO-B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 * 1,045,913,669 2003 GSC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 * 945,592,683 2006 AllWISE * 747,634,026 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13 WISE * 563,921,584 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14 USNO-A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 * 526,280,881 1998 2MASS * 470,992,970 2003 GSC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 * 455,851,237 2001 UCAC4 * 113,780,093 2012 APASS * 61,176,401 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 LAMOST DR9 * 10,907,516 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 LAMOST DR8 * 10,351,254 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 LAMOST DR7 * 9,846,793 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 Simbad * 8,322,647 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 LAMOST DR4 * 4,132,782 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 VSX-AAVSO V* 2,234,703 2022 Sep 24 Simbad V* 1,299,113 Variable stars ASAS-SN V* 378,861 of 687,696 stars Simbad Pu* 227,130 Pulsators SuperWASP Pu* 24,667 Pulsators OGLE-V dS* 24,488 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='30 ZTF dS* 16,709 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='17 VSX-DSCT 15,308 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 VSX-HADS 10,896 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 Simbad dS* 9,287 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 ASAS-SN DSCT 3,939 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 ASAS-3 DSCT 3,538 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 ASAS-SN HADS 2,231 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 Zhou (2015) 4,697 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01 Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013) 1,578 2013 GCVS DSCT 1,018 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 V5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2000) 636 2000 Simbad SX* 609 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 VSX-DSCTC 425 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 VSX-HADSB 273 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 VSX-SXPHE 292 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 GCVS SXPHE 267 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 ROTSE-I 61 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 †: Simbad/CDS and GCVS Classification Labels Used: *: general stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' V *: variable stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' dS*/DSCT/DSCTC: δ Sct stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Pu*: pulsating variable stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SX*/SXPHE: SX Phe-type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' HADS/HADSB: high-amplitude δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' around coordinates but considering variability property and use human inspection sometimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Therefore, some stars were double-checked manually through Simbad/CDS and MAST Portal web queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' After putting the δ Sct stars listed by Simbad, OGLE, VSX, ASAS-SN, ZTF, GCVS, and other sources together (see Table 3), we reached a sum of over 84,375 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Last but not least, we added three dozen sporadic new DSCTs picked up from recent publications based on ADS and IBVS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Kirmizitas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021) and 18 new discoveries in present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Surely, a known variable might be compiled in several catalogs such as GCVS, VSX, and Simbad, therefore there are lots of duplications in the combination of multiple source catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A computer program was written to check duplication and found more than 23,228 stars are duplicated in the combined list of published catalogs, for instance TIC 322378080 = OGLE GD-DSCT-6707 = ZTFJ183900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='91+074800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7, TIC 307185343 = OGLE GD- DSCT-7318 = ZTFJ190220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='03–040019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2, TIC 282617026 = OGLE GD-DSCT-0591 = ZTFJ065358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='89–034623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2, TIC 51835214 = ROTSE1 J223159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='77+135641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 = ZTFJ223159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='76+135641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8, TIC 57987993 = ROTSE1 J002103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='71+304215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 = ZTFJ002103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='69+304215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0, TIC 301909021 = ZTFJ024113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04+485846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 = UCAC4 695-018090 , IBVS No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6300 (detected on 10 November 2015, so IBVS re-reported) DW Psc = TIC 365157858 = ZTFJ013026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='96+084133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 =Gaia DR3 2566925171965557632, in table of Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Probably later catalogs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ZTF) could have re-reported discoveries regarding their publication dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For GSC 0762-2924 (2017, IBVS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='63 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6300), the Dec value suffered from a typo error, it should be 07:49:01 rather than 07:40:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Duplication largely attributes to combining the col- lection by Simbad and VSX, no attempt was made to distinguish whether the duplications are re-reported as new discoveries among multiple surveys, but for those duplicate entries, one of the sources is imported into the new catalog in the order of ID-selection priority: GCVS designations, the original discoverer used ID, Simbad main ID (HR, HD, 2MASS, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ), OGLE, ASAS, ZTF, and other surveys/project names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' By and large, the actual origin of variability discoverers and identification should be reflected in source identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then the duplicated entries were removed, last form an up-to-date comprehensive catalog of δ Sct stars that consists of 59,350 individual stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The number is almost 93 times of 636 DSCT in the catalog of Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2000) and 37 times 1578 DSCT from Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This new catalog has both TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 and Gaia DR3 identifiers cross-matched with source ID within a distance radius of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 arcseconds, a few cases with larger separation from given coordinates up to 2′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The present catalog of δ Sct stars focused on cross-identifications and parameters from the two space projects, such that did not gather all stellar parameters available in source catalogs but include four identifiers (the source catalog ID, TIC, Gaia DR3 and 2MASS/AllWISE or coordinates);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' parallax, distance, radial velocity, Gaia photometric magnitudes (G, BP, RP, BP −RP, G−RP) extracted from Gaia DR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' stellar atmospheric parameters effective temperature, luminosity, surface gravity, and radius retrieved from both TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 and Gaia DR3, magnitudes in B and V and mass from TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2, and absolute magnitude, which was calculated using Gaia distance directly according to definition MV = V + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 − 5 ∗ log(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is known that distance is not simply the inversion of the observed parallax, though they are reciprocal in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The naive approach of inverting parallax is a blind use and is just a biased estimator as noted in Gaia DR3 data release (Gaia Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 7 C o l l a b o r a t i o n e t a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2 0 2 2 , 2 0 1 6 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' T h e r e l i a b i l i t y o f t h e c a t a l o g w a s a c h i e v e d b y t h r e e f a c t o r s : ( 1 ) e x t r a c t i n g s o u r c e s g i v e n i n r e f e r e e d p u b l i c a t i o n s a n d w e l l r e c o g n i s e d s u r v e y s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ( 2 ) c r o s s m a t c h i n g b y T I C a n d Gaia DR3 identifiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (3) duplicate checking using TIC, Gaia ID, and coordinates by a computer program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The advantage of the catalog is cross-identifying all source IDs with the two largest catalogs of celestial objects TIC and Gaia DR3 along with up-to-date parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Nevertheless, there must be some stars missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Further updates will be complemented in the next version of the catalog with the strategy of automatedly dynamical updating 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Statistics of contribution from various sources are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1 Source Contributions to the 59350 δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Each entry in the catalog table is comprised of more than 26 columns: TIC, source ID, Gaia DR3 ID, RA Dec/2MASS ID, B TIC, V TIC, Teff TIC, Teff Gaia, log g TIC, log g Gaia, Mass TIC, R TIC, R Gaia, L TIC, L Gaia, parallax(mas), distance(pc), MV , log Teff TIC, log L TIC, G mag, BP mag, RP mag, BP − RP, BP − G, G − RP, Radial Velocity Gaia and errors for a few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is inconvenient to illustrate such a wide catalog table, it is too wide to be printed in A4-paper layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 4 demonstrated a simplified version of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The whole catalog in its entirety is provided as online materials in both machine-readable text format (CSV file) and human-friendly HTML version 22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Interested readers may enquire it from the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' More importantly, in the HTML version, each TIC ID is hyperlinked to STScI archive MAST in a default radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00033◦ of the target, each Gaia DR3 ID with hyperlink to Gaia data at VizieR/CDS, and main IDs or coordinates are linked to CDS Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The HTML version makes 21 Automation is imperatively needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Since the day commenced retrieving Simbad, the number of DSCTs in Simbad has increased by 585, from 8702 to 9287 by the time of writing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The current version used source data as of September 26, 2022 22 https://deltascuti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='wixsite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='com/delta/dsct-catalog consulting, checking and downloading TESS/Kepler/K2 and Gaia DR3 data unprecedentedly convenient and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is useful for follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Recalling the release dates interval of TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 and Gaia DR3, on September 2021 and 13 June 2022, respec- tively, there are discrepancies between TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 (where parallax adopted from Gaia DR2 by the time of this writ- ing) and Gaia DR3 for several parameters, for instance, TIC 213014556 = Gaia DR3 2524140189527507968, TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 gives plx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='127042±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='048995, Teff=6558±247 (DR2: 6529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3335), log g=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='82235, d=4063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='85±630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11, whereas DR3 updated them to plx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='075798, d=2618 pc, Teff=6318, log g=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We first queried TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 with a Python program, which was then modified to retrieve parameters from Gaia DR3 archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' An additional work in progress is adding results of LAMOST (spectral type, effective temperature, gravity and metallicity) and other stellar parameters, as reported for about 525 of 766 LAMOST-observed δ Sct variable stars by Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2018), who checked among 3,689 known δ Sct stars in VSX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Now VSX has indexed more than 27,194 DSCT, LAMOST should have observed more DSCT since then (Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Moreover, new DSCT from TESS, K2 and Gaia will be particularly gathered from a series of publications, for examples several in progress: Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Kahraman Alic¸avus¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fetherolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Kahraman Alic¸avus¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Benefited from the TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 and Gaia DR3 identifiers being cross-matched with source IDs in the new catalog, observers are able to easily examine a star manually or a number of stars programmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With the catalog, it is thus convenient to draw our new discoveries against the known ones in various parameters spaces (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' When retrieving parameters from TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2, some fainter stars were ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A few querying program exceptions are: – Eight δ Sct stars [MHN98] V1, V2,V3,V6–8,V11 and V19 were ignored due to fainter than V =22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m5 and no IDs of either TIC or Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – V4317 Sgr was listed in R00 but no any other ID given in Simbad (not in TIC and Gaia DR3 either), no color magnitude info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – [KPA2010] 36, B=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m42, V=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m01 only ID should correspond to TIC 283408312, Tmag=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m13 – [BMG2010] V046, V055–V159 are fainter than V =26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m0 were ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – [VM2013] DC-1, DC-10 – DC-99, 89 stars fainter than V =22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m0 were ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – Cl* NGC 7245 VHB 456 – no coord in Simbad except V =14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="m7: TIC query could not resolve the ID to a sky position As querying Gaia DR3 over such number of stars, we experienced a few failure cases and solved with human intervention, reported below: SE 52'800 4e'S Js'eo OCE 2A2A ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="20 400 J0'α0 22AMS Voso 1AC EbIC OfUGL2(CBI2'NCVC+'AWS0I3'COBOI*." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='.")8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou – TIC 63371156 = BD+41 3376: 2 stars within 1′′ sky region, identified to be Gaia DR3 2077694728707664128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – TIC 122717173 = TYC 3134-1328-1: 2 stars within 1′′ sky region, identified to be Gaia DR3 2052844254090672640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Actually no more data except G and BP RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – OGLE LMC-DSCT-164 = TIC 295070037: no Gaia DR3 designation star within 3′′ sky region – TIC 153264713 = OGLE-GD-DSCT-1418: 2 stars in 1′′ circle, discriminated the variable to be Gaia DR3 5698385763057621376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – alf Lyr: has four TIC IDs: TIC 471012052, TIC 70257116, TIC 471012017, TIC 157587146, no Gaia DR3 ID – bet Leo: =TIC 14725877, no Gaia DR3 identifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – V2988 Cyg: not resolved by Sesame Strasbourg, Gaia DR3 query cross-identified its TIC 278355617 to DR3 2180055374306851968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – CoRoT 101436549 = TIC 223887633 = UCAC4 453- 099296, no coordinates, 2 stars in 1′′ region, cross- identified DR3 4287505636453818752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' – TIC 319438599 = [VM2013] DC-86: no Gaia DR3 designation star within 3′′ sky region – TIC 82359935 = V1233 Her (EB) = ROTSE1 J165852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='87+391421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7: 2 DR3 designations in 1′′ sky region, to be Gaia DR3 1352082539737123840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Finally, we show three plots of the collected 59350 δ stars in multiple observational Hertzsprung-Russell diagrams together with our newly discovered variable stars in Figures 2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 9 Table 4 A Combined Catalogs of 59,350 δ Sct Stars (reduced demo version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 4 A Combined Catalogs of 59,350 δ Sct Stars (reduced demo version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TIC Source ID Gaia DR3 2MASS/RA Dec Bmag V mag Teff log g M/M⊙ R/R⊙ Luminosity Parallax(mas) Distance(pc) MV log Teff log L/L⊙ TIC 273871163 J19525891+4636506 2085540087774214656 J19525891+4636506 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='944 7021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='81715 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='54464 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1786737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='444449 2113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='319 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='846 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='152 TIC 273875324 J19530663+4736579 2086440961449151488 J19530663+4736579 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='731 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 7318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='53668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='63703 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='17894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='296576 3070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='164 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='864 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='534 TIC 273875361 J19531465+4736026 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00916 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4587107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='57316 624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='388 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='257 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='865 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='019 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 6 FREQUENCY ANALYSIS Fourier analysis was carried out for each data set by using PERIOD04 (Lenz & Breger 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The results are summa- rized in frequency solution tables for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, it is well known that stellar activities may not be periodic, and even irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Periodic signals including pulsation are not strictly sinusoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We analyze pulsation contents by resolving light curves into a sum of multiple sine or cosine waveforms with different frequencies and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Ground astronomy has various sampling intervals and gaps without data in time-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Uninterrupted space observations never mean no intervals of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS exposed every two seconds, HDU stacked images into groups of 2-minute cadence and 30-minute cadence for different science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' together with data-downloading periods and orbits or sectors switch, these are actual TESS data sampling intervals involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Because the observations even from space are still true discrete sampling in time- space, aliasing, as a data acquisition effect caused by sampling intervals is not yet completely eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A discussion on aliasing involved in multiple sites observing campaigns can be referred to Breger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In this reference, 75+ frequencies for FG Virginis were resolved using extensive photometric multisite campaign data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Guzik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2022) has re-visited FG Vir using Kepler K2 and TESS data and found around 30 significant frequencies in the K2 data, and more than 100 significant frequencies in the TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With TESS data, this number 75 will be easily overpassed as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Here we explored the TESS data aliasing for keeping in mind when picking up a peak frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The gaps in the middle of each sector’s time series are due to the data downlink separating the two physical orbits within each TESS sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The points showing severe residual uncorrected systematics in the fits are removed in the joint analysis presented in this paper, similar case as Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Under Nyquist sampling theorem, frequencies less than half the sampling rate will not alias: fmax = 1/2∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observations’ sampling intervals are 2 minutes and 30 minutes, that means the maximum possible frequencies should be less than 360 and 24 d−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' the Nyquist frequency), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, besides the regular sampling intervals, there are additional time gaps in a data set which was finally used in frequency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We know the single-site ground photometry suffered from daily aliases, an effect of daytime without observing and almost started observation 24 hours late at the same time the next night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' There are additional time gaps in TESS data between two consecutive orbits in a sector as well as between two successive sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In the case of ι Boo, current available TESS data are from sectors 22,23,49 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This set of data spanned 792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='19 days, starting on BJD 2458899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3216408 to 2459691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='51163725 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' between 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='19 19:43 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22 00:16), the gap between two orbits in each sector are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='96665 days in sector 23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90692 days in sec- tor 50, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='33205 days while the gaps between the two successive sectors are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6125 between 22–23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='956937 between 49–50, which act as two arbitrary TESS sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This would produce aliases around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0/(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6126), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0/(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='96665) or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0/(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='33205), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='065306, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='066524 d−1 aliases, similar to the ground observations’ daily aliases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As a composite effect, the spectral window reflects the aliasing structure: in the current case, Figure 7 depicts the aliases: two strong aliases at fa2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001384 and fa1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025752 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Nyquist sampling theorem indicates that a frequency peak will be accompanied by the combination side peaks at f ± n ∗ fa1 where each is further slightly aliased to be at fa1 ± m ∗ fa2 (n,m are integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' That is, f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13788, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21092, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28292, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='35526, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='42622, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='49787, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='56712 are the strongest among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then, f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002705, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004069, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In addition, the frequency resolution is restricted by the timebase length (T) of a dataset and is estimated by ∆f = 1/T, which in current case T=792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='19 days, which means ∆f= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001262 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, actually, four sectors, eight orbits, T ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7*8 days) — effective time span with data is only 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8% of that observing dates interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1/(8 ∗ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='009124 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7 RESULTS We report in this section our results for each analyzed star in tables of frequency solution and in graphs of light curves, periodograms and prewhitened residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Detailed works on several of the stars are underway and will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 HD 52788 = TIC 279361762 TESS observed HD 52788 in total of 24 sectors across 2 to 39 over 1035 days from 2018 August 22 through 2021 May 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves are available as a source of TESS-SPOC, HLSP-QLP, and HLSP-SPOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The TESS- SPOC 2-minute cadence SAP flux data were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We see the MIT HLSP-QLP team attempted resolving the data with multiple planets fittings, but no final positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The light curves exhibit a much more complex structure visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency analysis shows that the residual spectrum even with 130+ frequencies pre- whitened is still full of peaks in the range of 5–15 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With successive prewhitenings and multiple frequencies fittings, the residuals are improved slightly (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With a big number of parameters when simultaneous optimization for frequencies, amplitudes, and phases, non- linear least-squares fitting of PERIOD04 encountered a calculation issue of ’matrix cannot be inverted’, which failed to improve the three parameters of the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' So we have to stop prewhitening procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We computed noise levels based on both residuals and original data for comparison of significant peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Current preliminary Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2 Newly discovered pulsating variable stars of δ Sct and other types plotted contrast with the known 59350 DSCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Red squares refer to classical δ Sct domain: Teff in [6300, 9100], log Teff in [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='80, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='96], log g in [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3], log L/L⊙ in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='65, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90] consistent with the H-R diagram of pulsators given by Handler (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Jeffery (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Breger (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' magenta dashed squares refer to an extended domain: Teff in [3800, 10000], log Teff in [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='58, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00], log g in [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0], log L/L⊙ in [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90], MV in [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' δ Sct domain is observationally much enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' results are reported in Figure 9 and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Further study is underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 5 Fitting Residuals with Successive Prewhitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Number of Frequencies Fitting Residuals Imporvement 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00423889 – 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='003242 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5% 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0030626 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5% 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0030232 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='29% 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0029725 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='68% 158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0028472 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22% 169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0028147 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14% 191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0027301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 C1=HD 53166 = TIC 279431011 C1= HD 53166 (=TIC 279431011, V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m17, A1V, RA=07:00:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='60 Dec=-58:23:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27 J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0), its TESS light curves are available in sectors 2–13 in 2-minute cadence, and 24 sectors (2–13, 27–39) as HLSP- QLP products in cadence of 30-minute and 10-minute, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With TESS data we reveal the star to be a new δ Scuti-type pulsating variable star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Pulsation frequency contents are given in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 C2=HD 53349 = TIC 279476396 C2= HD 53349 (=HR 2662 = TIC 279476396, V =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m0, A8III, RA=07:01:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11 Dec=-58:56:23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='77) is a High Proper Motion Star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It was listed as NSV 3349 with F0V spectral type at the International Variable Star Index (VSX) of AAVSO but without identified variability type 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Analyses based on all available TESS data during 24 sectors 1 through 39 lead to multiple pulsational variation of light with two dominant frequencies at f1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='888139 and f2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='809503 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 11 and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 23 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='aavso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='org/vsx/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='php?' metadata={'source': 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2ae340 ku0mu Q 2cf 2f912 JS'2 JS'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="0r JO'O 12 1'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="己 52 5'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="0 5'212 A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 3 Newly discovered δ Sct stars and other variables plotted contrast with the known 59350 DSCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 4 Newly discovered δ Sct stars and other variables plotted contrast with the known 59350 DSCT in Gaia color- magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fig.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 6 59350 δ Sct stars compared with Gaia stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='16 Frequency (d−1) 0 200 400 600 800 1000 Normalized Amplitude (mmag) fa2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025752 d−1 fa2 + 2fa1 fa1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001384 d−1 4fa1 Spec(ra Window 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0 200 400 600 800 1000 Spec(ra window A iasing s(r)c()re for TESS sec(ors 22,23,49,50 combina(ion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7 Aliasing structure for the combination data of TESS sectors 22,23,49 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=" (Bb - Bb) (Bb - Bb) 3 s 0 丁 0 4 e J12 J2 J2'0 Julo2dA Julo2dA JS'2 JO JO'O gauifnqe 12 2 6 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content="2 (wga) (wg 0 p s'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2 И6M D2CI 2a320 D2Cl S424aa C919 2f912 S424aa C919 2f91214 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 8 Finder chart of HD 52788 and comparisons used in Kurtz(1979,1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 6 Frequency Solution of HD 53166 (=TIC 279431011) Based on 24 TESS Sectors 2–13 and 27–39.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 · · · · · The full 135 frequencies are provided in electronic version online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zeropoint: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='81185449 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00273008 mag Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 15 1630 1635 1640 1645 1650 Time (TBJD) 97000 98000 99000 100000 101000 102000 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Curve - TIC 279361762-S0012 5 10 15 20 25 Frequency (d−1) 0.' metadata={'source': 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Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0, noise based original data 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='30 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9776,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8548,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7193,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4664 d31 pre0hi−ened TIC 279431011 = HD 53166 in TESS ,ec−or, 2 and 12 5 10 15 20 25 Frequenc2 ( −1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='30 Amplitude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2 4 6 8 10 12 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 10 significant peaks prewhitened TIC279431011_S2739-HLSP_QLP_llc (S0027--S0039, 10-minute cadence) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 10 Amplitude spectrum of TIC 279431011: second panel to bottom: 2-minute cadence: sector 2, sectors 2 and 12, sectors 2–13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 30-minute cadence: sectors 27–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 17 Table 8 Frequency Solution of HD 53349 (=TIC 279476396) Based on 24 TESS Sectors Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mmag) Phase (0-1) SNR f0= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='888139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='476718 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90 f1= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='809503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='229343 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='35 f2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='538036∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000155 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='58 f3= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='287296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='773619 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='57 f4= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='148190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='131089 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='34 f5 =2f3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='458507 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='43 f6= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='074502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='458507 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='43 f7= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='251712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='215842 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='76 Zeropoint: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='43898411e-07 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000186609086 mag Accompanied with an equal amplitude peak at f2 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000794 within frequency resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0009656 d−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 NGC 6871 During the course of observing a known δ Sct star V1821 Cyg in the open cluster NGC 6871, one of the field star GSC 2683-3076 (=V2238 Cyg) was discovered to be a new δ Scuti star (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2001b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Now we are going to revisit this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed this field in sectors 14, 15, and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves are available in 2-minute cadence SPOC (S0041) and in 30-minute cadence HLSP- QLP (S0014 and S0015) products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A check led to the discoveries of four additional new variable stars nearby (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 NGC 6871: V1821 Cyg = TIC 90350726 V1821 Cyg (=HD 227695 = TIC 90350726 = 2MASS J20063348+3552420, B=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m56,V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m22, A5p) is a known δ Sct star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We provide the newly detected 39 pulsational contents based on TESS data in Table 10 and Figure 12 as updates to the previous two pulsation modes (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2001a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 NGC 6871: V2238 Cyg = GSC 2683-3076 = TIC 41195917 V2238 Cyg (= GSC 2683-3076 = TIC 41195917) was observed by TESS in sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SPOC light curves are available with cadence of 2 minutes only for sector 41, 30-minute cadence HLSP-QLP data for sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With TESS data, we resolved 38 pulsational frequency contents, see Figure 13 and Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 NGC 6871: TIC 89757305 TIC 89757305 (= HD 227658 = 2MASS J20061465+3551028,B=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m18, V =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m09, B2) was observed by TESS in sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SPOC light curves are available with cadence of 2 minutes only for sector 41, 30-minute cadence HLSP-QLP data for sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With TESS data, we identified the star to be a pulsating star of possibly β Cephei-type and resolved 8 pulsational frequency contents, see Figure 14 and Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 NGC 6871: TIC 41195818 TIC 41195818 (= 2MASS J20061931+3555462, V =14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m37, Tmag=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m6093) is a plain star with only one identifier, [MJD95] J200619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='32+355546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 and 2 references linked in Simbad/CDS (http://simbad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fr/simbad/sim- ref?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='bibcode=1995ApJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='.151M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves in sectors 14 and 15 are in 30-minute cadence without removal of systematic affects and detrending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves in sector 41 are in 2-minute cadence, which show evident periodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Based on sector 41 light curves, it is now identified to be a new W UMa-type eclipsing binary system with an orbit period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='370563 days (f0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='729626 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 15 and Tables 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' However, the Zwicky Transient Facility (ZTF 24 ) discovered this star (20 06 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='31 +35 55 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8, =ZTF J200619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='30+355545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8=[MJD95] J200619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='32+355546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4, spectral type K8) to be a BY Draconis-type variables with g-band period= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3702414 d (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020), which are emission-line dwarfs of dKe-dMe spectral type showing quasi-periodic light changes with periods from a fraction of a day to 120 days and amplitudes from several hundredths to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The light variability is caused by axial rotation of a star with a variable degree of non-uniformity of the surface brightness (spots) and chromospheric activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Some of these stars also show flares similar to those of UV Ceti stars, and in those cases they also belong to the latter type and are simultaneously considered eruptive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In the above point of view, light curves in sectors 14 and 15 did show some kind of emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We do not understand the periodic variability disappeared in the duration of sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 NGC 6871: TIC 41195988 TIC 41195988 (=GSC 02683-03318 =2MASS J20062044+3553079, B=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m32, V =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m99) is plain star with fewer identifiers and references in Simbad/CDS, where TESS ID (TIC) is not cross-matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS Light curves are available in HLSP-PATHOS and HLSP-QLP two products in sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No pulsation can be identified, but some kind of flare-like features are seen in sector 15 PATHOS light curves, see Figure16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 NGC 6871 22: TIC 41196013 TIC 41196013 = NGC 6871 22 (V =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m65, A5) was observed in TESS sector 41, the TESS PDCSAP light 24 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ztf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu/ 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='040 Amplitude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='010 8 significant peaks prewhitened TIC279476396-S0139-LC (S0001--S0039, 2-minute cadence) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 11 Amplitude spectrum of TIC 279476396 (=HD 53349) Table 9 A Group of 6 Stars in NGC 6871 Observed by TESS in Sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Stars Identifiers V , Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' RA, Dec(J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0)/Notes V1821 Cyg = TIC 90350726 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m37,Fm dD 20:06:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='388 +35:52:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='88 V2238 Cyg = TIC 41195917 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m53, A6 20:06:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='311 +35:54:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 V1=HD 227682 = TIC 41195891 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m4, F8 20:06:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='467 +35:54:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='88 V2=HD 227658 = TIC 89757305 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m09, B2 20:06:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='547 +35:51:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 V3=TIC 41195818 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m69, 20:06:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='276 +35:55:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='78 V4=TIC 41195988 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m99 20:06:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='439 +35:53:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07 curves demonstrated that the star is constant within the time scale of several weeks at least down to the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125 mmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fourier spectrum basically exhibits white noise except that a peak at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='576045 d−1 with amplitude of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='82 mmag (Figure 17), a little bit over significant level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This peak also appeared when only the data of the second orbit in this sector were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Direct use of ’to periodogram’ of LIGHTKURVE to the light curves (*llc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits) did show this peak too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The value is close to half of the ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' PDCSAP’s detrending should have removed most instrumental variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A further visual inspection on the other two TESS sectors 14 and 15 at 30-minute cadence as HLSP-QLP products did not support a positive light variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Thus, the star is non-variable at time scales less than days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The results support that it is a photometric standard star, and the color data listed in the uvby98 photoelectric photometric catalogue (Hauck & Mermilliod 1998) can still be referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 NGC 6871: HD 227682 = TIC 41195891 HD 227682 = TIC 41195891 (V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m4, F8) was observed in TESS sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The SPOC and QLP light curves data are available in 30-minute cadence as HLSP mission products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A frequency analysis applied to the two sectors data shows no variability over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 mmag level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Amplitude spectrum is just noise below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='035 mmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 NGC 6871: TIC 90350490 TIC 90350490 (= 2MASS J20064266+3550552 , V =14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m35, G=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m238817, Tmag=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m8359) is a plain star with fewer references (less than 5) in Simbad/CDS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No periodic variability was identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 NGC 6871: TIC 90350607 TIC 90350607 (=2MASS J20064698+3551472, B=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m15, V =13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m86) is a plain star with a couple of identifiers and one linked reference back to year 1995 in Simbad/CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves are available in 30-minute cadence for sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The uncorrected SAP light curves produced by HLSP-QLP from sectors 14 and 15 did not show periodic light variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We tried 4–6 order polynomial fittings to remove trends in each orbit but failed to detect variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No TPF images to download, FFI images could be the last chance to check any variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 HD 191025 = TIC 41189624 HD 191025 (=TIC 41189624 = 2MASS J20062426+ 3643560, V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m75, A5V) is a plain star in Simbad/CDS, it is now identified to be a new δ Sct-type pulsating variable star with TESS data in sector 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 18 and Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 19 2420 2425 2430 2435 2440 2445 Time (TBJD) 20400 20600 20800 21000 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Curve - TIC 90350726-S0041 10 20 30 40 50 Frequency (d−1) 0 1 2 3 4 5 Amplitude (1000 ppm) aliases Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='175 39 significant peaks prewhitened V1821Cyg-S141541 (TESS sectors 14, 15 and 41) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 12 TESS light curves and amplitude spectrum of V1821 Cyg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 TYC 2671-577-1 = TIC 90869850 TYC 2671-577-1 (=TIC 90869850 = 2MASS J20074805+ 3148345, V =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m44), a plain star in Simbad/CDS, is now identified to be a new δ Scuti pulsating variable star based on TESS light curves in sector 41 of 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 30-minute cadence light curves in sector 14 without detrendings were not analyzed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Results are reported in Figure 19 and Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' According to color magnitudes and indices in existing catalogs, B=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m93, V =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m44 (B − V =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='49 means about 6300 K according to the basic empirical formula (B − V ) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='865 + 8540/Teff);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' J=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m061, H=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m017, K=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m975, (J − H) means K3, (H − K) means A–F, and LAMOST spectrum we derived a spectral type in A0–F9 for the star, along with a consideration of effective temperature of 9058±423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='817 K and gravity log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='41791 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07722 in TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 TIC 40831024=HD 227647 HD 227647 (=TIC 40831024, V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m29, A2), was observed in TESS sector 41, it is identified to be a new δ Sct-type pulsating variable star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Figure 20 and Table 17 report the pulsation contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 V2455 Cyg = HD 204615 = TIC 266794067 V2455 Cyg (= HD 204615 =TIC 266794067, V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m84, F2), is a known HADS with f0=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='61 d−1 (Ostadnezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020) which was observed by TESS in sectors 15 and 16 with cadence of 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For this star, because of its high amplitude and short period, we used SAP fluxes other than PDCSAP, which removed long-term trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Our goal is to detect non- radial and long-term pulsation contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Now besides the harmonics of the fundamental radial frequency, non-radial contents are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 21 and Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2420 2425 2430 2435 2440 2445 Time (TBJD) 11100 11200 11300 11400 11500 11600 11700 PDCSAP Flux (e-/s) PDCSAP_Fluxes PDCSAP Light Curve - TIC 41195917-S0041 0 10 20 30 40 50 Frequenc1 (d21) 0 1 2 3 4 5 6 PDCSAP (mmag) Signi ican− level a− SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 40 significant peaks in 5--23 d−1 prewhi−ened TIC 41195917 = V2238 Cyg in TESS sector 41 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 13 TESS light curves and amplitude spectrum of V2238 Cyg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 NGC 6910: V2245 Cyg = HD 229196 = TIC 13876370 V2245 Cyg (=HD 229196 = TIC 13876370, V =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m59, O6II), a member of the young open cluster NGC 6910, is classified in Simbad/CDS as a pulsating variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Its variability exhibiting irregular light variations was confirmed by Karitskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is known to be a double-line spectroscopic binary early (Sanford 1949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This Galactic O-type blue giant star (Ma´ız-Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2004) at a distance of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 kpc, was previously known only from some peculiarities in the ultraviolet spectrum (Massa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Crawford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Gaia updated the parallaxes to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5535±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0324 mas corresponding to a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Hα line in emission reported by Kolaczkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2004) was not verified in the spectroscopic studies of Kub´at et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2007) who instead found variable Hα line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It was observed by TESS in sectors 15, 16 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SPOC light curves are available with a cadence of 2 minutes only for sector 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Both SAP and PDCSAP fluxes light curves show clear W UMa-type (a kind of close contact binary) featured with strong pulsation over the orbit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' So the primary component is pulsating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Fourier analysis exclusively showed just two significant frequencies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='52276988 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2613849 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Clearly, the second term is half of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The orbital period is then should be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9128875 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As pointed out by Kub´at et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2007) that these changes and the presence of the extended blue wing in the Hα profile, together with the rarely seen photometric variability in pulsating stars, make this star an interesting object for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 AL Tri = GSC 2293-1021 AL Tri = GSC 2293-1021 = TIC 61236485 (V =13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m98) is a W UMa-type eclipsing binary discovered by our group (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2000b), where they folded the light curves in an orbit period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='262 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' W UMa-type is a kind of contact binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is listed in the most recent catalogs of 9380 samples (Marsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2017), the earlier catalogs of 7216 stars in Avvakumova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2013) and 6330 stars in Malkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' AL Tri was observed in TESS sector 17 and the data are available to download in 30-minute cadence as the TESS High Level Science Products (HLSP mission) which are produced by the MIT Quick-Look Pipeline (QLP) team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' QLP light curves come in two forms as *llc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits and llc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='txt files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 21 2420 2425 2430 2435 2440 2445 Time (TBJD) 9150 9200 9250 9300 9350 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Cu ve - TIC 89757305-S0041 0 10 20 30 40 50 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 7 significant peaks in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125--6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='528 d−1 prewhitened TIC89757305-S0041 in TESS sector 41 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 14 TESS light curves (dots) with fitted solid lines (upper) and amplitude spectrum of TIC 89757305 Besides a strong frequency peak at f0=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='502196 d−1 that would correspond to orbit period of Porb= 2/f0= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2665886 days, there are several significant frequen- cies presented after removing the aliased peaks (fi ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='057814): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5f0, 2f0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5(f0 + f1) and f1 + f4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Table 19 and Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The presence of period-doubling (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5f0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5f1) can be referred to those detections in RR Lyrae stars (Plachy & Szab´o 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Szab´o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2014) and other cases (Kemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This period doubling bifurcation means a chaotic pulsation behaviour of the star, being worthwhile to further investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11 IT Dra = SAO 16394 IT Dra (= SAO 16394 =HD 127411 = TIC 166177270) is a δ Scuti star first discovered by our group (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1998, 2000a) to be pulsating with two detected frequencies (f1=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8493, f2=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0613 d−1), which are confirmed in TESS data with more additional pulsation contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed IT Dra in 7 sectors (15, 16, 22, 23, 48, 49 and 50) in 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The data are summarized in Table 20, pulsational frequency analysis results are presented in Table 21 and Figures 24 and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 AD Ari AD Ari (= HD 14147 = HIP 10701 = TIC 246938869, F0, V =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m43) was misclassified and collected in the δ Scuti star catalog (Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2000), Zhou (2002) observed the star on four nights with a result of 991 photometric points through two photoelectric 3-channel and 4-channel photometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Due to actual longer periodic light variations, the author only saw a portion of light variation at different phases each night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With the data from STEREO TRansiting Exoplanet and Stellar Survey (STRESS), Sangaralingam & Stevens (2011) showed full periodic light curves with two distinct amplitudes and suspected this star could be a binary system rather than a δ Sct star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Ziaali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2019) derived a period–luminosity pair values of log P = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='57, MV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 using Gaia DR2 parallaxes, they quoted the star as being an ellipsoid variable following Handler & Shobbrook (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No updates on the star’s variability until TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed AD Ari in two sectors 42 and 43 in 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Uninterrupted monitors during four successive ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7-day orbits, TESS collected perfect phase-coverage light curves which show distinct binary nature and the QLP team paid close attention to the light curves’ fitting for identifying an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As shown now in Figure 26, it is ultimately an eclipsing binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Table 10 Frequency Solution of V1821 Cyg (=TIC 90350726) Based on TESS Sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='081879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='296366 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 f1= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='817484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='989323 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f2= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='242338 0.' 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+page_content='8 Gyr, 172±47 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Here we checked a few of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We will address the detection of pulsation frequencies of these stars and leave mode characterization in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 NGC 2632: BR Cnc BR Cnc (= HD 73175 = TIC 307678320, F0Vn, V =8.' metadata={'source': 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+page_content='000577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='651683 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 f6= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='955200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='034838 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 f7= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='528000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='322140 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 Zeropoint: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0829341 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00254047713 mag It was not a TESS Objects of Interest (TOI,Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021), but SPOC checked it as a potential transiting planets by searching for periodic flux decreases, known as threshold-crossing events (TCEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Suspicion of hosting an exoplanet was not confirmed either on the NASA Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 23 Table 13 Astronomic Parameters of TIC 41195818 from TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Parameter TIC 41195818 B 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='535±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='176 V 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='136±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 J 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='283 pm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='028 H 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='641±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='031 K 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='465±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='026 G 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6009±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000972 Tmag 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6093±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='007496 Teff 3814±157 log g 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='66061±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0107662 R/R⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='588903±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0177 M/M⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='57894±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0203863 Table 14 Frequency Solution of TIC 41195818 Based on TESS Sector 41 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='729627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='030967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='323611 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='498328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='330653 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f2= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='190000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='192861 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f2 =3f0± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 f4= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='504000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='143969 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f4 =2f0 + 2f2± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='143969 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 Zeropoint: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4511123 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0134016886 mag Table 15 Frequency Solution of HD 191025 (=TIC 41189624) Based on TESS Sector 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='294803 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f4= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='611954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='634321 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 f5= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='877480 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+page_content='5517624 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00074233798 mag 2420 2425 2430 2435 2440 2445 Time (TBJD) 980 1000 1020 1040 1060 1080 1100 1120 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Cu ve - TIC 41195818-S0041 0 10 20 30 40 50 Frequency (d−1) 0 5 10 15 20 25 30 PDCSAP (mmag) S gn f cant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 P(e−h tened Re) dual) Ampl tude )pect(um 0 10 20 30 40 0 1 2 3 4 5 ) gn f cant peak) n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1--15 d−1 pre−h tened TIC41195818-S0041 in TESS sector 41 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 15 TESS light curves and amplitude spectrum of TIC 41195818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Exoplanet Catalog 25 or on the NASA Exoplanet Archive (Exoplanet and Candidate Statistics) at Caltech 26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Results 25 https://exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='gov/discovery/exoplanet-catalog/ 26 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ipac.' metadata={'source': 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+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 Normalized Flux HLSP_QLP_S0014_41195988*_llc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits TIC 41195988_S0014 Normalized SAP_flux Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 16 TESS light curves and amplitude spectrum of TIC 41195988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 16 Frequency Solution of TYC 2671-577-1 (=TIC 90869850) Based on TESS Sector 41.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 Zeropoint: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4195204 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00294914535 mag of an attentive pulsational frequency analysis are presented in Table 22 and Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 NGC 2632: EX Cnc EX Cnc (= TIC 437039231, B9V, V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m92, B=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m17) is one of the members of NGC 2632, a known δ Sct star (Zhou 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed EX Cnc in four sectors 42, 44, 45, and 46 at 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Results of an attentive pulsational frequency analysis are presented in Table 23 and Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 NGC 2632: BU Cnc = EPIC 211936696 BU Cnc (= HD 73576 = EPIC 211936696 = TIC 175240124 (V =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m65, A7Vn) is a known δ Sct star in the Praesepe cluster which was once observed at five international observatories during a multi-site photoelectric photometry campaign during 1989 February 2 through 26 and resulted in five frequencies of pulsation with millimag amplitudes (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='76, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='36, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='69, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='62 Table 17 Frequency Solution of HD 227647 (= TIC 40831024) Based on TESS Sector 41 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='624848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='947638 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f1= 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='912960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='207245 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f2= 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='992391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='724975 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f3= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='396062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='611759 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f4= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='258787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='063170 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f5= 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='615047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='865100 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f6= 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='678784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='269233 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f7= 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='332758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='464750 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f8= 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='352195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='128891 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f9= 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='259636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='981180 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f10= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='766517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='800103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 f11= 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='598521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='322200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 f12= 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='134476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='142620 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 f13= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='715744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='251687 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 f14= 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='720986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='780971 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f15= 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='647648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='808365 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f16= 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='557385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='087259 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f17= 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='453724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='957313 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 Zeropoint: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='95198413 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00126494106 mag Table 18 Frequency Solution of V2455 Cyg (=TIC 266794067) Based on TESS Sectors 15 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 20 40 60 80 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 Amplitude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 25 significant peaks prewhitened TIC 41189624 - S0041 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 18 TESS light curves and amplitude spectrum of HD 191025 (=TIC 41189624).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 25 30 35 40 45 50 55 60 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 Amplitude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 5 significant peaks prewhitened TIC 90869850 - S0041 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 19 TESS light curves and amplitude spectrum of TYC 2671-577-1 (=TIC 90869850) and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='87 d−1, Breger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This is one of the highest cited works in the δ Sct field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In addition, the two δ Scuties in the Praesepe cluster, namely BN Cnc and BU Cnc, were observed during the 1992 STEPHI IV campaign (lasting 3 weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Five and six frequency peaks were detected, respectively (Perez Hernandez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A re-analysis of the published photoelectric photometry showed that using statistical weights results in a dramatic 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 30 40 50 60 70 80 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Amplitude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 18 significant peak prewhitened TIC 40831024 - Sector S0041 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 20 TESS light curves and amplitude spectrum of HD 227647 (=TIC 40831024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 20 TESS Data of IT Dra (=SAO 16394 = TIC 166177270).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Term Notes Sectors 15, 16, 22, 23, 48, 49, 50 fa1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001203 (primary alias fa2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='026936 (secondary alias) Time span BJD 2458711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3629–2459691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='51 (2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15–2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22) Data length 23523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 hours, 127968 points Days 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7*7*2= 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 days reduction in the noise level along with the detection of seven frequencies in BU Cnc and eight in BN Cnc, with three of the latter being previously unknown(Arentoft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' K2 observed the Praesepe cluster (=NGC 2632) area during 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27–2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 in Campaigns c05, c16 and c18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A total of 9450 data points were collected over 1162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='75 days time span with actual 208 observing days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency or spectral resolution is down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00086 d−1, which ensures closely spaced frequencies can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' After removing outliers, the rest 9244 points were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 24 lists the main stellar parameters for four of the Praesepe members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Now with TESS and K2 data, the pulsation frequencies are updated to 28 and 10 for BU Cnc and BN Cnc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' That additional frequency at 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='91 d−1 suspected by Breger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (1993) was not confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The current frequency solution of pulsation fits the observations to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 mmag, even worse than that of ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 mmag in the first campaign work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Figure 29 and Table 25 summarized the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 NGC 2632: BN Cnc and BV Cnc BN Cnc (= EPIC 211933524) and BV Cnc (= EPIC 211931309) are two δ Sct stars in NGC 2632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No observational study on them in the past 20 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Using Kepler K2 data we resolved 10 and 27 pulsation frequencies for each of them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Tables 26 and 27, demo light curves and Fourier analyses are given in Figures 30 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 NGC 2632: HD 73712 = EPIC 211941583 = TIC 175261925 HD 73712 (= TIC 175261925, V =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m78, A9V) is a spectroscopic binary system with the primary component being a pulsating star of δ Sct type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 44 and 46 at 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' K2 observed it in campaigns c05, 16 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We applied frequency analyses separately to TESS and K2 data, and their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We resolved 30 significant pulsational frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Final results are reported in Tables 28 and 29 and Figure33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 27 0 10 20 30 40 50 Frequency (d−1) 0 25 50 75 100 125 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 10 20 30 40 0 1 2 3 f0=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='615119 c/d and its Harmonics prewhitened V2455 Cyg=TIC 266794067 - TESS sectors 15+16 0 10 20 30 40 50 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 PDCSAP (mmag) Significant l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' l at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f0 2 5f0 Pr whit n d R siduals V2455 C0g=TIC 266794067 - TESS sectors 15+16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 21 TESS light curves (dots) with fitted solid lines (upper) and amplitude spectrum of TIC 266794067 (=V2455 Cyg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 NGC 2632: TIC 175261942 TIC 175261942 is an eruptive variable in NGC 2632 (=HSHJ 295, M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Simbad/CDS lists V =18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m02 and B=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m63, which are quite faint for TESS if it was staying at that brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 44 and 46 at 2-minute cadence, and the 5-minute binned SAP light curves show clear stable unique mono-periodic variation with a period of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='62299 days (f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3812437 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Eruptive variable stars show irregular or semi-regular brightness variations caused by material being lost from the star, or in some cases being accreted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' They may vary in brightness because of the violent progresses such as flares that occur on the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The changes in luminosity coincide with shell events or mass outflow in the form of stellar wind, or interaction with the outside interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This star may be R Coronae Borealis sub-type (RCB) which are high luminosity, simultaneously pulsating and eruptive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Slow non-periodic fading or eruptive changes up to several hundred days are superposed on cyclic pulsations, with periods in the range of 30-100 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' RCB stars light-curves show variation in the luminosity when it erupts or pulsates (Bruch 2021) 27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14 BL Cam = TIC 392774261 BL Cam (= GD 428 = TIC 392774261, B=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m10, G=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m015898) is a typical SX Phe-type pulsating variable with fruitful photometric investigations (Fauvaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Hintz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It was observed in TESS sector 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves are available for the public in three formats including HLSP QLP, HLSP SPOC, and the general 2-minute cadence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' After a comparison among them, we selected the SAP Flux (*-0164-s lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='fits) which have no visible instrumental systematic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A frequency analysis led to the disclosure of more new pulsation contents in the range of 25–60 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 35 and Table 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 TIC 309661089 and TIC 356473060 TYC 3413-228-1 (= TIC 309661089, V =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m82, G1V) and TYC 3413-187-1 (= TIC 356473060, V =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m06) are two high proper motion stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The variability of TYC 27 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='uk/workx/superwasp-variable- stars/Eruptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html 28 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2426 2427 2428 2429 2430 Time (TBJD) 166000 167000 168000 169000 170000 171000 172000 173000 SAP F (x (e−/s) SAP_F (xes SAP Light C(r)e - TIC 13876370-S0041 - a portion zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 22 TESS light curves (dots) with fitted solid lines (mid-panel) of TIC 13876370 (=V2245 Cyg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 3413-228-1 (=TIC 309661089) is clear from the PDCSAP light curves of TESS sector 47 and the second orbit of TESS sector 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We resolved a primary variation period of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1235 d (f0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='076199 d−1), which is much close to the ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 days TESS orbit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The profile of light variations in middle panel of Figure 36 and G1V spectral type (with Teff=5814) would suggest a rotating variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Table 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For TYC 3413-187-1, variations are fundamentally at a period of about 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='80346 days (f0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1020048 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The dip seen in the 30-minute binned light curves around BJD 2451848 deserves further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Regarding its effective temperature of 4851 (spectral type K2), it probably is a Mira-type variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='16 HD 48270 = TIC 307618601 HD 48270 (= TIC 307618601, V =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m64, G5) is a bright giant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' G and K giants have often been used in variable star research as photometric comparison stars because they are bright, relatively numerous, and not expected to be intrinsically variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' But Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2000) found low- amplitude photometric variability on timescales of days to weeks for 81 of 187 selected giants, where HD 48270 was listed as K2 III star without detected light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It was observed in TESS sector 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Two kinds of light curves data available: 2-minute cadence and 30-minute cadence (as HLSP-QLP light curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency analysis showed irregular or quasi-periodic light variations exceeding a day timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A dominant period would be around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 days (f1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='553853 d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 38 and Table 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' All stars, including red giant stars, with convective outer layers, can exhibit solar-like oscillations, which are standing waves excited by the turbulent motion of convective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' These waves are not coherent, instead, these waves are stochastic and damped, which means that the lifetimes and amplitudes of the waves are limited and variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' So what we got here is a normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='17 TIC 309661100 TIC 309661100 (V =13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m09, M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0Ve, =PM J07472+5020) is a high proper motion star in Simbad/CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is an M dwarf (L´epine & Gaidos 2011) which was selected to be a planet-hosting Candidate by both CARMENES exoplanet survey (D´ıez Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019) and TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The stellar parameters had been recently updated: Teff = 3284 ± 109 K, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='307 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='067R⊙, log L/L⊙ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='224 (Khata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ASAS-SN Catalogs of Variable Stars V and X identified the star to be ROT (spotted stars showing rotational variability, Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This star was observed in TESS sectors 14 and 20 at the cadence of 2 minutes, and late in sector 47 at 20 seconds cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The 20-second data unexpectedly do not show clear variability patterns, but the low-rate cadence data exhibit evident periodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 29 1779 1780 1781 1782 1783 1784 1785 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 sap_flux TIC 61236485 -- Normalized SAP Flux 0 5 10 15 20 25 Frequency (d−1) 0 20 40 60 80 100 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 5 10 15 20 25 0 2 4 6 8 10 12 14 7 significant peaks in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125--6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='528 d−1 prewhitened AL Tri = TIC 61236485 - TESS sectors 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 23 TESS light curves (upper) and amplitude spectrum of TIC 61236485 (=AL Tri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 Frequency (d−1) 0 1 2 3 4 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 18 significant peaks prewhitened IT Dra = TIC 166177270 - TESS sectors 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 24 TESS light curves (upper) and amplitude spectrum of TIC 166177270 (=IT Dra = SAO 16394).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' we used binned time series data to resolve the light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A frequency analysis based on 5-minute and 10-minute binning fluxes consistently showed that it is likely a pulsating variable star with merely a dominant fundamental period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='31280 days and the second and fourth harmonics: f0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='761729 d−1, 2f0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='517933 d−1, and 4f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The odd-order harmonic of 3f0 was not present in a Fourier power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 39 and Table 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='18 ι Boo = HR 5350=TIC 310381204 ι Boo is a bright δ Sct star (V =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m75, A7V, =21 Boo =HR 5350 =HD 125161=TIC 310381204).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Liakos & Niarchos (2017) listed it as a suspected pulsator in 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 0 10 20 30 40 50 60 70 Frequency (d−1) 0 1 2 3 4 SAP (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14 35 Significant peaks prewhitened TIC 166177270 - Sectors 15+16+22+23+48+49+50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 25 TESS light curves (upper) and amplitude spectrum of TIC 166177270 (=IT Dra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Time (TBJD) 207500 210000 212500 215000 217500 220000 222500 SAP Flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='/() SAP_Fluxe( SAP Lig ) Curve - TIC 246938869-S0043 - a portion zoomed 0 10 20 30 40 50 Frequency (d−1) 0 5 10 15 20 25 30 PDCSAP (mmag) Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Prewhitened Residuals Amplitude spectrum 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 Significant peaks prewhitened AD Ari = TIC 246938869-S42+S43 - TESS sectors 42+43 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 26 TESS light curves (upper) and amplitude spectrum of TIC 246938869 (=AD Ari).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 31 2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Time (TBJD) 94000 94500 95000 95500 96000 SAP Flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='/() SAP_Fluxe( SAP Lig ) Curve - TIC 307678320-S0044 - a portion zoomed 0 10 20 30 40 50 60 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 PDCSAP (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 Significant peaks prewhitened TIC 307678320 - Sectors S44+S46 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 27 TESS light curves (upper) and amplitude spectrum of TIC 307678320 (=BR Cnc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 0 10 20 30 40 50 60 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 PDCSAP (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 Significant peaks prewhitened TIC 437039231 - Sectors S44+45+46 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 28 TESS light curves (upper) and amplitude spectrum of TIC 437039231 (=EX Cnc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' eclipsing binary system and adopted a dominant frequency of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='750±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001 d−1 according to the result of 1995– 1998 photoelectric observations by Kiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (1999) and The Washington Double Star Catalog (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star is too bright to be a good target for telescopes sized over 50 cm with CCD photometry due to saturation issues concerning a reasonable CCD exposure time and restricted readout rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in four sectors 22, 23, 49, and 50 at 2-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The TESS stacked 2-minute cadence observations are good for a pulsation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' For this star, SAP fluxes are lined up and look like little systematic variations, thus SAP fluxes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As seen now, it is absolutely true that a one- component sinewave function is not enough for account- ing for the light variations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We update the pulsation contents with 65 frequencies in Table 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='19 CD-58 1608 =TIC 279476440 CD-58 1608 (=TIC 279476440, B=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m97, V = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m36 G= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m763089) is a plain star with which there is no reference linked in Simbad/CDS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 29 Kepler K2 mission light curves (upper) and amplitude spectrum of BU Cnc (=EPIC 211936696).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 14 16 18 20 22 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 31 Kepler K2 Mission light curves (upper) and amplitude spectrum of BV Cnc (=EPIC 211931309).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' light curves (HLSP-QLP products) in 30-minute cadence (sectors 1–13) and in 10-minute cadence (sectors 27– 39) exhibit clear light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' But the combination of all 24 sectors’ data led to confusing results due to Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 Amplitude (1000 ppm) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='150 30+ significant peaks prewhitened EPIC211941583-c051618r (K2 Campaigns 05, 16 and 18) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 32 Kepler K2 Mission light curves (upper) and amplitude spectrum of HD 73712 (=EPIC 211941583) 2555 2560 2565 2570 2575 2580 Time (TBJD) 409000 409500 410000 410500 411000 411500 412000 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Cu ve - TIC 175261925-S0046 0 5 10 15 20 25 30 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 Observed (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='175 30 Significant peaks prewhitened TIC 175261925 - Sectors S4446 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 33 TESS light curves (upper) and amplitude spectrum of TIC 175261925 (=HD 73172).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' undetrended SAP light curves without the removal of spacecraft’s systematic effects and variation from sector to sector were not cleanly removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With only sector 39 we derived a 7 low frequencies solution with the goodness of fitting residual less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00037526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then we selected the data that suffered from less scatter and abnormal variations from sectors 1,2,7,36,37 and 39 as a subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We summarized our final results from sector 39, the 6-sector subset, and all 24-sector full datasets in Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star is a longer multi-periodic pulsating star of type mostly γ Doradus, which is pulsating in non-radial gravity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 TYC 8549-1255-1=TIC 279476423 TYC 8549-1255-1 (=TIC 279476423 =2MASS J07010497-5854230, B= 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m71, V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m78, G= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m552827) is a plain star with a null reference linked in Simbad/CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency analysis showed two twin peaks at (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='941631,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='940385) and (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07122,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='072548) d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' They should be aliases and prewhitening can not eliminate the closely spaced peaks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Overall, concerning the long-cadence sampling, these high frequencies are wrongly introduced, mostly by the undetrended SAP light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No intrinsic pulsational 34 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2555 2560 2565 2570 2575 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 Normalized pdcsap_flux PDCSAP: 10-min binned 0 5 10 15 20 25 30 Frequency (d−1) 0 500 1000 1500 2000 Normali2ed PDCSAP_Fl−0 (e −s−1) Am(lit−de s(ectr−m Prewhitened Resid−als Si nificant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 0 100 200 300 400 One significant peak prewhitened TIC 175261942 - Sectors S4446 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 34 TESS light curves (upper) and amplitude spectrum of TIC 175261942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 20 25 30 35 40 45 50 55 60 Frequency (d−1) 0 20 40 60 80 Ampli)ude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 50 significant peaks prewhitened 20 30 40 50 60 70 80 90 0 1 2 3 4 5 f0 and 2f0 prewhitened TIC 392774261 - S0019 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 35 TESS light curves (upper) and amplitude spectrum of BL Cam (=TIC 392774261).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' frequency was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This star seemed to vary in brightness irregularly over a time scale of days to weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21 IRAS 20069+3648 = TIC 42247329 IRAS 20069+3648 (= TYC 2683-2241-1 = TIC 42247329, V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m43, B=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m02) is a plain star with none reference Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 35 0 1 2 3 4 5 Frequency (d−1) 0 1 2 3 4 Normalized PDCSAP_Flux (e −s−1) Amplit−de spectr−m Prewhitened Resid−als Si nificant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 Significant peaks prewhitened TIC 309661089 - Sector S47 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 36 TESS light curves (upper) and amplitude spectrum of TYC 3413-228-1 (=TIC 309661089).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' in Simbad/CDS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS sectors 14, 15, and 41 observed the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Variability on a time scale of days is evident, but it seems irregular or non-periodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A frequency based on these three sectors’ data failed to resolve a good fit for the light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22 HD 227681 = TIC 41194739 HD 227681 (=TIC 41194739, V =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m807, B=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m798) is a plain star with just 3 references where the latest was back to 1993 according to CDS/Simabd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ASAS archived its light curves (as AP13958649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='csv) on HJD 2457069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='16843 – 2458327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='89257 with 184 data points at a mean photometric error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m02 but no variability was visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 14, 15, and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is now surely identified to be an eclipsing binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Table 35 and Figure 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='23 TYC 2682-229-1 = TIC 89755468 TYC 2682-229-1 (= TIC 89755468, V =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m77, B=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m04) is a plain star with merely two references linked in Simbad/CDS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is a binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sectors 14, 15, and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A preliminary analysis based merely on the data in sector 41 shows strong variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Detailed analyses will be given elsewhere when combining the FFI images’ light curves on sectors 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Table 36 and Figure 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='24 21 Com = TIC 393819751 21 Com (= TIC 393819751, B=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m496, V =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m436, A2pv) is a classical magnetic chemically peculiar (Ap/CP2) star showing increased abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 21 Com has been extensively studied in the past, with widely differing and sometimes contradictory results, concerning the occurrence of short-term variability (Paunzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star exhibits rotational light variability with a period of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05219(2) d, with no significant frequencies were found beyond 5 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Their radial velocity data also do not indicate short-term variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Pulsational models assuming different metallicities and ages, which do not predict the occurrence of unstable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star 18 Com, employed as a comparison star for 21 Com in the past, had been identified as a periodic variable (P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='41645 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This is similar to the case of HD 52788, that is previous differential photometry of both 21 Com and HD 52788 had used a variable as comparison star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We checked the TESS 2-minute cadence data (S0022 and S0049) for this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The 30-minute cadence data provided by HLSP-QLP and HLSP-SPOC looks nice – a few outliers are gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We found that the rotational period 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 1845 1850 1855 1860 1865 1870 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='003 Normalized pdcsap_flux PDCSAP: 30-min binned 1845 1850 1855 1860 1865 1870 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='003 Normalized pdcsap_flux PDCSAP: 30-min binned 0 1 2 3 4 5 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Normalized PDCSAP_Flux ( −s−1) Amplitud −p ctrum Pr 1hit ( d R −idual− Sig(ifica(t l v l at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 One significant peak prewhitened TIC 356473060 - Sector S20-HLSP-QLP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 37 TESS light curves (upper) and amplitude spectrum of TYC 3413-187-1 (=TIC 356473060).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' becomes to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='046107(=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='488733) from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05219 (=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='487284).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves can be well-fitted with 10 low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 46 and Table 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No trace of short-term variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 TIC 99180739 TYC 6672-772-1 (= TIC 99180739, B=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m66, V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m91) is a plain star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS light curves from sectors 10 and 37 at the cadence of 2-minute and 10-minute are available, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We obtained a 6-frequency solution to fit the light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Regarding to the B − V =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m743 value (could roughly correspond an effective temperature of 5310 K) and other astronomical parameters from TIC catalog v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2: Teff= 5644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 K, log g= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='53865, M/M⊙= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001, and R/R⊙= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='891099, this star is probably a solar-like oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Check results in Table 38 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 CD-54 7154=TIC 173503902 CD-54 7154 (=TIC 173503902 =Gaia DR3 5923586600813106816, B=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m65,V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m26) is a known δ Scuti star with GCVS designation of V0952 Ara (Kazarovets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We list its parameters by combining Gaia DR3 parallax and TIC v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star’s light curves are wonderful for its conspicuous long-time periodic light variations at a period of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='81 days, superimposed by shorter periodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Such dumbbell-shaped profile (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 49) is strongly reminiscent of amplitude modulation presented in RR Lyr-type pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It also conjured up the demonstration theoretically expected in overcontact eclipsing binary systems, where both stellar components have overfilled their Roche lobes, resulting in a dumbbell-shaped shared envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' As we know that a significant number of RRab stars (up to 50% according to Jurcsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' (2009)) exhibit long-term modulations of the amplitudes and Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 37 1845 1850 1855 1860 1865 1870 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0010 Normalized pdcsap_flux PDCSAP: 10-min binned 0 5 10 15 20 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 Observed (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 6 Significant peaks prewhitened TIC 307618601 - Sectors S0020-LC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 38 TESS light curves (upper) and amplitude spectrum of TIC 307618601 (=HD 48270) phases of their light curves – a phenomenon first discovered by Sergey Blazhko in 1907 and the origin of this effect remains a mystery to the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The Blazhko modulation of light curves may be strictly periodic (with periods ranging from days to years), multi-periodic, or irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Demo examples given in OGLE project 28 , for instances OGLE discovered: RRab stars (just fundamental-mode pulsators) OGLE-BLG- RRLYR-09193 (17:58:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='32 −27:37:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2), OGLE-BLG- RRLYR-11419 (18:02:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27 −28:47:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9), OGLE-BLG- RRLYR-11992 (18:03:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='50 −29:10:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' RRc star (pulsate in the first-overtone mode) OGLE-BLG-RRLYR- 12135 (18:03:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='67 −29:07:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' RRd star (with fun- damental and first overtone double-mode pulsations) OGLE-BLG-RRLYR-05762 (17:52:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 −29:33:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7, Pf=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4663027,P1O=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3472630 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' They all exhibit strong Blazhko modulation over 100 days period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The Blazhko effect may shed light on the cause of the light variation profile of TIC 173503902, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' periodic amplitude modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Furthermore, this is also similar to the observed pulsation amplitude changes with the orbital phase in OGLE-SMC-T2CEP-28, an eclipsing binary with a pulsating primary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' From the light curve being subtracted the eclipsing modulation, the residual pure pulsations, the amplitudes of pulsations change with 28 http://ogle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='pl/atlas/RR Lyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html half the orbital period, which reflects complex oscillations of a star that is distorted by tidal interactions from its companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The observed pulsation amplitudes change depending on the angle at which we observe the star 29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' On the other hand, The longer periodic variation of brightness may be due to its rotation, so might be a rotational brightness variation with a period of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='81 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Last, resonance occurs when an external oscillation is exerted on the star, with a frequency in the neighborhood of a certain resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to the natural frequency of the system on which it acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' When an oscillating force is applied at a resonant frequency of a dynamic system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies The two strongest peaks at f0=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='162285 and f1=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='36902 d−1 probably can interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The star deserves further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Check Figure 49 and Table 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 29 http://ogle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='pl/atlas/W Vir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='html 38 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 0 10 20 30 40 50 60 Frequency (d−1) 0 5000 10000 15000 20000 25000 SAP_Flux (e 1s11) Amplitude spect)um P)ew itened Residuals Significant le−el at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 0 2000 4000 6000 8000 10000 3 Significant peaks prewhitened TIC 309661100_S20 - Sectors 5m-bin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 39 TESS light curves (upper: with the fitted line) and amplitude spectrum of TIC 309661100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27 GSC 04040-01606 = UCAC4 771-012013 GSC 04040-01606 (= UCAC4 771-012013 = TIC 372724683 = Gaia DR2 512143690071260800) is a known δ Sct star which variability was first claimed by Handler & Meingast (2011) during a search for new β Cephei stars in the young open cluster NGC 637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Due to fewer observations, it was suspected to be multiple- frequency pulsation with apparent multiple frequencies between 6 and 8 d−1 with amplitudes around 12 mmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' No more research since discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in Sectors 18, 24, and 25 at the 30-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Light curves are available as HLSP-QLP products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We resolved 15 significant pulsation frequencies, see Table 40 and Figure 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28 Six Photometric Standards We checked a few Landolt photometric standard stars (Landolt 2013, 1992) as an inauguration of another project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Since CoRoT, with unprecedented photometry precision, variability is becoming basically universal for a huge population of celestial objects down to the sub- milli-magnitude level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In turn, it becomes hard to confirm a star to be non-variable or constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' So we go to check a group of photometry standards for variability with data from space telescopes (see Table 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 SA 41-660 = UCAC4 678-116341 = TIC 455553286 As Landolt photometric standard star, SA 41-660 (V =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m804) is a high proper motion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='WG (IBID) 18e2 J810 J82 J820 822 J8e0 asoo as20 a320 a400 a420 a200J bDC2Vb_LInx62 bDc2vb r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='aμf Cnla6 - lIC 30aeeJJ00-20050Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 39 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Time (TBJD) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='630 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='635 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='640 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='645 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='650 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='655 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='660 SAP Fl)x (e−/s) 1e6 SAP_Fl)xes SAP L gh( C)rve - TIC 310381204-S0022 - a portion zoomed 0 10 20 30 40 50 60 70 80 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 SAP (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 64 Significant peaks prewhitened TIC 310381204 - Sectors 22+23+49+50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 40 TESS light curves (middle: red dots with the fitted line) and amplitude spectrum of TIC 310381204 (=ι Boo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' it in sectors 15 and 16, light curves are extracted as HLSP-QLP results in fits and txt formats, where the latter has deleted unusual light variations caused by external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Instead, we used the fits file and did custom detrending, that is, divide each sector into two portions, which were separately removed from a 6-order polynomial fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Then we analyzed the residual light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The result of a frequency analysis makes clear the star’s longer periodic variability, see Figure 50 and Table 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 SA 26-93 = IRAS 06415+4434 = TIC 307650624 Simbad/CDS classifies SA 26-93 (= IRAS 06415+4434 = TIC 307650624) to be a long periodic variable candidate (also refers to Gaia DR2: Mowlavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ASAS-SN Catalogs of Variable Stars V and X identified the star to be SR (semi-regular variable, Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS observed the star in sector 20, light curves are available as HLSP-QLP products in 30-minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This sector data suffered from severe instrumental issues (see top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 51), After detrending for each orbit – removing a third-order polynomial fitting from the data, the residual light curves confirmed clear variability, being consistent with Simbad/CDS classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A Fourier analysis shows a couple of peaks lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 d−1 with the highest peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21317 d−1, corresponding to a period of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='69 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Significant frequencies are not a good 40 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Amplitude (1000 ppm) No ) gn f cant peak) la(ge( than 1 d−1 Ampl tude )pect(um P(e−h tened Re) dual) S gn f cant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 7 ) gn f cant peak) p(e−h tened TIC279476440_S0039 (S0039 at 30-minute cadence) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='enc1 (d21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 Am)li−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='de (mmag) N( signi ican− )eaks larger −han 1 d21 Am)li−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='de s)ec−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m Prewhi−ened Resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='als Signi ican− level a− SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 4 signi ican− )eaks )rewhi−ened TIC279476440_S0139 (S0001--0013, 10-minute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' S0027--0039, 30-minute cadence) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 41 TESS light curves and amplitude spectrum of TIC 279476440 (sectors 1–13, 27–39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 41 10 20 30 40 50 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0, noise based original data 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='175 4 significant peaks prewhitened TIC279476423_S0139 (S0001--0013, 10-minute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' S0027--0039, 30-minute cadence) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 42 TESS light curves and amplitude spectrum of TIC 279476423 (sectors 1–13, 27–39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1685 1690 1695 1700 1705 1710 Time (TBJD) 88000 88200 88400 88600 88800 89000 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Curve - TIC 42247329-S0014 1715 1720 1725 1730 1735 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='003 Normalized pdcsap_flux PDCSAP: 10-min binned Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 43 TESS light curves of TIC 42247329 in TESS sectors 14, 15 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' fit for the light variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Higher precision photometry over a longer time is needed to disclose the true nature of the variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 SA 41-654 = BD+44 3978 =TIC 455517315 BD+44 3978 (V =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='m01, F8) is a Landolt photometric standard star without light variability reported before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' A frequency analysis of the TESS data in sectors 15 and 16 showed that the star is constant down to 78 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Long time-scale variations may be involved in systematic issues, detrending was not applied to derive any long periodic variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 SA 26-96 = UCAC4 673-048808 = TIC 307650596 This Landolt photometric standard star is confirmed constant in TESS photometry down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00108 in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Though an enforced frequency analysis was applied and three significant peaks were presented in the low domain, we are hardly certain light variations in a time scale over days or weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 42 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 2420 2425 2430 2435 2440 2445 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='008 Normalized pdcsap_flux PDCSAP: 10-min binned 1685 1690 1695 1700 1705 1710 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='020 sap_flux TIC 41194739_S0014-- Normalized SAP_flux 1715 1720 1725 1730 1735 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='020 sap_flux TIC 41194739_S0015-- Normalized SAP_flux 10 20 30 40 50 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Amplitude (mmag) Aliasing Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 5 significant peaks prewhitened TIC 41194739 - S141541 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 44 TESS light curves and amplitude spectrum of TIC 41194739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 SA 26-95 = UCAC4 673-048807 = TIC 307650637 This Landolt photometric standard star is confirmed constant in TESS photometry down to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='188 mmag, the standard deviation of the residuals after removing polynomial fittings individually to the SAP fluxes which suffered from instrumental variations during each orbit in TESS sector 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' See Figure 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 8 SUMMARY AND ENDING REMARKS With the uninterrupted high-precision photometry from TESS and Kepler K2, we are able to insight the real nature of those stars which would have been confused or misunderstood due to short-term poor-quality ground observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The author has explored, further processed, and analyzed the TESS data for a small sample of 50 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This paper reports our preliminary TESS discoveries Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 43 1715 1720 1725 1730 1735 Time (TBJD) 81200 81400 81600 81800 82000 82200 82400 PDCSAP Flux (e−/s) PDCSAP_Fluxes PDCSAP Light Curve - TIC 89755468-S0015 1 2 3 4 5 Frequency (d−1) 0 50000 100000 150000 200000 250000 Ampli)ude (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 2000 4000 6000 8000 10000 12000 5 significant peaks prewhitened TIC 89755468 - S0015 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 45 TESS light curves and amplitude spectrum of TIC 89755468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12 significant peaks prewhitened TIC372724683_S182425 (TESS Sectors 18, 24 and 25 ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 48 TESS light curves and amplitude spectrum of TIC 372724683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 45 10 20 30 40 50 Frequency (d−1) 0 5 10 15 20 25 Amplitude (1000 ppm) aliases Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0, noise based original data 10 20 30 40 50 0 1 2 3 30+ significant peaks prewhitened TIC173503902-S1239 (TESS Sectors 12 and 39) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 49 TESS light curves and amplitude spectrum of TIC 173503902.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 3 significant peaks prewhitened TIC455553286 (TESS Sectors 15 and 16 ) 0 1 2 3 4 5 Frequency (d−1) 0 2 4 6 8 Normali2ed PDCSAP_Fl−0 (e −s−1) Am(lit−de s(ectr−m Prewhitened Resid−als Si nificant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 1 2 3 4 5 6 One significant peak prewhitened TIC 455553286 - Sectors S1516-llc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 50 TESS light curves (upper) and amplitude spectrum of TIC 455553286 (UCAC4 678-116341).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' resulting from frequency analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We unveiled the variability types and pulsation contents down to an unprecedented precision for the selected targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We discovered a sum of 20 new pulsating variable stars 46 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Table 21 Frequency Solution of IT Dra (= SAO 16394 = TIC 166177270) Based on TESS Sectors 15, 16, 22, 23, 48, 49 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency(d−1) Amplitude (mag) Phase (0–1) SNR f0= 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='850786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='681244 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f1= 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='062767 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='858786 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f33= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='768254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='569036 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Zeropoint: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0015644164 and eclipsing binaries in the sample, including 5 δ Sct stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 4 EB/EW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1 BCEP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1 gDor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 1 solar-like oscillator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 2 pulsators in SB+EB/EW systems and 6 unclassified variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Table 43 outlines the results of the selected targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Among them, the two comparison stars HD 53349 and HD 53166 used in the differential photometry of Table 23 Frequency Solution of EX Cnc (= TIC 437039231) Based on TESS Sectors 42, 44, 45 and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='536208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002487 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' AD Ari, formerly known as δ Sct due to fewer data, actually is now identified to be an eclipsing binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The pulsation frequency spectrum of HD 52788 over 130 frequencies is an outstanding record and unique among the δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In addition, we compiled a comprehensive catalogue consisting of more than 59,350 individual δ Sct stars based on existing catalogs, data releases of various surveys as well as recent publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This is the largest collection for this class of pulsating stars by far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' With TIC and Gaia DR3 cross-identifiers and stellar parameters extracted from the TESS Input Catalog and Gaia DR3, the catalog is much more useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' The H-R diagrams using this biggest amount of DSCT show a much extended DSCT domain than that demonstrated in earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' It is interesting and necessary to further examine those members outside of the extended domain for a confirmed observational region of δ Sct pulsators, because real observational borders will strictly constrain and impact on the theories of stellar evolution and pulsation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' In special, the HTML version of the catalog provides hyperlinks to MAST Portal, CDS portal, and Gaia data archive, which play the role of observation planning and follow-up investigation portal for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 47 Table 24 Cataloged Parameters for BU, BV, BN Cnc and HD 73712 in NGC 2632 (Praesepe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ID & Parameters V* BU Cnc V* BV Cnc V* BN Cnc HD 73712 TIC TIC 175240124 TIC 175264756 TIC 175264749 TIC 175261925 HD HD 73576 HD 73746 HD 73763 HD 73712 2MASS J08394466+1916308 J08403296+1911395 J08403924+1913418 J08402013+1920564 EPIC EPIC 211936696 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='830386 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 f34= 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='833269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='386790 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f35= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='810244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='869261 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 f36= 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='647385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='276940 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 f37= 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='275041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='192975 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 f38= 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='310527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='953923 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f38 =f0 + f3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='953923 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 f39= 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='129373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='841000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 Zeropoint: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6150921 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00696801437 mag Table 31 Frequency Solution of TYC 3413-228-1 (=TIC 309661089) Based on TESS Sector 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency Amplitude Phase f0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='076199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='004246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='546173 f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='145024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='266956 f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='172062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='734760 f3 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5f0± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='306296 f4 =f2 + f3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='899568 f5 =3f0 + f3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='292986 f6 =3f1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='912223 Zeropoint: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000165870177 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000774585274 mag 50 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Table 32 Frequency Solution of HD 48270=TIC 307618601 Based on TESS Sector 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Frequency (d−1) Amplitude (mag) Phase (0-1) SNR f0= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='161778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='861076 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 f1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='982090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='896451 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 f2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='553853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='674589 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 f3= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='052511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='249170 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='7 f4= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='430140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='713577 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 f4 =f1 − f2± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='713577 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 f5= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='810795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='306757 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 f5 =5f0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='306757 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 Zeropoint: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9495583 mag Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00037776471 mag Table 33 Frequency Solution of TIC 309661100 Based on TESS Sectors 20 and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Label Frequency (d−1) Amplitude (ppt) Phase f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='761729 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='961933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='614266 2f0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='517933 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='926124 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 Zeropoint: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000154424229 Residuals: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00599494917 Table 43 Discoveries and Updates Presented in Current Work on a Sample of 50 Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='SN TIC ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='Simbad Identification ' metadata={'source': 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HD 191025 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New DSCT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='06 TIC 89757305 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: HD 227658 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New BCEP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='07 TIC 279476440 CD-58 1608 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New gDor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='08 TIC 99180739 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TYC 6672-772-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New solar-like oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='09 TIC 356473060 TYC 3413-187-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New Pu* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='10 TIC 309661089 TYC 3413-228-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New Pu* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='11 TIC 309661100 PM J07472+5020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New Pu* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='12 TIC 307618601 HD 48270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New Pu*: Irregular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='13 TIC 279510617 TYC 8549-1773-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='14 TIC 89755468 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TYC 2682-229-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New EB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='15 TIC 246938869 AD Ari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New EB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='16 TIC 41194739 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: HD 227681 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New EB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='17 TIC 41195818 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: UCAC4 630-092212 New EW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='18 TIC 42247329 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='IRAS 20069+3648 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='New SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='19 TIC 175261925 NGC 2632: HD 73712 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='SB+EB+dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='20 TIC 13876370 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6910: V2245 Cyg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='SB+EW+Pu* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='21 TIC 41195917 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: V2238 Cyg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='22 TIC 90350726 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: V1821 Cyg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='23 TIC 27936176 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='V383 Car = HD 52788 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='24 TIC 166177270 IT Dra = SAO 16394 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='25 TIC 307678320 NGC 2632: BR Cnc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='26 TIC 175240124 NGC 2632: BU Cnc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='27 TIC 175264749 NGC 2632: BN Cnc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='28 TIC 175264756 NGC 2632: BV Cnc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='29 TIC 437039231 NGC 2632: EX Cnc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='30 TIC 173503902 CD-54 7154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='31 TIC 372724683 GSC 04040-01606 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='32 TIC 310381204 ι Boo = HR 5350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='33 TIC 266794067 V2455 Cyg = HD 204615 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='HADS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='34 TIC 392774261 BL Cam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='SX* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='35 TIC 175261942 2MASS 08400416+1924502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='ER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' mono-periodic 33 TIC 307650624 IRAS 06415+4434 SR/LPV 36 TIC 393819751 21 Com Ap/CP2 37 TIC 61236485 AL Tri = GSC 2293-1021 EW 38 TIC 279476423 TYC 8593-1255-1 uncertain 39 TIC 41195988 GSC 02683-03318 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='40 TIC 455553286 UCAC4 678-116341 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='41 TIC 307650596 UCAC4 673-048808 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='42 TIC 307650637 UCAC4 673-048807 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='43 TIC 455517315 BD+44 3978 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='44 TIC 41195891 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: HD 227682 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='45 TIC 41196013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='NGC 6871: UCAC4 630-092237 constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='46 TIC 90350490 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2MASS J20064266+3550552 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='47 TIC 90350607 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2MASS J20064698+3551472 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TIC 279361762 HD 52788 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='dS* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TIC 402338608 IP Vir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='no TESS LC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TIC 416302408 V577 Oph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='no TESS LC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1842 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1844 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1846 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1848 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1850 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1852 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1854 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='TBJD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='04 Amplitude (mmag) Orbit-1 of TESS Sector 20 Observed 3th-order polyfit 1844 1846 1848 1850 1852 1854 TBJD −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 Amplitude (mmag) Orbit-1 of TESS Sector 20 Residuals 1858 1860 1862 1864 1866 1868 TBJD 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0 10 20 30 40 50 60 4 Significant peaks prewhitened TIC 307650624 - Sectors S20-Res-LC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 51 TESS light curves (upper) and amplitude spectrum of TIC 307650624 (IRAS 06415+4434).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 54 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou 1845 1850 1855 1860 1865 TBJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='003 Normalized flux TIC 307650596 in TESS Sector 20 Observed 0 1 2 3 4 5 Frequency (d−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 Observed (mmag) Amplitude spectrum Prewhitened Residuals Significant level at SNR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='5 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 3 Significant peaks prewhitened TIC 307650596 - Sectors S20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 52 TESS light curves (upper) and amplitude spectrum of TIC 307650596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Catalog of 59 Thousand δ Sct Stars and Dozen Discoveries with TESS 55 1845 1850 1855 1860 1865 1870 Time - 2457000 [BTJD days] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='3 sap_flux TIC 307650637_S0020-- Normalized SAP_flux 1842 1843 1844 1845 1846 1847 1848 TBJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='98 Amplitude (mag) 1st-Orbit of TESS Sector 20 Observed 3th-order polyfit 1848 1849 1850 1851 1852 1853 1854 1855 TBJD 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='01 Amplitude (mag) 2nd-Orbit of TESS Sector 20 Observed 4th-order polyfit 1845 1850 1855 1860 1865 TBJD −15 −10 −5 0 5 10 15 Amplitude (mmag) TIC 307650637 in TESS Sect r 20 Observed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 53 TESS light curves (upper) and amplitude spectrum of TIC 307650637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' 56 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Zhou Acknowledgements This paper includes data collected with the TESS mission and by the Kepler mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Funding for the TESS mission is provided by the NASA Explorer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Funding for the Kepler mission is provided by the NASA Science Mission Directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' STScI is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=', under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' We acknowledge the use of TESS data, which are derived from pipelines at the TESS Science Processing Operations Center (SPOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' TESS High Level Science Products (HLSP) produced by the Quick-Look Pipeline (QLP) at the TESS Science Office at MIT, which are publicly available from the Mikulski Archive for Space Telescopes (MAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This research has made use of the SIMBAD/VizieR databases, operated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This research has made use of the International Variable Star Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Special acknowledgements go to all the projects and surveys mentioned in this work including but not limited to: OGLE-V, ASAS-3, ASAS-SN, GCVS, ZTF, SuperWASP, 2MASS, ROTSE, CRTS/CSS, LAMOST, and other surveys and websites from where catalogs of δ Sct stars and relevant data were downloaded and compiled into the new catalog in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' Some of them are footnoted in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia ( h t t p s : / / w w w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' c o s m o s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' e s a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' i n t / g a i a ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, h t t p s : / / w w w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE_T4oBgHgl3EQf4xzd/content/2301.08355v1.pdf'} +page_content=' c o s m o s .' metadata={'source': 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b/mdA0T4oBgHgl3EQfJf-W/content/tmp_files/2301.02091v1.pdf.txt @@ -0,0 +1,2384 @@ +Fast-Scrambling and Operator Confinement Using an Auxiliary Qubit +Joseph C. Szabo1 and Nandini Trivedi1 +1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA +(Dated: January 6, 2023) +We introduce a minimal model for realizing a fast-to-slow scrambling transition mediated by an auxil- +iary central qubit (c-qubit). The c-qubit is coupled to a spin-1/2 Ising model with local Ising interactions +and tunable c-qubit-spin coupling. Each spin becomes next-nearest neighbor to all others through the +c-qubit, which mediates effective all-to-all interactions. As the interaction with the c-spin increases, +we find a surprising transition from super-ballistic scrambling and information growth to continuously +restricted sub-ballistic entanglement and operator growth. This slow growth occurs on intermediate +timescales that extend exponentially with increasing coupling and system size, indicative of logarith- +mic entanglement growth. We find that in the slow-scrambling regime, the c-qubit Ising interaction +allows commuting operators to grow support on all sites rapidly, while operators orthogonal to the in- +teraction become echoed out. This projects local operators to lie in a restricted subspace and prevents +extensive operator entanglement growth. We provide exact dynamics of small systems working with +non-equilibrium, effective infinite temperature states, and additionally contribute analytic early-time +expansions that support the observed rapid scrambling to quantum Zeno-like crossover. Tracing out +the central qubit provides a unique translation from the full, closed unitary dynamics to a simple open +system construction consisting of a typical spin-chain with hidden qubit degree of freedom. +I. +INTRODUCTION +Operator +scrambling +and +entanglement +entropy +spreading are unambiguous discriminators of purely +quantum mechanical nonequilibrium dynamics; +fasci- +nating properties underlying quantum thermalization, +dynamical phase transitions, and topological order [1–7]. +In the Heisenberg picture, quantum operator scrambling +details how initially localized operators propagate over +spatiotemporal degrees of freedom due to noncommu- +tative many-body interactions. +In the complementary +Schrodinger +picture, +entanglement +entropy +captures +growing information complexity: from initially classical +states to those with nontrivial entangled structure. +Quantum information dynamics bridge both theoretical +and experimental communities as primary measures for +quantum complexity and expressivity [8–12]. These con- +cepts combined with quantum simulation/circuit devices +have coalesced into many enriching, recent experiments +[10, 13–15]. The accelerating pace of results and drive +to continually advance the corresponding theory extend +these successes to further research at the intersection +of quantum chaos, thermalization, and computability, +extending from qubits to black-holes and quantum +gravity [16–24]. +The primary research thrusts among the quantum in- +formation dynamics community fall along the lines of un- +covering the minimal mechanisms behind myriad infor- +mation dynamical phases and understanding the fate of +quantum to classical thermalization. In studying gener- +alized quantum information dynamics, there are typically +two disparate perspectives: closed and open quantum sys- +tems. Closed quantum systems exhibit rich scrambling +physics ranging from frozen [25] to fast [18] dynamics, +with the typical questions relating to how well-preserved +is such physics under driving and dissipation contribu- +tions from an external environment [26–29]. The envi- +ronment is oftentimes reduced to a memoryless, effective +Markovian description, which hinges on assumptions in- +cluding weak-coupling and a separation in the timescales +associated with system and environment [30, 31]. Though +solving for the exact dynamics for a full complex environ- +ment is beyond the capabilities of current devices, taking +into account the structure and interaction with the en- +vironment poses interesting research questions: what is +the fate of entangled information within the system, how +can a structured environments drive effective interactions +and information dynamics, and how does the environment +serve as a probe in an information theoretic/entropic ca- +pacity? +A simple avenue for exploring the impact of a struc- +tured quantum environment is by considering compos- +ite, unitary models. +Focusing on a particular subsys- +tem of a full closed quantum system and tracing over +the additional degrees of freedom (DoFs), then termed +the environment, captures the subsystem’s effective dy- +namics/interactions. +This is a popular focus of study +as it provides an open quantum system perspective for +the subsystem and allows full consideration of the envi- +ronment’s structure and interaction topology. This con- +struction allows us to specifically evaluate how variable +structured environments impact the overall quantum in- +formation dynamical phase as expressed by the under- +lying subsystem. +Previous work considered the valid- +ity of Markovian assumptions provided variable system- +environment coupling, and here we are looking to add +an information scrambling perspective. +Considering a +system-environment construction in this manner directly +applies to those codes/models investigating the infor- +mation physics of auxiliary bits or those systems with +inherent auxiliary DoFs such as mechanical or optical +modes [32–35]. +Studying composite systems in this manner provides an +interesting framework extending current quantum infor- +arXiv:2301.02091v1 [quant-ph] 5 Jan 2023 + +2 +mation dynamic research. Significant recent results focus +on the range of interactions, the speed and nature of infor- +mation propagation, the role of inherent symmetries, and +the effect of emergent symmetries in the cases of many- +body localization, Floquet periodic driving, etc. The same +phenomena can be similarly cast as an environment me- +diated effect. The tunability of the system-environment +network topology and the inherent environment structure +and interaction symmetries then allows for systematic in- +vestigation into the particular contribution on the overall +dynamics. +In this paper we explore these aforementioned ques- +tions by considering the simplest environment extension; +an auxiliary central qubit (c-qubit) coupled to 1-d chain +system of interacting spin-1/2 (qubit) objects. Tracing out +the c-qubit and considering the dynamics of the 1-d sys- +tem provides a translation from a full, unitary model, to +an effective long-range, non-Hermitian spin chain with a +hidden qubit degree of freedom. This provides a single +long-range quantum channel for transmitting informa- +tion but at the same time imposes a shared two-fold DoF +across all spins. Though only a small addition to well- +understood nearest-neighbor qubit model, we observe an +abundance of exciting repercussions. +The system-environment coupling expresses various +regimes: in the weakly coupled regime, the c-qubit pro- +vides little feedback and acts as a free channel for infor- +mation to pass unimpeded; while when strongly coupled +to the low dimensional qubit environment, the c-qubit +acts as a strong drive and imposes an effective hidden +symmetry on the underlying spin system and generates +disorder-free localization. We liken the physics observed +here to that seen in systems undergoing quantum mea- +surement or strong Floquet driving. Considering the c- +qubit as a hidden degree of freedom provides unique in- +sight into how the quantum scrambling dynamics of the +underlying spin chain maps to an extended unitary model +provided one additional qubit. Central qubit or a higher +dimensional qudit/cavity/register are popular theoretical +and experimental tools for providing non-invasive many- +body measurements [36–39], evaluating Hermitian and +non-Hermitian response [38, 40], generating effective in- +teractions [41], and studying the fundamentals of deco- +herence and information transport [42, 43]. +Here we particularly focus on the dual effect of a tun- +able central qubit by investigating the operator and en- +tanglement growth in a nonintegrable, ring-star Ising +model. The model includes homogeneous spin-spin inter- +actions in a mixed magnetic field. We find an extremely +surprising fast-to-slow quantum information spreading +transition that occurs due to the nonlocal and coher- +ent nature of the c-qubit. We summarize this result in +Fig.1(d), where in the weakly coupled regime (regime +I), the central qubit mediates rapid scrambling with a +timescale that decreases with system size (green, upper +curve in). In the strong coupling regime (regime II), the +scrambling time increases exponentially with system size +(red, lower curve). The mechanism behind this transition +is the interplay between the noncommuting, extensive c- +qubit Ising interactions and transverse field hc. The met- +ric here presented for scrambling is eSvN(t)/2L+1, which +provides a measure of the span of the quantum wavefunc- +tion throughout the full Hilbert space. As we detail in +what follows, in the strong coupling regime the central +qubit rapidly saturates its entanglement with the sur- +rounding spin-chain environment and becomes strongly +driven by this extensive interaction. This strong inter- +action rapidly aliases operators orthogonal to the cen- +tral qubit Ising interaction on the central qubit and even +more surprisingly within the spin chain. The long life- +time of states and operators that commute with the cen- +tral Ising interaction leads to slow multi-particle entan- +glement growth and operator complexity. +Our work agrees with previous research that finds +an extensively scaling, +nonlocal interaction leads to +rapid scrambling, where the rate increases with system +size [44–46]. +At the same time we find a surprising +limit where the purely quantum nature of the c-qubit +imparts a coherent effect that slows operator decoher- +ence/entanglement and subsequent spreading. This phe- +nomena mirrors what is seen in strongly driven Floquet +systems, where periodic driving can impart an effective +symmetry in all eigenstates and leads to prethermaliza- +tion and correspondingly slow entanglement spreading. +We liken the projective action of the central qubit in this +time independent Hamiltonian to the quantum Zeno ef- +fect where quantum measurement leads to a ballistic to +sub-ballistic entanglement growth transition. Here the +c-qubit imparts a highly nonlocal effect on operator pro- +jection in contrast to a local purification/disorder network +that redefines local spreading dynamics. We illuminate +this c-qubit physics by examining the growth of out-of- +time-order correlators (OTOCs) and the von Neumann +entanglement entropy for sufficiently high-energy initial +product states. +II. +SCRAMBLING METRICS +Many recent works have made significant progress on +establishing the family of scrambling dynamics that oc- +cur in various lattice models and geometric random cir- +cuit designs, as characterized by the growth of OTOCs. +The OTOC generically given as +CVW = ⟨[ ˆW(j,t), ˆV(i,0)]†[ ˆW(j,t), ˆV(i,0)]⟩, +(1) +examines how an initially prepared unitary operator ˆV on +site−i commutes after Heisenberg evolution with operator +ˆW after time t (here assumed a local operator on site−j). +The operator spreading picture is unique to quantum sys- +tems, where in working with pure states, no information +is truly lost but transforms into many-body degrees of +freedom that become increasingly inaccessible provided +control over an initial localized region. +Studying OTOCs and the timescales associated with +scrambling dynamics provides a conjugate perspective as + +3 +compared to entanglement entropy measures and transi- +tions. Where OTOCs and specifically infinite temperature +OTOCs examine the light cone established by Heisen- +berg evolution and depend more strongly on the com- +mutivity graph, entanglement entropy examines how the +wavefunction over a bipartition of Hilbert space spreads +throughout. Here we specifically focus on the von Neu- +mann bipartite entanglement entropy given as +SvN = −∑ +k +λk log(λk), +(2) +where λk are eigenvalues of the reduced density matrix +ρA∣B (RDM) obtained by integrating out subsystem A or B +with corresponding Hilbert spaces HA,HB. OTOCs and +entanglement identify similar physics and previous collo- +quial conceptions of the two established quantum scram- +bling as a unifying framework behind them; +where, +scrambling represents the time for an OTOC between ar- +bitrary sites to become O(1) and entanglement entropy +to become O(L). In the case of OTOCs, this limit is not +rigorous enough and only provides a best-case scenario +for operators traversing the system rather than provid- +ing a timescale for nontrivial operator strings to span +the system [47] (extensive operator entanglement). Rig- +orous relationships between OTOCs and Renyi−2 entropy +have been established [48–50] and special cases have been +studied in particular optical Hamiltonians [24, 51, 52]. +Unitary scrambling physics generally falls into two cat- +egories: systems that thermalize rapidly and those that +fail to do so. The former are known as fast-scramblers; +ergodic systems typically with variable all-to-all range +interactions that spread quantum information through- +out the full Hilbert space in tsc ∼ log(N). Models such +as Sachdev-Ye-Kitaev (SYK) and non-integrable infinite +range Ising and XY models are known to exhibit fast- +scrambling physics [45, 46, 53–55]. +Systems that fail +to thermalize with tsc ∼ eL are slow-scramblers, non- +ETH obeying systems and candidates for highly coherent +quantum information storage. +There are multiple vec- +tors through which non-ETH physics occurs: integrability, +disorder-free localization [56, 57], quantum scarring [58– +61], and/or higher order exact or proximate conservation +laws [62–64]. The origins of much of this work stems from +the dramatic quantum correlations observed in quantum +simulation experiments. +Typical models accessible to simulation and experi- +ment are semiclassical in nature with infinite or long- +range interactions. These models exhibit the characteris- +tic unitary scrambling features we detailed previously, yet +continue to enrich the discussion with new puzzling re- +sults. In Lipkin-Meshkov-Glick (LMG) model or the Dicke +model exhibit strict conservation of the total spin moment +ˆS2 such that the effective number of degrees of freedom +is O(L), compared to O(2L) [24, 65, 66]. These systems +have been observed to spread information rapidly, while +the complexity saturation value remains low. This is in +stark contrast to fast scrambling models like SYK where +infinite-range connectivity allows for rapid and complex +quantum information scrambling. One immediately puz- +zling question is: how do long-range interactions, tend- +ing toward generating semiclassical behavior, compete +with local chaotic quantum dynamics to allow a fast-to- +slow scrambling transition? A complete understanding of +quantum information physics not only hinges on under- +standing the unitary dynamic contribution to experimen- +tal results, but similarly understanding non-Hermitian +processes. +These are inherent to quantum simulation +platforms and represent the an exotic next frontier for +theory and experiment as we move toward fully expres- +sive quantum circuits and computation. +More generalized quantum dynamical behavior has +been +explored +in +recent +studies +consisting +of +non- +Hermitian operations: +composite system-environment +undergoing quantum measurement [67–72], light-matter +interactions [24], dissipative and driven systems [73–76]. +The most extraordinary findings reveal that these non- +unitary dynamics generate effective inter-system interac- +tions and impose effective static long-lived symmetries: +Floquet periodically driven systems are akin to various +unitary scrambling phases. +III. +MODEL +We consider the Hamiltonian for the c-qubit or ring-star +Ising model: +H = +L−1 +∑ +i=0 +λσz +i σz +c − Jσz +i σz +i+1 + hσx +i + gσz +i + hcσx +c + gcσz +c, +(3) +where λ represents the uniform spin-c-qubit interaction, +and J,h, g represent the much-studied nonintegrable +mixed-field Ising model. Here we take J = 1.0,h = hc = +1.05, g = gc = 0.45 unless otherwise noted. This is a well +characterized nonintegrable point for polarized state evo- +lution and operator dynamics allowing us to benchmark +the impact of the c-qubit [46, 77]. +Previous works examined operator dynamics and en- +tanglement growth in random unitary circuits (RUCs) +applied in a star-graph network. +RUCs on the star- +graph generated OTOC dynamics that saturated at t ∝ +log(L) [47, 78]. +In contrast, an interesting section +of [78], examines dynamics of time-independent, star- +Ising Hamiltonians which consist of Ising interactions +and a non-commuting field hc strictly on the central +qubit. In contrast to fast scrambling RUC networks, the +time-independent c-qubit Hamiltonian dynamics gener- +ate novel operator confinement on the c-qubit with a co- +herent lifetime τ ∝ h2 +λL. +This defines the time for sig- +nificant operator weight to decohere into operator sub- +spaces that no longer commute with the Ising interaction, +subsequently allowing slow scrambling to all other sites. +Provided this contrast between the fast-scrambling RUCs +and the slow Hamiltonian dynamics, we ask: can the c- +qubit coupling mediate rapid scrambling in a truly time- +independent nonintegrable model and, if so, when/how +does this picture break down? + +4 +FIG. 1. (a) Illustration of the ring-star model achieved in (b) optically dressed trapped-ion experiments and (c) through nonlocal +unitary gates in a circuit realization. (d) Central qubit mediated dynamics: local interaction (black), c-qubit induced fast scrambling +(green), and c-qubit inhibited scrambling (red). Information scrambling approximated as wavefunction spread in Hilbert space or ∼ +eSvN(t) to show dramatic prethermal-like approach to full thermalization. (e) Qualitative depiction of (d) or how a well defined initial +state grows to fill state/operator Hilbert space with a varying rate depending on dynamical phase. (f) von Neumann entanglement +entropy SvN(t) following a quench from polarized state ∣+ y⟩ for the ring-star Ising model under varying λ. +IV. +ANALYTIC INSIGHTS +Before discussing the exact numerics on the full nonin- +tegrable ring-star Ising model, we first motivate the rea- +sons underlying a dynamical transition from fast-to-slow +scrambling and the corresponding mechanism. We first +summarize typical operator growth in local spin models +and then review the Ising star graph dynamics. +A. +Local Ising Model +In the completely local Ising chain limit λ = 0, we can +gain a heuristic understanding of the characteristic light +cone spreading by performing an early-time expansion +of the dynamical correlator ⟨σz +i (t)σz +j⟩. Using the early- +time expansion of the Heisenberg evolution we can write +ei ˆHtσz +i e−i ˆHt using the Baker–Campbell–Hausdorff (BCH) +expansion +e−i ˆHtσz +i e−i ˆHt = +∞ +∑ +m +(it)m +m! [H,Sz +i ]m +[A,B]m = [A,[A,B]m−1];[A,B]0 = B +[H,Sz +i ]1 = it[hSx +i ,Sz +i ] = −ithS y +i +[H,Sz +i ]2 = (it2) +2 +[H,S y +i ] += −t2 +2 (−λSz +0Sx +i + hSz +i − gSx +i − J(Sz +i−1Sx +i + Sz +i+1Sx +i )). +(4) +Before continuing this expansion to higher orders we ob- +serve a trend in operator growth and can approximate the +weight as +[Sz +i (t),Sz +j] ∝ t∣i−j∣ +∣i − j∣!O(1) +(5) +Naively, we then expect the OTOC, or squared commuta- +tor, to generically follow +Czz(i − j,t) ∼ +t2∣i−j∣ +(∣i − j∣!)2 O(1) +(6) +This approximation is valid for times t ∼ ∣i − j∣/vB, where +vB is the characteristic butterfly velocity. +Before this +time, operator growth is suppressed with an exponent +that grows with distance, and allows for optimized simu- +lations using matrix product operator dynamics (MPO) by +just keeping track of the operator wavefront [79]. Though +many extensive theoretical works provide rigorous esti- +mates on the form of operator growth for translationally +invariant models, here we simply examine the asymptotic +form. This exponentially growing operator weight with +exponent like r = ∣i− j∣ leads to the development of a linear +light cone with butterfly velocity exactly calculated as 2eJ +(e being Euler’s number) for h/J > 1 [80]. Following this +light cone in the nonintegrable Ising model, any localized +operator spans the full operator Hilbert space { ˆX, ˆY , ˆZ, ˆI} +within region r. +An interesting extension of the OTOC is the integrated +OTOC (iOTOC), which is the integral over r and the bi- +partite OTOC [50, 81]. Both provide a novel characteriza- +tion of the operator complexity within a region r. This also +provides an insightful way to then understand the cor- +responding entanglement dynamics for pure state wave- +functions with energy density kBT as in accordance with +ETH the states and operators should exhibit ergodic equi- +libration. +In the nonintegrable Ising model, an opera- +tor saturates the reduced Hilbert space 4∣i−j∣ after a time + +e +Hilbert Space +a) +c) +d) +A +Initial +0.6 +Ancilla free +1 +state: to +Regime I +Lsc +8 +VL +Regime II +B +C-gubit.enhance +Scrambling ( +time +Localqubit-qubit +T S +b) +f) +0.2 +interaction +6 +C-qubit +α logt +protected +0.0 +100 +101 +102 +position +100 +101 +102 +time +time (tJ)5 +tsat = ∣i−j∣ +vB , so simply integrating over Eq.6 provides an +exponentially growing iOTOC for all Cvw. +This opera- +tor growth then guarantees ballistically growing entan- +glement entropy SvN ∼ ∑v,w log[iOTOC(v,w)]. +B. +Star-Ising model +In the star limit, we set J = 0 and tune external fields +h, g to allow more expansive operator evolution. +First +working with only λ ≠ 0, we then add complexity to ar- +rive at the full, nonintegrable ring-star Ising model. For +∣λ∣ > 0, operator growth between leaves of the graph is +trivial. The dynamics are exactly solvable as [σz +i , ˆH] = 0 +for all sites i. Similarly, the commutivity graph represent- +ing the Hamiltonian and how local operators propagate is +completely disconnected with vertices σz +i ,σz +L for all i ∈ L +and no bonds in between. Operators evolve simply under +the central Ising interaction and the two-time correlator +behaves as: +⟨Sx +i (t)∣Sx +i ⟩ = cos2λt. +(7) +And, again, exactly in this case, the OTOC goes as: +Cxx(i, i,t) = 2sin2(2λt), +(8) +Though this limit admits a trivial result, it allows us +to understand the key coherent property of the c-qubit +qubit. We see that for all times, operators orthogonal to +the Ising interaction λ, initially prepared on the leaves, +propagate to the c-qubit after a time t = π/λ. Because all +terms in the Hamiltonian commute with this interaction +[σz +i σz +0,σz +jσz +0] = 0, the action of operator development from +leaves to c-qubit does not decohere into nontrivial, orthog- +onal operators { ˆX, ˆY}. Due to this coherence, operators +oscillate between the initial node and the c-qubit. If the +operator was initially prepared on the c-qubit, it fluctu- +ates onto all nodes, but instead of becoming a many-body +operator, it is a collective superposition of L unique two- +body operators ∑j σx,y +0 σz +j ∈ H(4⊗L). This superposition, +though nonlocal on a timescale τ = O(1), has minimal +operator entanglement and the complexity of operators +strings is fixed to be a maximum of 2. In this scenario, +similarly, minimal entanglement entropy develops as the +number of unique operator states is of L in a Hilbert space +of 4L. Quench experiments on such systems with nonzero +homogeneous magnetic fields will only permit half-system +entanglement to grow like logL since we have effectively +a large semiclassical spin system coupled to a single two- +level qubit [82]. +The simplest extension we can make is to now intro- +duce a transverse field on the c-qubit spin, hc ≠ 0. This +was similarly analyzed in a disordered star-Ising system +for small values of h/λL [78]. For ∣hc∣ > 0, the model re- +tains the same integrability, as any {z1,...zL} is an eigen- +state of the system and will not evolve under unitary +dynamics. +We can then reduce the problem to solving +for the evolution of the central qubit in a mixed x − z +field where the effective longitudinal strength given by +∑i σz +i . Though fully solvable, we find illuminating oper- +ator dynamics in the infinite temperature limit. Solving +exactly for ⟨σx +i (t)σx +i ⟩ as was first provided in [78], we ob- +serve how the coherent oscillation ⟨σx +i (t)σx +i ⟩ = cos(2Jt) +for hc = 0, evolves for increasing hc and leads to operator +decoherence. Operator decoherence, in this sense, is that +as auxiliary operators such as σz +c → σx +c,σy +c, they then no +longer commute with the c-qubit Ising interaction and op- +erator weight then grows on sites j ≠ i. The superposition +of these growing dynamical correlations on sites j ≠ i simi- +larly decreases the probability of finding σx +i on site−i after +time t, leading to a decay in the autocorrelation function +on site i. +For λL >> h, it was shown numerically that early time +operator dynamics go as +⟨σx +i (t)σx +i ⟩ ∼ cos(2λt)e− +√ π +2L +h2 +λ t +(9) +using the memory matrix formalism [78]. Regardless of +whether λ is a Gaussian random variable or homoge- +neous, we arrive at the same exponential dependence on +system size. In the case of Gaussian random variables, +this approximation was observed to be faster than the +true decay rate. The key physics being that the L−site in- +teractions with the c-qubit lead to an extensive coherence +time, which we can evaluate explicitly by tracing over the +set of eigenstates Z: +⟨σx +i (t)σx +i ⟩ = 1 +2L TrZ[⟨zc,...zL∣ei ˆHtσx +i e−i ˆHtσx +i ∣zc,...zL⟩] +(10) += ⟨zc∣⊗⟨Zm∣ei ˆHtσx +i e−i ˆHt∣Zm⟩⊗∣zc⟩ +(11) +ˆH∣Zm⟩⊗∣zc⟩ = ei(hcσx +c+λ∑i>0 ziσz +c)t∣zc⟩ +(12) += I cos(2ωZm t)+ (hcσx +c +λZmσz +c) +ωZm +sin(2ωZm t)∣z0⟩. +(13) +We use same simplified notation as [78], where ωZ = +√ +h2 +λ2Z2m, with Zm = ∑i>0 zi and Z = Z[m] is just +z−eigenstate with magnetization m ∈ [−L,L]. As the set +of states {z1...zL} commute with the Hamiltonian, we re- +duce the Heisenberg evolution to that of a 2-level system +and averaging over the respective density of states, which +is simply a binomial distribution in the infinite tempera- +ture limit. Going back to the full evolution we then have + +6 +⟨σx +i (t)σx +i ⟩ = 1 +2L ∑ +z0 +∑ +m +( L +∣m∣ )⟨zc∣[I cos(2ωZm t)+ (hcσx +c +λZ+σz +c) +ωZm +sin(2ωZm t)]× +[I cos(2ωZ−m t)+ (hcσx +c +λZ−σz +c) +ωZ−m +sin(2ωZ−m]t)]∣zc⟩ +(14) += +1 +2L−1 ∑ +m +( L +∣m∣ )cos(2ωZm t)cos(2ωZ−m t)+ (h2 +c +λ2ZmZ− +m) +ωZmωZ−m +sin(2ωZm t)sin(2ωZ−m t). +(15) +σx +i flips the single spin state zi and leads to two unique +frequencies wZm and wZ−m, separated like λ for h = 0, +Z− +m = ∑j≠i>0 z j − zi. +hc = 0 provides the exact, non- +scrambling result ⟨σx +i (t)σx +i ⟩ = cos(2λt), and for λ = 0 we +have ⟨σx +i (t)σx +i ⟩ = cos(2λt)2+sin(2λt)2 = 1. Starting in the +hc = 0 limit and moving toward high transverse magnetic +field h0 >> λL, we numerically integrate Eq.15 and study +the autocorrelation function on site−i (Fig.2). In the low +field limit, we expect exponentially small modifications to +pure cosine oscillations as we have recreated from [78], +and in the high field limit, σx +i should completely break +down as σz +c decoheres and operator weight spreads to all +sites rapidly. +As σz +c → [σy +0,σx +0], operator weight can be +distributed to all sites j ≠ i, and the rapid rotation of op- +erators on the c-qubit aliases away coherent growth on +site-i. +In Fig.2(a) we see that coherent oscillations are appar- +ent for h/λL O(1) out to tλ ∼ 30 and slow decay with in- +creasing h. Above log[h/λL] > 0, oscillations are barely +visible and dynamics are dominated by exponential decay +with no revival even out to time tλ ∈ [0,200]. This is more +clearly depicted in Fig.2(b), where we take the long-time +average of the autocorrelation function: +A(i,t0) = ∫ +t0 +t>0 ∣⟨σx +i (t)σx +i ⟩∣dt. +(16) +The autocorrelation function transitions sharply at h/L ∼ +O(10−1), where σx +i no longer has significant weight on +the initial site. +For small h, the autocorrelation func- +tion is a nearly pure cos2λt and decay that grows with +h. Near the transition, A(i,t0) sharply decays to ∼ 1/L for +h/λL > O(10−1). We provide a Fourier analysis of Fig.2(a) +in (c), where the transition is more clearly resolved. The +oscillatory part (ϵ0) decreases (elongates in time) above +h/λL = 1 with an exponential decay rate that peaks at the +transition point. Above the transition point, the dynam- +ics are no longer captured by a decaying cosine function, +but crossover to exponentially damped operator dynam- +ics. The large magnetic field on the central site initially +prevents operator weight from leaving site−i as shown by +the elongating decay in (a). At this point the magnetic +field is dominating the operator dynamics and rapidly ro- +tates σz +c into eventual two-body operators that live in a +superposition on all sites. In this regime, operator weight +on the initial site saturates with a scaling like 1/L. +Adding a mix of on-site frustrating magnetic fields +is not sufficient to fully scramble information. +Apply- +ing fields along the x, z-direction, local operators do not +spread throughout full operator Hilbert space as the +system can still be described as a collective S = L/2- +spin interacting with a qubit. +Similarly the steady- +state entanglement entropy remains independent of λ +and system size, while the growth rate is determined +by min[1/λ,1/hc] and does not slow when when deep +into the coherent regime λL > hc [see Supplemental for +greater details]. +V. +NUMERICAL RESULTS +After discussing the nearest neighbor mixed-Ising +chain and the star Ising model, we now seek to under- +stand the dynamical behavior in the full, ring-star sys- +tem. Here we study the dynamics of the star-local model +using exact diagonalization for systems up to L + 1 = 13 +and Krylov subspace expansion techniques for evaluating +the Schrodinger ODE for sizes up to L+1 = 22. We employ +periodic boundary conditions i = L = 0. In the previous sec- +tion we revealed that a dynamical operator growth tran- +sition occurs as a function of hc/λ on the Ising star graph +that allows for rapid operator growth from the central +qubit or coherently protected operator dynamics on the +central qubit. When the operator dynamics are coherently +oscillating on the central site, this leads to slow two-time +correlator growth. +Here we investigate the fate of this +transition, its effect on the secondary, local Ising chan- +nel for quantum scrambling, and whether fast-scrambling +is achievable in simple local-nonlocal construction. +We +know that the mixed-field Ising model is nonintegrable +and capable of scrambling information throughout the full +operator Hilbert space, but does a simple nonlocal qubit +rapidly enhance this process? +Firstly, we calculate the average adjacency level ratio +⟨¯r⟩ of the full Hamiltonian Eq. 3 [see Appendix Fig.S4]. +We solve the full spectrum exactly considering the parity +conserving and k = 0 sector of the Ising chain (L = 15) and + +7 +FIG. 2. +Autocorrelation ⟨σx +i (t)σx +i ⟩: +(a) Eq.15 vs. +time (tλ) +as a function of h,L = 40,λ = 1.0. (b) Long-time average of (a) +A(i, t0 = 200) as a function of h and system size L. (c) Curve +fit of smoothed results in (a) consisting of coherent oscillating +component with frequency ϵ0 and exponential decay parameter +ϵ1. ϵ0 shown in the top, solid color palette remains equal to 2λ +roughly until reaching the critical point λ = h/L. Vertical lines +in (c) correspond to critical point hc = Lλ = [0.5,1.0,1.5,2.0]. +confirm that for the ring-star model and nonintegrable +Ising model that all points in phase space we consider in- +deed follow GOE random matrix statistics with ⟨¯r⟩ ≈ 0.53. +Though the level spacing provides a first check of noninte- +grable dynamics, it is not sufficient in capturing strongly +coherent effects inherent to fractionalized regions of the +Hilbert space or rare states that exhibit confinement or +slow growth [83]. +A. +Entanglement Growth +The first test as to the general information scrambling +capacity of this model is to understand the entanglement +entropy dynamics. We numerically investigate entangle- +ment growth SvN(t) as a function of c-qubit coupling. To +understand the infinite temperature information dynam- +ics of the system, we work with a product state with en- +FIG. 3. +Entanglement spreading in the ring-star model: +(a) +SvN(t) as a function of λ, (b) plotted on a semilog axis in time +(tJ), and (c) λ reparameterized to λ/ +√ +L, system size L. (d) Poly- +nomial fit of the intermediate time entanglement growth behav- +ior SvN ∝ tα. System size L+1 = 22 and J,h, g = [1.0,1.05,0.45]. +ergy density equivalent to that of an infinite temperature +state ⟨Hi⟩ = 0. As the system exhibits GOE random ma- +trix statistics for all parameters of the Hamiltonian tuned +here, we expect that any such effective infinite tempera- +ture pure state will indeed obey ETH. The state we work +with here is the polarized ∣+ y⟩ spin-state, which has been +used to exemplify high-energy quench dynamics of the +mixed-field Ising chain previously [46]. +In the limit λ = 0, we have the local, nonintegrable Ising +model which exhibits ballistic entanglement growth for +times up to L/vk, where vk is the maximal dispersion as- +sociated with quasiparticle momentum k. This timescale +is upper-bounded by L/(2J) which provides the charac- +teristic timescale of nearest-neighbor interactions multi- +plied by the length of the system. The factor of two comes +from periodic boundary conditions and simply captures +the longest path between spins. +In the λ = 0 curve in +Fig.3(c) we see that for increasing system size, entangle- +ment saturation occurs on a timescale that scales linearly +with L and the maximal quasiparticle velocity remains +independent of system size. Entanglement saturates like +Llog(2) as the effective infinite temperature state ex- +plores the full Hilbert space of the system. With the struc- +ture of correlations reaching the scale of the Hilbert space +O(2L) we then expect entanglement to be similar to the +log of full operator space complexity. +As we slowly increase the nonlocal coupling to the cen- +tral c-qubit, we expect that nonintegrability is maintained +and now the shortest path between sites is next-nearest +neighbor, mediated by the central qubit. The timescale for +the c-qubit mediated entanglement spread is 2/λ, where + +a) +2.5 +hc/L = 0.00 +2.0 +hc/L = 0.10 +1.5 +hc/L = 0.25 +1.0 +hc/L =1.00 +hc/L =2.50 +0.5 +0.0 +0.5 +1.0 +5 +10 +15 +20 +25 +30 +35 +b) +0.6 += 10 +decoherent +=1 +0.4 +regime +L = 20 +?)V +coherent +L = 30 +regime +L = 40 +0.2 +L = 50 +L = 60 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +101 +h/L +c) +100 +Eo +入= 0.5 +E1 +入= 1.0 +10-1 +22 +入= 1.5 +入 = 2.0 +10-2 +10-2 +10-1 +100 +101 +h/ La) +b) +6 +8 +3 +6 +4 +入= 0.00 +4 +入 = 0.28 +入 = 0.56 +α logt +1 +2 +2 +入 = 0.84 +入 = 2.34 +0 +入 = 3.28 +0 +100 +101 +102 +c) +d) +tJ += 11 +5.0 +1.2 += 13 +入=0 += 15 +1.0 +2.5 +=17 +2.5 += 19 +α 0.8 +vN +vL +0.0 +S +0.6 +Su(t) α tα +-2.5 +6.5 +0.4 +入= +VL +-5.0 +10-1 +100 +入 +0 +5 +10 +15 +20 +tJ8 +the factor of 2 comes from the second order interaction +with the central c-qubit that allows correlations to de- +velop between sites i, j ≠ 0. In the extremely weak regime +λ << J,h, g, the c-qubit qubit can similarly be thought of +as a cavity, where the finite size Hilbert space can be dis- +regarded. +Including the c-qubit modifies the early-time entangle- +ment growth, as operators grow ballistically due to lo- +cal transport and super-ballistically with an additional +nearest-neighbor c-qubit mediated contribution. We see +in Fig.3(a) that for L + 1 = 20 as λ → h the linear coeffi- +cient of entanglement growth grows continuously and the +saturation value is modified by the addition of a single +qubit. In this enhanced rapid scrambling regime, if we +reparameterize the central qubit coupling λ → λ/ +√ +L such +that the effective spin-spin interaction is not extensive +Jeff ∼ λ2.0/L, we find that the coefficient for linear growth +is roughly independent of system size (Fig.3(b) λ = 2.5 +√ +L). +Entanglement entropy growth rate that increases with +system size is indicative of fast-scrambling behavior [46]. +For λ > h, we see a surprising entanglement transition. +The early time growth defined as t < 2 1 +λ exhibits rapid +entanglement growth but becomes sub-ballistic at inter- +mediate times 2 1 +λ < t < tsat. In Fig.3(b) the width of this +intermediate timescale grows exponentially with increas- +ing λ as exhibited by the logarithmic scale in time, while +SvN(t → ∞) remains largely unchanged. We perform a +polynomial fit of the entanglement growth in Fig.3(d) over +the region log[2] < SvN(t) < SvN;sat and find that in the +local and fast-scrambling regime, entanglement contin- +ues to grow ballistically with exponent α ∼ 1.0− 1.1. For +λ > λc, α monotonically decreases with λ as expected if +SvN ∝ 1 +λ log[t]. +In order to determine the location of the phase transi- +tion and relevant scalings, we analyze the entanglement +entropy at a fixed time t∗. We choose a time such that the +SvN(t∗,λ = 0) is roughly 1 +2 SvN(t → ∞). In Fig.4(a) we plot +SvN(t∗) as a function of two magnetic fields h depicted +by solid/dashed curves and as a function of system size +L. We extract the maxima of Fig.4(a) and perform a size +and field scaling and find that the fast-to-slow scrambling +transition value occurs like λc ∝ L(−0.52±0.05)h(0.52±0.02) +c +, +(b,c) respectively. In the high hc limit, the transition be- +comes nearly independent of system size (hc = 2.5 curve in +Fig.4(c)) and in the low field limit, is dominated by system +size: hc = 0.05 − 0.75 curves in Fig.4(c). Greater details +on identifying the entanglement transition are included +in the Supplemental Information. This contrasts the dy- +namical transition observed in the star Ising model with +a transition that occurs at λc = hc/L. +The novel entanglement transition mediated by the c- +qubit dynamics is extremely surprising in that not only +does the nonlocal coupling provide a secondary channel +for distributing entanglement, but in the strong coupling +regime it inhibits growth of even the local Ising interac- +tions with which it commutes. The problem is similarly +interesting in that it is completely disorder-free, so the +slow information growth in the system can be attributed +FIG. 4. Identifying entanglement dynamics crossover: (a) Entan- +glement entropy calculated at time t⋆ ∼ 3.5tJ plotted as a func- +tion of c-qubit coupling λ (SvN(t⋆)/t⋆ ∼ vb) for three transverse +magnetic fields and various system sizes (hc ∈ [1.05,2.5];L ∈ +[11−21,2]. For nonzero λ, entanglement growth rate varies as +a function of hc,L. Crossover point λc determined as maxima of +the curves in (a) with λc(L) and λc(hc) plotted in (b,c), respec- +tively. In the respective hc dominant and L dominant regimes, +the critical point find λc ∝ Lγhκ +c with γ = −0.52 ± 0.02(1) and +κ = 0.52±0.05(1). +to purely coherent effects. In order to shed more light on +how this central coupling serves to rapidly/slowly scram- +ble quantum information we examine the transverse and +longitudinal OTOCs (Cxx,Czz). +B. +Operator Spreading +Starting from an initial state selected from the Haar +measure, as to approximate infinite temperature state, +we examine how operators initially prepared on site i = 0 +spread under the influence of Ising interactions, where +site c represents the central qubit. When calculating the +OTOC we simplify Eq.1, as the local ˆX, ˆZ operators are +Hermitian and initially commute on all sites at t = 0. We +then explicitly evaluate +CVW = 1−⟨ ˆW(j,t) ˆV(i,0) ˆW(j,t) ˆV(i,0)⟩ +(17) +In Fig.5(a), when considering only local interactions we +see that operator weight develops according to a ballistic + +a) +h= 0.00 +h= 1.25 +4 × 100 +h= 2.00 +3 × 100 +(*7) +2 × 100 +100 +10-1 +100 +入 +b) +2 × 100 +L-0.3 +2 +L-0.5 +hz = 0.00 +100 +h, = 2.50 +101 +2 × 101 +L +c) +2 × 100 +00'6 = 7 +L = 19.00 +100 +6 × 10-1 +100 +2 × 1009 +FIG. 5. Space-time OTOC spreading: +(a-c) OTOC for longitu- +dinal spin component Czz for λ = 0,0.6,2.6, respectively. +Ini- +tial linear light cone spreads on top of c-qubit mediated op- +erator weight that spreads super-ballistically on all sites af- +ter an initial wait time that scales with min[1/hx,1/λ]. For +(λ/ +√ +L)/h ≥ 1, Czz is suppressed on all sites beyond a weak ini- +tial light cone. (d) In contrast, for λ = 2.6, transverse OTOC Cxx +rapidly fluctuates on the initial site-i and spreads to nearest +neighbors i+1 and the central site L on the order of t = 1/λ. The +light cone profile is no longer visible and Cxx rapidly oscillates +with frequency 2λ for sites ∣i − j∣ > 1. System size L +1 = 14 and +J = 1.0,h = 1.05, g = 0.45. +light cone with a bubble-like profile that saturates on all +sites following the wavefront: Czz(r,t) ∝ t2r +r!2 . With small, +nonzero λ (b) the light cone remains apparent but now +spreads on top of growing operator weight distributed by +the central qubit. +The central qubit super-ballistically +distributes operator weight to all sites on a timescale like +2 +λ. The growth timescale on the c-qubit is half that of the +bulk spins, as the σz operator must decohere on the initial +site and again on the central site and hence no longer com- +mutes with the Ising interaction. For λ << hc, 1/λ sets the +timescale for this process. In the strong coupling regime +(c), a weak light cone remains but transfers a small frac- +tion of the original operator weight. Operator spreading +becomes increasingly restricted where the lightcone pro- +file is only visible for few tJ and Czz(j,t) grows increas- +ingly slowly on sites far from the initialized operator. This +behavior is puzzling; the highly nonlocal coupling c-qubit +with increasing interaction leads to unintuitively slow lo- +cal operator spreading. +Here we can draw insight from the analytic results on +the star-Ising model (Fig.2). The L−body interaction on +the central qubit acts to coherently project operators into +σz, while rapidly aliasing orthogonal operators. Once σz +grows on the central qubit, it develops a coherent life- +time with operator weight decaying like ∼ e−1/λ. +Czz +characterizes how operators no longer commute with σz, +and in the same vain as the star-Ising model, we ex- +pect operators on the central site to be strongly projected +into the z−subspace and coherently oscillate like 2λ. As +FIG. 6. +Cxx(∣i − j∣, t) spreading in the ring-star Ising model: +Real-time evolution of Cxx(i − j, t) for varying c-qubit coupling +λ, plotted on a log-log scale. λ = 0 exhibits lightcone spreading +Cxx ∝ t2(i−j) +(i−j)!2 . In the fast-scrambling regime (λ < λc) (b,c), oper- +ator weight grows homogeneously across all sites at half the rate +as the on the c-qubit site (Cxx(c, t), blue). (b,c). (d) As λ ∼ O(1), +Cxx(c, t) becomes ∼ O(1) at t ∝ λ−1. Once Cxx(c, t) becomes +O(1), the rapid growth on spin sites ∣i − j∣ > 1 slows and coher- +ent oscillations develop. For λ = 3.0 and focusing on distant sites +(purple) and tJ ∈ [1,5], the operator growth after c-qubit satura- +tion becomes slower than that observed for λ = 1.0. System size +L+1 = 15 and J = 1.0,h = 1.05, g = 0.45. +the central c-qubit is highly/fully entangled with the re- +maining chain, the coherent projection on the central site +then similarly restricts how rapidly the many-body state +of the L−spins decoheres from the z−subspace. We ex- +pect Czz(t) to exhibit slow behavior as z−operators are +strongly driven on all sites. +We then examine Cxx(t), +which captures the rapid growth of ˆZ. +In Fig.5(d) we +see that Cxx(t) continues to grow like O(1) across all +sites, with strong, oscillatory behavior on the central +qubit (i − j = c). Cxx(j,t) on all sites similarly oscillates +at frequency 2λ but shifted by half a period compared to +Cxx(c,t). +On the initial site i, as σz +i decoheres under the trans- +verse magnetic field it no longer commutes with itself nor +the Ising interaction. This leads to a nonzero Czz(t) on +site−i and subsequent operator weight growing on near- +est neighbors and the central qubit. Once operator weight +grows on the c-qubit, the associated decoherence time of +σz +c is exponential in λ and as weak operator weight leaks +onto bulk spins, the same coherent oscillations and slow +decay is imparted on the local spin-chain. We gain further +insight by examining the individual curves that make up +the density plot in Fig.5 as Fig.6. We see the polynomial +growth associated with the light-cone spreading on top of +the exponential c-qubit mediated growth (a-c).In the slow +scrambling regime (Fig.6(d)), we find that for λ = 3.0 that +Cxx(c,t) becomes O(1) at tc = π/2λ (dashed, red) and af- +ter time tλ = π/λ the rate of change of Cxx(i − j,t) de- +creases dramatically (solid). Once σz exists on the central + +Czz(i,j, t) +Czz(i,j, t) +1.5 +a)=0 +b) = 0.6 +6 +1.0 +0.5 +2 +0 +0.0 +0 +5 +10 +C +0 +5 +10 +C +Czz(i,j, t) +Cxx(i,j, t) +1.5 +c) = 2.6 +d) α = 2.6 +6 +1.0 +4 +0.5 +2 +0 +0.0 +0 +5 +10 +0 +5 +10 +C +C +i-j +i-ja) +b +100 +i=0 +-i=2 +10-1 +^=0 +-i=4 +i-i=6 +10-2 +^ = 0.5 +i-i=8 +i-i=10 +10-3 +i-j=12 +10-4 +c) +100 +10-1 +1.5 +t +10-2 +^= 1.0 +^ = 3.0 +1.0 +0.5 +10-3 +0.0 +0 +1 +2 +3 +10-4 +4 +tJ +10-1 +100 +10-1 +100 +tJ +t)10 +FIG. 7. Czz(t) spreading in the ring-star Ising model: Longitu- +dinal OTOC Czz(t) on (a) site j = 10 and (c) c-qubit j = c with +real-time evolution plotted on a log-log scale. (b, d) Polynomial +fit at intermediate times Czz(t) ∝ αtβ. As λ increases, coef- +ficient of longitudinal operator growth becomes exponentially +suppressed (top to bottom curve on log scale) logα ∝ λ/ +√ +Lh2. +Operator growth on distant spin sites is approximately linear +β ∼ 1 in the slow scrambling regime and coefficient α = e +− 3 +2 +λ +√ +Lh2 . +(d) Czz the c-qubit site grows with exponent βc ∼ 0.5 and coeffi- +cient αc = e− λ +h2 . Operator growth on bulk spin sites is expected +to behave like Czz(j, t) = Czz(L, t)2, as the effective c-qubit me- +diated spin-spin interaction is second order process in Hλ. +qubit, orthogonal operators on the c-qubit and bulk sites +rapidly fluctuate and lead to an effective wait time, slow- +ing the amount of operator growth under the action of the +Ising−zz interaction. +A complementary perspective for operators orthogonal +to ˆZ is observed by Czz (Fig.7. Looking at the OTOC in +the slow scrambling regime, operator weight on the c- +qubit and bulk sites grows like h2t6, as captured by early- +time expansion, up to time tλ = π/2λ [see Supplemental +for greater details]. For t > tλ, operators become essen- +tially projected onto z−operator subspace due to the co- +herent lifetime of operators on the central qubit. Fitting +the OTOC to αtβ shows that operator weight decoheres +on the c-qubit with exponent βc ∼ 0.45, while on bulk sites +it is nearly linear with βj =∼ 0.85. On all sites, the coef- +ficient of growth becomes exponentially suppressed in λ +with log[α] ∝ −λ. This can be understood heuristically: +once operator weight lies on the central site, it coherently +oscillates between sites like 2λ and leads to a waiting +time during which little to no operator weight has deco- +hered from the z−subspace as it oscillates between z − I. +The amount of operator weight that decoheres in-between +these waiting periods is ∫ +tc2 +tc1 e−λ so the rate of operator +growth into this orthogonal subspace is independent of +time and leads generically to Cxx(j! = c,t) ∝ e−λt. This +slow growth for simple two-body body operators outside of +the z−subspace continues to be exponentially suppressed +for greater complexity many-body operators. The sublin- +ear growth of OTOCs into the bulk of operator Hilbert +space then generically provides log[t] entanglement en- +tropy growth, as we have observed for quantum quenches +from the ∣+ y⟩ state. +VI. +DISCUSSION +Here we have shown how the collective interactions +between many constituent spins and a central auxiliary +qubit is both able to rapidly scramble information across +all degrees of freedom and restrict state/operator growth +from exploring the full Hilbert space. From an entangle- +ment entropy picture, the central bit entangles rapidly +with its surrounding environment, the state space of the +qubit simply being q = 2. +Treating it as a 2-level sys- +tem in an effective bath, the extensive bath interactions +and on-site magnetic field frustrate the central qubit and +lead to a rapidly fluctuating spin-moment when Nλ ∼ hc. +In the operator picture this leads to rapidly decohering +operators that quickly lie in a superposition of states +{ ˆX, ˆY , ˆI, ˆZ}. +For λ >> hc, { ˆZ} operators are strongly +driven to the qubit and therefore do not fully decohere +while { ˆX, ˆY} become essentially echoed-out. +This picture is more clearly developed when we again +turn-off local interactions, J = 0. In this case only the cen- +tral bit is able to mediate entanglement throughout the +system. +As a function of λ/hc the half-chain entangle- +ment entropy grows with a velocity like 1/λ until λ hc +and surprisingly saturates with an entropy that is in- +dependent of system size. The entropy growth rate and +saturation value do not change when crossing over into +the strong-coupling regime. Though the half-system en- +tropy value is identical, we know from an operator pic- +ture that scrambling occurs slowly on the leaves of the +star graph, leaving the central qubit to scramble rapidly +while other sites scramble slowly. From both the opera- +tor and entropy picture, we know that the central bit is +highly entangled with the system and the saturation en- +tropy remains fixed regardless of coupling; therefore once +the central bit becomes highly entangled, it similarly can- +not scramble quantum information throughout. The sat- +uration in information capacity of the star-graph remains +fixed as a function of λ, so by tuning λ we tune whether +entanglement is frozen in a highly entangled auxiliary bit +or distributed globally. +Understanding the information +dynamics of the star-graph then informs the surprising +behavior observed in the fully interacting star-local Ising +system. Once the central qubit becomes highly entangled +with it’s environment, it similarly acts to project operators +into the z−subspace, inhibiting the local, nonintegrable +Ising interactions from fully scrambling the system and +reaching full thermalization quickly. +As the commutivity graph that translates operators +from site−i to site−j requires operation of the Ising−zz +interactions, which similarly commutes with the central +spin coupling, local operator growth spreading is inhib- +ited on the same decoherence timescale. The local channel +is effectively turned off by the slow operator decoherence. +If the local interactions were orthogonal to the central + +a) +100 +(q +Czz(L, t) α αth +t +0 +0.4 +B +α入 +0.2 +-4 +10-2 +10-1 +100 +101 +0 +2 +4 +6 +tI +入 +c) +d) +100 +: 10, +入 = 1.28 +0 +入= 2.57 +[0]60] +1.0 +10-1 +入= 3.85 +B +入= 5.13 +10-2 +0.5 +5 +ZZ. +10-3 +0.0 +C +10-1 +100 +101 +0 +2 +4 +6 +t/ +入11 +FIG. 8. Information spreading in the transverse ring-star model +(λ∑L−1 +i +σx +i σx +c, J = 1.0): +(a) Early-time entanglement entropy +growth crosses over from ballistic to sub-ballistic with increas- +ing λ, plotted on a semi-log scale. Entropy depicts a prether- +mal phase with entanglement that saturates around t, then +exhibits characteristic MBL, logarithmic growth ∝ log(t) out +to late times tJ = 200. +Early-time entropy growth increases +smoothly with λ until reaching a discontinuity at λ ∼ 1. (c, d, +e, f) OTOC for transverse spin component for λ = 0,0.84,2.0,3.0, +respectively. In contrast to the parallel ring-star Ising model, +the light cone becomes increasingly visible as λ increases. (c) +In the nonintegrable, purely local case, operator scrambling +saturates following the light cone. As c-qubit coupling grows, +operators orthogonal to σx rapidly fluctuate, so that following +the light cone, σx operator weight dominates each site. +(e) +The light cone boundary eventually becomes the only region +where operators are orthogonal to the c-qubit coupling and prop- +agate under local Ising interactions. +System size L = 13 and +J = 1.0,h = 1.05, g = 0.45. +coupling, we expect different operator spreading results. +Here we take a simple extension of our model and change +the form of the auxiliary coupling to be Ising−xx which we +term a compass, ring-star Ising model. Now the the auxil- +iary coupling projects the system into ˆX subspace, which +no longer commutes with the local −zz interactions. This +should allow many-body operators to propagate through- +out but inhibit the growth of operator strings contain- +ing ˆY , ˆZ. Translating the operator picture to the entropy +picture, we are allowing higher order many-body opera- +tors to develop through local channels, so the intermedi- +ate time saturation value should be significantly larger +and scale extensively with system size, the logarithmi- +cally slow growth of orthogonal operator strings should +then extend to long times as operators can decohere to +larger regions of the Hilbert space. +In Fig. +8 we observe exactly this behavior. +Noting +that the magnitude of the orthogonal field on the c-qubit +changes from h → g = 0.45, the inhibited operator growth +goes as λx/g2. The entanglement entropy (a) following a +quench from ∣+ y⟩ transitions from local spreading, fast- +spreading, to confined for λx > 0.5. +For times tJ < 10, +the system exhibits ballistic operator growth regardless +of coupling but reaches an intermediate saturation value +SvN ≈ 3 with slow growth extending out to late times. The +OTOC agrees identically, where we have the same ballis- +tic spreading lightcone with fully scrambled operators fol- +lowing the wavefront (b). With increasing coupling (c) we +observe a small amount of operator weight that spreads +super-ballistically across all sites, a well defined wave- +front, and a suppression in the OTOC behind the wave- +front. This becomes more dramatic deeper into the con- +fined regime (d,e), where the wavefront becomes the only +region where operators orthogonal to ˆX may be found. As +the local Ising interaction is orthogonal to the operator +subspace in which the c-qubit protects, the wavefront may +freely propagate but operators behind the wavefront are +aliased to predominantly span strings of ˆX and ˆI opera- +tors. +VII. +CONCLUSION +We have interrogated the quantum entanglement and +operator dynamics in a central spin-bath design. The local +Ising chain acts as a structured, thermalizing bath that +when coupled to a nonlocal central qubit, is able to rapidly +scramble quantum information across the system up un- +til the information capacity of the c-qubit is quenched. +When the central bit is rapidly quenched in the strong +coupling regime, it leads to operator confinement in sec- +tors of the Hilbert space spanned by ˆI and operators par- +allel to the central qubit coupling. Entanglement entropy +grows slowly out to late times as observed in effective in- +finite temperature states. +This model admits unique limits; where, for weak cou- +pling it exhibits super-ballistic OTOC spreading and fast- +scrambling as supported by previous works examining +a similar star-graph structure, and a confined quantum +Zeno-like limit in which the complexity of operator growth +is inhibited by extensive, coherent interaction with the +two level central mode. This confined regime mirrors a +larger class of models which admit prethermal or local- +ization physics: disorder driven many-body localization, +projective or weak quantum measurement, floquet-driven +prethermalization. Our star-local work is similar in that +the c-qubit drives operator fluctuations on the leaves on +a timescale like 1/λ once λ is greater than noncommut- +ting fields on the central site. In contrast, the qubit re- +tains its infinite-range information sharing capacity com- +pared to the external drive scenario in floquet physics. In +the measurement theory sense, once the central qubit be- +comes nearly maximally entangled with the Ising spin +chain bath, its as if the bath projects the central spin + +a) +10 +4 +8 +3 +6 +2 +4 +2 +1 +0 +100 +101 +102 +tJ +Cxx(i, j) +Cxx(i,j) +15 +b) +^=0 +c) = 0.6 +10 +1.5 +5 +1.0 +0 +15 +d) = 1.7 +e) +^ = 3.5 +- +0.5 +- +10 +- +- +5 +- +0.0 +- +- +- +0 +0 +5 +10 +C +0 +5 +10 +c +i-j +i-j12 +which in turn freezes the information sharing dynamics +with which the bath is entangled. And finally in the MBL +case, instead of local integrals of motion that are nearly +integrable with exponentially small overlap, here we have +global operators orthogonal to the auxiliary coupling that +have a logarthimically slow thermalization timescale. +This work provides a novel mechanism for exploring a +host of quantum information dynamics. Here we outline +the existence of a dynamical transition that occurs under +static Hamiltonian dynamics, and in future work it would +be fruitful to investigate the nature of the transition and +how similar physics can be observed in RUCs. Tracing out +the central qubit to produce non-Hermitian physics may +similarly illuminate how this model relates to strong peri- +odic driving prethermalization. As the inverse problem, it +would be interesting if exotic non-Hermitian physics has +a hidden unitary quantum system analog, where multiple +ancillary qubit degrees of freedom produce the observed +non-Hermitian dynamics. Future work should also seek +to understand how this transition persists with larger c- +qudit towards an infinite bosonic fields or multiple dissi- +pative modes. 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[[H,σx +i ]m,σx +j] +[A,B]m = [A,[A,B]m−1];[A,B]0 = B +[H,σx +i ]1 = it[σz +i ,σx +i ] = itλσz +0σy +i +[H,σx +i ]2 = (it2) +2 +[H,λσz +0σy +i ] = −t2 +2 (−λ2σx +i + hλσy +0σy +i ). +[H,σx +i ]3 = (it3) +6 +[H,...] = −it3 +6 (−λ3σz +0σy +i + hλ2(∑ +j≠i +σx +0σz +jσy +i +σx +0σx +i )− h2λσz +0σy +i ) +In the limit h/λ = 0,λ = 1 we have: +[σx +j(t),σx +i ] = [∑ +m +(it)2m +2m! σx +i + (it)2m+1 +(2m+1)!σz +0σy +i ,σx +j] += [cos(2t)σx +i − isin(2t)σz +0σy +i ,σx +j]. +Then the two-time correlator behaves as: +⟨σx +i (t)∣σx +i ⟩ = cos2t. +(19) +And the OTOC goes as: +Cxx(i,1,t) = 2sin2(2t), +(20) +strictly for i ∈ c,1. Under the operation of the central Ising interaction, the operator weight is completely preserved to +live in the projective space as we have seen before. When adding a non-commuting term that rotates states away from +this space, how does this then modify growth on the remaining sites? Looking back, we see that +⟨σx +j(t)∣σx +i ⟩ ∼ O(1) ∝ t3 +6 h2 +(21) +Cxx(j, i,t) ∝ h2t6. +(22) +As exhibited in Fig.S1, we see that for fixed λ = 1, that h2t6 provides an excellent approximation to the early-time growth +behavior of OTOC on next nearest neighbors to the initial site. This rapid growth on the central site and corresponding +neighbors at early-times in essence captures the extreme early-time entanglement growth that is observed to grow +steadily with growing h/λ. The interesting phenomena occurs when t ∼ π +2λ, where in Fig.S1(a) the operator space q is +completely preserved, allowing C(0, i,t) to decay to zero. On this same timescale we see in (b, c) and most clearly in +(d) that this leads to an effective wait time that is proportional λ and similarly corresponds to the amount of operator +weight outside of the space q on the c-qubit. Also, any weight operator weight transferred to edge spins then undergoes +similar oscillations seen in (a) during this wait time. In the weakly decoherent regime h << λ, the wait time leads to +essentially linear OTOC growth with oscillations that similarly grow in time. + +16 +FIG. S1. OTOC Cxx(i, j, t) for (a) h = 0, (b) hc > 0, (c, d) h > 0 with L + 1 = 13, λ = 1. (a) OTOC is only nonzero on the edge and +central c-qubit as coherent evolution under the star-coupling does not allow operators to propagate beyond the c-qubit. (b,c) early- +time growth behavior is accurately captured by BCH expansion of the two-time correlator. (d) At intermediate times, t, as operator +weight transitions back to the initial site, there is then essentially a waiting time that occurs before more operator weight may grow +on sites j ≠ [i,0]. This leads to a polynomial growth in time of the two-time correlators. +II. +ENTANGLEMENT SPREADING ON THE STAR GRAPH +Here we examine the corresponding entanglement entropy growth on the Ising star graph. Starting from the same +polarized product state used in the main results (∣ + y⟩) we plot the half-system entanglement entropy following a +quantum quench. We treat λ as the variable and set h,hc = 1.05 and g, gc = 0.45. The entanglement entropy saturates +to a length independent value Fig.S2(a), constant for system sizes L = 11 − 19. Though the mixture of onsite fields +allows local operators to explore the operator space of single sites, the collective magnetic field and collective c-qubit +interaction still limit the system to behave semiclassically. In Fig.S2(b) we note that the entanglement entropy growth +rate and saturation value become independent for λ > 1. This is an interesting finding as we know in the large λ >> h +limit that operators simply scramble between the initial site and the cqubit. The corresponding entanglement entropy +between halves is dominated by the system-qubit entanglement as entanglement generated between spins grows slowly. +Once the system rapidly entangles with the cqubit, the entanglement entropy of the system remains fixed but becomes +uniformly distributed between all sites. In the OTOC picture operators are scrambled with the cqubit and then leak +into the remaining sites. +III. +OTOC GROWTH VS. SYSTEM SIZE SCALING +In the main text we examined the longitudinal OTOC and found that it goes like Czz(t) ≈ e−λt. In Fig.S3 we perform +the same analysis and vary the system size. We see that close to the transition λ ≤ 1 that the coefficient on OTOC growth +is roughly independent of system size. For increasing λ we see in Fig.S3(top) that there is weak decrease in log(α) with +system size. Looking at the 1/α to highlight the slight difference in system size, we see that into the localized operator +phase λ > 3−4 that larger system sizes exhibit increasingly slow operator growth. + +a) +b) +2.0 +10-1 +α h2t6 +1.5 +t +10-3 +1.0 +hc/^ = 0.26 +li-j=2 +10-5 +hc/^ = 0.53 +li-jlo +0.5 +hc/△ = 0.79 +li-jl=c +10-7 +hc/△ = 3.68 +sin2(2t) +hc/^ = 5.00 +0.0 +10-9 +5 +10 +10-2 +10-1 +100 +101 +t +t +c) +d) +10-1 +10-1 +t +h;^ = 0.26 +hj^ = 0.38 +hj^ = 0.13 +10-3 +10-3 +h;^ = 0.51 +h;^ = 0.26 +h;^ = 1.79 +h;^ = 0.38 +h;^ = 2.44 +cQubit:hi/^ = 0.13 +10-5 +10-5 +10-1 +100 +101 +10-1 +100 +101 +t +t17 +FIG. S2. Information spreading star model (J = 0): Real-time evolution of the 1/2−chain entanglement entropy initialized in ∣+ y⟩ (a, +b), and OTOC following quench from a Haar random initial state. Information spreading in the star-Ising model (J = 0): Real-time +evolution of the the 1/2−chain entanglement entropy initialized in ∣+ y⟩ (a,b) and Czz(t, i− j) with initial state chosen randomly from +the Haar distribution (c,d). (a) Entanglement entropy plotted as a function of length for λ = 1.0. Growth is independent of system +size for fixed λ and saturates to a value independent of system size. (b) Entanglement growth rate and steady state value for λ > h +saturate and become independent of λ. (c, d) OTOC Czz for λ/ +√ +L = {0,0.78}, respectively. (c) Initial operator simply fluctuates +under mixed-fields and does not propagate. (d) OTOC grows on the c-qubit at a rate that grows with λ after an initial wait time that +scales with min[1/h,1/λ]. Operator weight then spreads to the remaining leaves after a time 2 min[1/h,1/λ] corresponding to two +processes of operator transfer that occur on the c-qubit. +IV. +SPECTRAL STATISTICS OF THE STAR-ISING MODEL +We calculate the spectral statistics for the central qubit Ising model studied in the main text. These statistics provide +insight into the random nature like behavior of the density matrix or whether there are conservation laws present. +Anything beyond random statistics manifests as prethermalization, MBL, or integrable behavior when a system is +quench from a highly energetic state. We calculate the spectral statistics using 100 states surrounding the middle of +the energy spectrum. The ratio is between two levels ri is calculated as +ri = min[Ei − Ei−1 +Ei + Ei−1 +, Ei+1 − Ei +Ei + Ei+1 +] +(23) +We then take the average spacing between all levels. We work in the k = 0 sector as well as the paritiy conserving sector, +as the translational invariance leads to degeneracies between k−sectors when only one particular sector is meaningful +under dynamics. We take the tensor product of this Ising spin-chain Hilbert space with the central qubit. In Fig.S4 +we observe that the only point in phase space that does not exhibit GOE random matrix statistics ⟨r⟩ ≈ 0.53 as for +λ = 0, g = 0, where we have the integrable TFIM. Here the system exhibits sub-Poissonian statistics. + +2.0 +a) +b) +1.5 +1.5 +入=0.00 +入=0.37 +1.0 +1.0 +L = 11 +入=0.75 +L = 13 +入=1.40 +0.5 +L = 15 +0.5 +入=2.34 +L = 17 +入=3.28 +L = 19 +入= 4.59 +0.0 +0.0 +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +t +t +Czz(i, j) +Czz(i, j) +7.5 + c) = 0 +7.5 +1.5 +d) ^ =0.78 +5.0 +5.0 +- 1.0 +2.5 +2.5 +0.5 +3 +0.0 +0.0 +0.0 +0 +5 +10 +15 +0 +5 +10 +15 +i-j +i-i18 +FIG. S3. +Finite size scaling of the coefficient of linear growth αtβ (see Main text Fig.7) in Czz(∣i − j∣ = 6, t) for system sizes +L = [8,10,12,14]. (Top) log[α] decreases linearly with λ as exponentially less operator weight decoheres on all sites and spreads +throughout the system. (Bottom) Plot of 1/α for better resolution of the system size dependence. We observe a small decrease in α +with increasing system size, corresponding to the weak coefficient in λc ∼ +√ +L. +FIG. S4. +Average adjacency level spacing of energy eigenvalues r(Ei,Ei+1) for the ring-star Ising Hamiltonian for L +1 = 16 with +periodic boundary conditions for fixed momentum and Z-reflection symmetry blocks of the Hamiltonian. (a) Spacing as a function of +λ, g for fixed h, (b) h, lambda, g = 0, and (c) h, g,λ = 0. We find strong evidence of the integrable case of h ≠ 0, g = 0,λ = 0 in (b,c). + +0.0 +1.0 +log[α] +-2.5 +B +L=8 +0.5 +L=10 +L=12 +-5.0 +L=14 +0.0 +0 +2 +4 +L=8 +L=10 +L=12 +200 +L=14 +1/α +0 +0 +2 +42.0 +0.550 +h =1.05 +1.5 +0.525 +1.0 +0.500 +0.5 +0.475 +9 +0.0 +0.450 +0.5 +0.425 +1.0 +0.400 +1.5 +0.375 +2.0 +0.350 +2.0 +2.00 +1.5 +1.50 +1.2 +1.0 +1.00 +0.75 +0.5 +0.50 +0.25 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +g \ No newline at end of file diff --git a/mdA0T4oBgHgl3EQfJf-W/content/tmp_files/load_file.txt b/mdA0T4oBgHgl3EQfJf-W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28dc7baa4ce0cd588bc63b7362f5656a63912dae --- /dev/null +++ b/mdA0T4oBgHgl3EQfJf-W/content/tmp_files/load_file.txt @@ -0,0 +1,977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf,len=976 +page_content='Fast-Scrambling and Operator Confinement Using an Auxiliary Qubit Joseph C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Szabo1 and Nandini Trivedi1 1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA (Dated: January 6, 2023) We introduce a minimal model for realizing a fast-to-slow scrambling transition mediated by an auxil- iary central qubit (c-qubit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The c-qubit is coupled to a spin-1/2 Ising model with local Ising interactions and tunable c-qubit-spin coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Each spin becomes next-nearest neighbor to all others through the c-qubit, which mediates effective all-to-all interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the interaction with the c-spin increases, we find a surprising transition from super-ballistic scrambling and information growth to continuously restricted sub-ballistic entanglement and operator growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This slow growth occurs on intermediate timescales that extend exponentially with increasing coupling and system size, indicative of logarith- mic entanglement growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We find that in the slow-scrambling regime, the c-qubit Ising interaction allows commuting operators to grow support on all sites rapidly, while operators orthogonal to the in- teraction become echoed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This projects local operators to lie in a restricted subspace and prevents extensive operator entanglement growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We provide exact dynamics of small systems working with non-equilibrium, effective infinite temperature states, and additionally contribute analytic early-time expansions that support the observed rapid scrambling to quantum Zeno-like crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Tracing out the central qubit provides a unique translation from the full, closed unitary dynamics to a simple open system construction consisting of a typical spin-chain with hidden qubit degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' INTRODUCTION Operator scrambling and entanglement entropy spreading are unambiguous discriminators of purely quantum mechanical nonequilibrium dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' fasci- nating properties underlying quantum thermalization, dynamical phase transitions, and topological order [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the Heisenberg picture, quantum operator scrambling details how initially localized operators propagate over spatiotemporal degrees of freedom due to noncommu- tative many-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the complementary Schrodinger picture, entanglement entropy captures growing information complexity: from initially classical states to those with nontrivial entangled structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Quantum information dynamics bridge both theoretical and experimental communities as primary measures for quantum complexity and expressivity [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' These con- cepts combined with quantum simulation/circuit devices have coalesced into many enriching, recent experiments [10, 13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The accelerating pace of results and drive to continually advance the corresponding theory extend these successes to further research at the intersection of quantum chaos, thermalization, and computability, extending from qubits to black-holes and quantum gravity [16–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The primary research thrusts among the quantum in- formation dynamics community fall along the lines of un- covering the minimal mechanisms behind myriad infor- mation dynamical phases and understanding the fate of quantum to classical thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In studying gener- alized quantum information dynamics, there are typically two disparate perspectives: closed and open quantum sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Closed quantum systems exhibit rich scrambling physics ranging from frozen [25] to fast [18] dynamics, with the typical questions relating to how well-preserved is such physics under driving and dissipation contribu- tions from an external environment [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The envi- ronment is oftentimes reduced to a memoryless, effective Markovian description, which hinges on assumptions in- cluding weak-coupling and a separation in the timescales associated with system and environment [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though solving for the exact dynamics for a full complex environ- ment is beyond the capabilities of current devices, taking into account the structure and interaction with the en- vironment poses interesting research questions: what is the fate of entangled information within the system, how can a structured environments drive effective interactions and information dynamics, and how does the environment serve as a probe in an information theoretic/entropic ca- pacity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' A simple avenue for exploring the impact of a struc- tured quantum environment is by considering compos- ite, unitary models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Focusing on a particular subsys- tem of a full closed quantum system and tracing over the additional degrees of freedom (DoFs), then termed the environment, captures the subsystem’s effective dy- namics/interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This is a popular focus of study as it provides an open quantum system perspective for the subsystem and allows full consideration of the envi- ronment’s structure and interaction topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This con- struction allows us to specifically evaluate how variable structured environments impact the overall quantum in- formation dynamical phase as expressed by the under- lying subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Previous work considered the valid- ity of Markovian assumptions provided variable system- environment coupling, and here we are looking to add an information scrambling perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Considering a system-environment construction in this manner directly applies to those codes/models investigating the infor- mation physics of auxiliary bits or those systems with inherent auxiliary DoFs such as mechanical or optical modes [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Studying composite systems in this manner provides an interesting framework extending current quantum infor- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='02091v1 [quant-ph] 5 Jan 2023 2 mation dynamic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Significant recent results focus on the range of interactions, the speed and nature of infor- mation propagation, the role of inherent symmetries, and the effect of emergent symmetries in the cases of many- body localization, Floquet periodic driving, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The same phenomena can be similarly cast as an environment me- diated effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The tunability of the system-environment network topology and the inherent environment structure and interaction symmetries then allows for systematic in- vestigation into the particular contribution on the overall dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In this paper we explore these aforementioned ques- tions by considering the simplest environment extension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' an auxiliary central qubit (c-qubit) coupled to 1-d chain system of interacting spin-1/2 (qubit) objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Tracing out the c-qubit and considering the dynamics of the 1-d sys- tem provides a translation from a full, unitary model, to an effective long-range, non-Hermitian spin chain with a hidden qubit degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This provides a single long-range quantum channel for transmitting informa- tion but at the same time imposes a shared two-fold DoF across all spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though only a small addition to well- understood nearest-neighbor qubit model, we observe an abundance of exciting repercussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The system-environment coupling expresses various regimes: in the weakly coupled regime, the c-qubit pro- vides little feedback and acts as a free channel for infor- mation to pass unimpeded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' while when strongly coupled to the low dimensional qubit environment, the c-qubit acts as a strong drive and imposes an effective hidden symmetry on the underlying spin system and generates disorder-free localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We liken the physics observed here to that seen in systems undergoing quantum mea- surement or strong Floquet driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Considering the c- qubit as a hidden degree of freedom provides unique in- sight into how the quantum scrambling dynamics of the underlying spin chain maps to an extended unitary model provided one additional qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Central qubit or a higher dimensional qudit/cavity/register are popular theoretical and experimental tools for providing non-invasive many- body measurements [36–39], evaluating Hermitian and non-Hermitian response [38, 40], generating effective in- teractions [41], and studying the fundamentals of deco- herence and information transport [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we particularly focus on the dual effect of a tun- able central qubit by investigating the operator and en- tanglement growth in a nonintegrable, ring-star Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The model includes homogeneous spin-spin inter- actions in a mixed magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We find an extremely surprising fast-to-slow quantum information spreading transition that occurs due to the nonlocal and coher- ent nature of the c-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We summarize this result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='1(d), where in the weakly coupled regime (regime I), the central qubit mediates rapid scrambling with a timescale that decreases with system size (green, upper curve in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the strong coupling regime (regime II), the scrambling time increases exponentially with system size (red, lower curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The mechanism behind this transition is the interplay between the noncommuting, extensive c- qubit Ising interactions and transverse field hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The met- ric here presented for scrambling is eSvN(t)/2L+1, which provides a measure of the span of the quantum wavefunc- tion throughout the full Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As we detail in what follows, in the strong coupling regime the central qubit rapidly saturates its entanglement with the sur- rounding spin-chain environment and becomes strongly driven by this extensive interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This strong inter- action rapidly aliases operators orthogonal to the cen- tral qubit Ising interaction on the central qubit and even more surprisingly within the spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The long life- time of states and operators that commute with the cen- tral Ising interaction leads to slow multi-particle entan- glement growth and operator complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Our work agrees with previous research that finds an extensively scaling, nonlocal interaction leads to rapid scrambling, where the rate increases with system size [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' At the same time we find a surprising limit where the purely quantum nature of the c-qubit imparts a coherent effect that slows operator decoher- ence/entanglement and subsequent spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This phe- nomena mirrors what is seen in strongly driven Floquet systems, where periodic driving can impart an effective symmetry in all eigenstates and leads to prethermaliza- tion and correspondingly slow entanglement spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We liken the projective action of the central qubit in this time independent Hamiltonian to the quantum Zeno ef- fect where quantum measurement leads to a ballistic to sub-ballistic entanglement growth transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here the c-qubit imparts a highly nonlocal effect on operator pro- jection in contrast to a local purification/disorder network that redefines local spreading dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We illuminate this c-qubit physics by examining the growth of out-of- time-order correlators (OTOCs) and the von Neumann entanglement entropy for sufficiently high-energy initial product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' SCRAMBLING METRICS Many recent works have made significant progress on establishing the family of scrambling dynamics that oc- cur in various lattice models and geometric random cir- cuit designs, as characterized by the growth of OTOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The OTOC generically given as CVW = ⟨[ ˆW(j,t), ˆV(i,0)]†[ ˆW(j,t), ˆV(i,0)]⟩, (1) examines how an initially prepared unitary operator ˆV on site−i commutes after Heisenberg evolution with operator ˆW after time t (here assumed a local operator on site−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The operator spreading picture is unique to quantum sys- tems, where in working with pure states, no information is truly lost but transforms into many-body degrees of freedom that become increasingly inaccessible provided control over an initial localized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Studying OTOCs and the timescales associated with scrambling dynamics provides a conjugate perspective as 3 compared to entanglement entropy measures and transi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Where OTOCs and specifically infinite temperature OTOCs examine the light cone established by Heisen- berg evolution and depend more strongly on the com- mutivity graph, entanglement entropy examines how the wavefunction over a bipartition of Hilbert space spreads throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we specifically focus on the von Neu- mann bipartite entanglement entropy given as SvN = −∑ k λk log(λk), (2) where λk are eigenvalues of the reduced density matrix ρA∣B (RDM) obtained by integrating out subsystem A or B with corresponding Hilbert spaces HA,HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' OTOCs and entanglement identify similar physics and previous collo- quial conceptions of the two established quantum scram- bling as a unifying framework behind them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' where, scrambling represents the time for an OTOC between ar- bitrary sites to become O(1) and entanglement entropy to become O(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the case of OTOCs, this limit is not rigorous enough and only provides a best-case scenario for operators traversing the system rather than provid- ing a timescale for nontrivial operator strings to span the system [47] (extensive operator entanglement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Rig- orous relationships between OTOCs and Renyi−2 entropy have been established [48–50] and special cases have been studied in particular optical Hamiltonians [24, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Unitary scrambling physics generally falls into two cat- egories: systems that thermalize rapidly and those that fail to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The former are known as fast-scramblers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ergodic systems typically with variable all-to-all range interactions that spread quantum information through- out the full Hilbert space in tsc ∼ log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Models such as Sachdev-Ye-Kitaev (SYK) and non-integrable infinite range Ising and XY models are known to exhibit fast- scrambling physics [45, 46, 53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Systems that fail to thermalize with tsc ∼ eL are slow-scramblers, non- ETH obeying systems and candidates for highly coherent quantum information storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' There are multiple vec- tors through which non-ETH physics occurs: integrability, disorder-free localization [56, 57], quantum scarring [58– 61], and/or higher order exact or proximate conservation laws [62–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The origins of much of this work stems from the dramatic quantum correlations observed in quantum simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Typical models accessible to simulation and experi- ment are semiclassical in nature with infinite or long- range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' These models exhibit the characteris- tic unitary scrambling features we detailed previously, yet continue to enrich the discussion with new puzzling re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Lipkin-Meshkov-Glick (LMG) model or the Dicke model exhibit strict conservation of the total spin moment ˆS2 such that the effective number of degrees of freedom is O(L), compared to O(2L) [24, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' These systems have been observed to spread information rapidly, while the complexity saturation value remains low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This is in stark contrast to fast scrambling models like SYK where infinite-range connectivity allows for rapid and complex quantum information scrambling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' One immediately puz- zling question is: how do long-range interactions, tend- ing toward generating semiclassical behavior, compete with local chaotic quantum dynamics to allow a fast-to- slow scrambling transition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' A complete understanding of quantum information physics not only hinges on under- standing the unitary dynamic contribution to experimen- tal results, but similarly understanding non-Hermitian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' These are inherent to quantum simulation platforms and represent the an exotic next frontier for theory and experiment as we move toward fully expres- sive quantum circuits and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' More generalized quantum dynamical behavior has been explored in recent studies consisting of non- Hermitian operations: composite system-environment undergoing quantum measurement [67–72], light-matter interactions [24], dissipative and driven systems [73–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The most extraordinary findings reveal that these non- unitary dynamics generate effective inter-system interac- tions and impose effective static long-lived symmetries: Floquet periodically driven systems are akin to various unitary scrambling phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' MODEL We consider the Hamiltonian for the c-qubit or ring-star Ising model: H = L−1 ∑ i=0 λσz i σz c − Jσz i σz i+1 + hσx i + gσz i + hcσx c + gcσz c, (3) where λ represents the uniform spin-c-qubit interaction, and J,h, g represent the much-studied nonintegrable mixed-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we take J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,h = hc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05, g = gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45 unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This is a well characterized nonintegrable point for polarized state evo- lution and operator dynamics allowing us to benchmark the impact of the c-qubit [46, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Previous works examined operator dynamics and en- tanglement growth in random unitary circuits (RUCs) applied in a star-graph network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' RUCs on the star- graph generated OTOC dynamics that saturated at t ∝ log(L) [47, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In contrast, an interesting section of [78], examines dynamics of time-independent, star- Ising Hamiltonians which consist of Ising interactions and a non-commuting field hc strictly on the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In contrast to fast scrambling RUC networks, the time-independent c-qubit Hamiltonian dynamics gener- ate novel operator confinement on the c-qubit with a co- herent lifetime τ ∝ h2 λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This defines the time for sig- nificant operator weight to decohere into operator sub- spaces that no longer commute with the Ising interaction, subsequently allowing slow scrambling to all other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Provided this contrast between the fast-scrambling RUCs and the slow Hamiltonian dynamics, we ask: can the c- qubit coupling mediate rapid scrambling in a truly time- independent nonintegrable model and, if so, when/how does this picture break down?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (a) Illustration of the ring-star model achieved in (b) optically dressed trapped-ion experiments and (c) through nonlocal unitary gates in a circuit realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) Central qubit mediated dynamics: local interaction (black), c-qubit induced fast scrambling (green), and c-qubit inhibited scrambling (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Information scrambling approximated as wavefunction spread in Hilbert space or ∼ eSvN(t) to show dramatic prethermal-like approach to full thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (e) Qualitative depiction of (d) or how a well defined initial state grows to fill state/operator Hilbert space with a varying rate depending on dynamical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (f) von Neumann entanglement entropy SvN(t) following a quench from polarized state ∣+ y⟩ for the ring-star Ising model under varying λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ANALYTIC INSIGHTS Before discussing the exact numerics on the full nonin- tegrable ring-star Ising model, we first motivate the rea- sons underlying a dynamical transition from fast-to-slow scrambling and the corresponding mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We first summarize typical operator growth in local spin models and then review the Ising star graph dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Local Ising Model In the completely local Ising chain limit λ = 0, we can gain a heuristic understanding of the characteristic light cone spreading by performing an early-time expansion of the dynamical correlator ⟨σz i (t)σz j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Using the early- time expansion of the Heisenberg evolution we can write ei ˆHtσz i e−i ˆHt using the Baker–Campbell–Hausdorff (BCH) expansion e−i ˆHtσz i e−i ˆHt = ∞ ∑ m (it)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' [H,Sz i ]m [A,B]m = [A,[A,B]m−1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='[A,B]0 = B [H,Sz i ]1 = it[hSx i ,Sz i ] = −ithS y i [H,Sz i ]2 = (it2) 2 [H,S y i ] = −t2 2 (−λSz 0Sx i + hSz i − gSx i − J(Sz i−1Sx i + Sz i+1Sx i )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (4) Before continuing this expansion to higher orders we ob- serve a trend in operator growth and can approximate the weight as [Sz i (t),Sz j] ∝ t∣i−j∣ ∣i − j∣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='O(1) (5) Naively, we then expect the OTOC, or squared commuta- tor, to generically follow Czz(i − j,t) ∼ t2∣i−j∣ (∣i − j∣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' )2 O(1) (6) This approximation is valid for times t ∼ ∣i − j∣/vB, where vB is the characteristic butterfly velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Before this time, operator growth is suppressed with an exponent that grows with distance, and allows for optimized simu- lations using matrix product operator dynamics (MPO) by just keeping track of the operator wavefront [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though many extensive theoretical works provide rigorous esti- mates on the form of operator growth for translationally invariant models, here we simply examine the asymptotic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This exponentially growing operator weight with exponent like r = ∣i− j∣ leads to the development of a linear light cone with butterfly velocity exactly calculated as 2eJ (e being Euler’s number) for h/J > 1 [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Following this light cone in the nonintegrable Ising model, any localized operator spans the full operator Hilbert space { ˆX, ˆY , ˆZ, ˆI} within region r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' An interesting extension of the OTOC is the integrated OTOC (iOTOC), which is the integral over r and the bi- partite OTOC [50, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Both provide a novel characteriza- tion of the operator complexity within a region r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This also provides an insightful way to then understand the cor- responding entanglement dynamics for pure state wave- functions with energy density kBT as in accordance with ETH the states and operators should exhibit ergodic equi- libration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the nonintegrable Ising model, an opera- tor saturates the reduced Hilbert space 4∣i−j∣ after a time e Hilbert Space a) c) d) A Initial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 Ancilla free 1 state: to Regime I Lsc 8 VL Regime II B C-gubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='enhance Scrambling ( time Localqubit-qubit T S b) f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 interaction 6 C-qubit α logt protected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 100 101 102 position 100 101 102 time time (tJ)5 tsat = ∣i−j∣ vB , so simply integrating over Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 provides an exponentially growing iOTOC for all Cvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This opera- tor growth then guarantees ballistically growing entan- glement entropy SvN ∼ ∑v,w log[iOTOC(v,w)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Star-Ising model In the star limit, we set J = 0 and tune external fields h, g to allow more expansive operator evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' First working with only λ ≠ 0, we then add complexity to ar- rive at the full, nonintegrable ring-star Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For ∣λ∣ > 0, operator growth between leaves of the graph is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The dynamics are exactly solvable as [σz i , ˆH] = 0 for all sites i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Similarly, the commutivity graph represent- ing the Hamiltonian and how local operators propagate is completely disconnected with vertices σz i ,σz L for all i ∈ L and no bonds in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operators evolve simply under the central Ising interaction and the two-time correlator behaves as: ⟨Sx i (t)∣Sx i ⟩ = cos2λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (7) And, again, exactly in this case, the OTOC goes as: Cxx(i, i,t) = 2sin2(2λt), (8) Though this limit admits a trivial result, it allows us to understand the key coherent property of the c-qubit qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We see that for all times, operators orthogonal to the Ising interaction λ, initially prepared on the leaves, propagate to the c-qubit after a time t = π/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Because all terms in the Hamiltonian commute with this interaction [σz i σz 0,σz jσz 0] = 0, the action of operator development from leaves to c-qubit does not decohere into nontrivial, orthog- onal operators { ˆX, ˆY}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Due to this coherence, operators oscillate between the initial node and the c-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' If the operator was initially prepared on the c-qubit, it fluctu- ates onto all nodes, but instead of becoming a many-body operator, it is a collective superposition of L unique two- body operators ∑j σx,y 0 σz j ∈ H(4⊗L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This superposition, though nonlocal on a timescale τ = O(1), has minimal operator entanglement and the complexity of operators strings is fixed to be a maximum of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In this scenario, similarly, minimal entanglement entropy develops as the number of unique operator states is of L in a Hilbert space of 4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Quench experiments on such systems with nonzero homogeneous magnetic fields will only permit half-system entanglement to grow like logL since we have effectively a large semiclassical spin system coupled to a single two- level qubit [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The simplest extension we can make is to now intro- duce a transverse field on the c-qubit spin, hc ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This was similarly analyzed in a disordered star-Ising system for small values of h/λL [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For ∣hc∣ > 0, the model re- tains the same integrability, as any {z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='zL} is an eigen- state of the system and will not evolve under unitary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We can then reduce the problem to solving for the evolution of the central qubit in a mixed x − z field where the effective longitudinal strength given by ∑i σz i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though fully solvable, we find illuminating oper- ator dynamics in the infinite temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Solving exactly for ⟨σx i (t)σx i ⟩ as was first provided in [78], we ob- serve how the coherent oscillation ⟨σx i (t)σx i ⟩ = cos(2Jt) for hc = 0, evolves for increasing hc and leads to operator decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator decoherence, in this sense, is that as auxiliary operators such as σz c → σx c,σy c, they then no longer commute with the c-qubit Ising interaction and op- erator weight then grows on sites j ≠ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The superposition of these growing dynamical correlations on sites j ≠ i simi- larly decreases the probability of finding σx i on site−i after time t, leading to a decay in the autocorrelation function on site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λL >> h, it was shown numerically that early time operator dynamics go as ⟨σx i (t)σx i ⟩ ∼ cos(2λt)e− √ π 2L h2 λ t (9) using the memory matrix formalism [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Regardless of whether λ is a Gaussian random variable or homoge- neous, we arrive at the same exponential dependence on system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the case of Gaussian random variables, this approximation was observed to be faster than the true decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The key physics being that the L−site in- teractions with the c-qubit lead to an extensive coherence time, which we can evaluate explicitly by tracing over the set of eigenstates Z: ⟨σx i (t)σx i ⟩ = 1 2L TrZ[⟨zc,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='zL∣ei ˆHtσx i e−i ˆHtσx i ∣zc,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='zL⟩] (10) = ⟨zc∣⊗⟨Zm∣ei ˆHtσx i e−i ˆHt∣Zm⟩⊗∣zc⟩ (11) ˆH∣Zm⟩⊗∣zc⟩ = ei(hcσx c+λ∑i>0 ziσz c)t∣zc⟩ (12) = I cos(2ωZm t)+ (hcσx c +λZmσz c) ωZm sin(2ωZm t)∣z0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (13) We use same simplified notation as [78], where ωZ = √ h2 +λ2Z2m, with Zm = ∑i>0 zi and Z = Z[m] is just z−eigenstate with magnetization m ∈ [−L,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the set of states {z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='zL} commute with the Hamiltonian, we re- duce the Heisenberg evolution to that of a 2-level system and averaging over the respective density of states, which is simply a binomial distribution in the infinite tempera- ture limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Going back to the full evolution we then have 6 ⟨σx i (t)σx i ⟩ = 1 2L ∑ z0 ∑ m ( L ∣m∣ )⟨zc∣[I cos(2ωZm t)+ (hcσx c +λZ+σz c) ωZm sin(2ωZm t)]× [I cos(2ωZ−m t)+ (hcσx c +λZ−σz c) ωZ−m sin(2ωZ−m]t)]∣zc⟩ (14) = 1 2L−1 ∑ m ( L ∣m∣ )cos(2ωZm t)cos(2ωZ−m t)+ (h2 c +λ2ZmZ− m) ωZmωZ−m sin(2ωZm t)sin(2ωZ−m t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (15) σx i flips the single spin state zi and leads to two unique frequencies wZm and wZ−m, separated like λ for h = 0, Z− m = ∑j≠i>0 z j − zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' hc = 0 provides the exact, non- scrambling result ⟨σx i (t)σx i ⟩ = cos(2λt), and for λ = 0 we have ⟨σx i (t)σx i ⟩ = cos(2λt)2+sin(2λt)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Starting in the hc = 0 limit and moving toward high transverse magnetic field h0 >> λL, we numerically integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='15 and study the autocorrelation function on site−i (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the low field limit, we expect exponentially small modifications to pure cosine oscillations as we have recreated from [78], and in the high field limit, σx i should completely break down as σz c decoheres and operator weight spreads to all sites rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As σz c → [σy 0,σx 0], operator weight can be distributed to all sites j ≠ i, and the rapid rotation of op- erators on the c-qubit aliases away coherent growth on site-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2(a) we see that coherent oscillations are appar- ent for h/λL O(1) out to tλ ∼ 30 and slow decay with in- creasing h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Above log[h/λL] > 0, oscillations are barely visible and dynamics are dominated by exponential decay with no revival even out to time tλ ∈ [0,200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This is more clearly depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2(b), where we take the long-time average of the autocorrelation function: A(i,t0) = ∫ t0 t>0 ∣⟨σx i (t)σx i ⟩∣dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (16) The autocorrelation function transitions sharply at h/L ∼ O(10−1), where σx i no longer has significant weight on the initial site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For small h, the autocorrelation func- tion is a nearly pure cos2λt and decay that grows with h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Near the transition, A(i,t0) sharply decays to ∼ 1/L for h/λL > O(10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We provide a Fourier analysis of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2(a) in (c), where the transition is more clearly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The oscillatory part (ϵ0) decreases (elongates in time) above h/λL = 1 with an exponential decay rate that peaks at the transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Above the transition point, the dynam- ics are no longer captured by a decaying cosine function, but crossover to exponentially damped operator dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The large magnetic field on the central site initially prevents operator weight from leaving site−i as shown by the elongating decay in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' At this point the magnetic field is dominating the operator dynamics and rapidly ro- tates σz c into eventual two-body operators that live in a superposition on all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In this regime, operator weight on the initial site saturates with a scaling like 1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Adding a mix of on-site frustrating magnetic fields is not sufficient to fully scramble information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Apply- ing fields along the x, z-direction, local operators do not spread throughout full operator Hilbert space as the system can still be described as a collective S = L/2- spin interacting with a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Similarly the steady- state entanglement entropy remains independent of λ and system size, while the growth rate is determined by min[1/λ,1/hc] and does not slow when when deep into the coherent regime λL > hc [see Supplemental for greater details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' NUMERICAL RESULTS After discussing the nearest neighbor mixed-Ising chain and the star Ising model, we now seek to under- stand the dynamical behavior in the full, ring-star sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we study the dynamics of the star-local model using exact diagonalization for systems up to L + 1 = 13 and Krylov subspace expansion techniques for evaluating the Schrodinger ODE for sizes up to L+1 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We employ periodic boundary conditions i = L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the previous sec- tion we revealed that a dynamical operator growth tran- sition occurs as a function of hc/λ on the Ising star graph that allows for rapid operator growth from the central qubit or coherently protected operator dynamics on the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' When the operator dynamics are coherently oscillating on the central site, this leads to slow two-time correlator growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we investigate the fate of this transition, its effect on the secondary, local Ising chan- nel for quantum scrambling, and whether fast-scrambling is achievable in simple local-nonlocal construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We know that the mixed-field Ising model is nonintegrable and capable of scrambling information throughout the full operator Hilbert space, but does a simple nonlocal qubit rapidly enhance this process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Firstly, we calculate the average adjacency level ratio ⟨¯r⟩ of the full Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 3 [see Appendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We solve the full spectrum exactly considering the parity conserving and k = 0 sector of the Ising chain (L = 15) and 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Autocorrelation ⟨σx i (t)σx i ⟩: (a) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='15 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' time (tλ) as a function of h,L = 40,λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (b) Long-time average of (a) A(i, t0 = 200) as a function of h and system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (c) Curve fit of smoothed results in (a) consisting of coherent oscillating component with frequency ϵ0 and exponential decay parameter ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ϵ0 shown in the top, solid color palette remains equal to 2λ roughly until reaching the critical point λ = h/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Vertical lines in (c) correspond to critical point hc = Lλ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' confirm that for the ring-star model and nonintegrable Ising model that all points in phase space we consider in- deed follow GOE random matrix statistics with ⟨¯r⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though the level spacing provides a first check of noninte- grable dynamics, it is not sufficient in capturing strongly coherent effects inherent to fractionalized regions of the Hilbert space or rare states that exhibit confinement or slow growth [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entanglement Growth The first test as to the general information scrambling capacity of this model is to understand the entanglement entropy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We numerically investigate entangle- ment growth SvN(t) as a function of c-qubit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' To understand the infinite temperature information dynam- ics of the system, we work with a product state with en- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entanglement spreading in the ring-star model: (a) SvN(t) as a function of λ, (b) plotted on a semilog axis in time (tJ), and (c) λ reparameterized to λ/ √ L, system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) Poly- nomial fit of the intermediate time entanglement growth behav- ior SvN ∝ tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' System size L+1 = 22 and J,h, g = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ergy density equivalent to that of an infinite temperature state ⟨Hi⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the system exhibits GOE random ma- trix statistics for all parameters of the Hamiltonian tuned here, we expect that any such effective infinite tempera- ture pure state will indeed obey ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The state we work with here is the polarized ∣+ y⟩ spin-state, which has been used to exemplify high-energy quench dynamics of the mixed-field Ising chain previously [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the limit λ = 0, we have the local, nonintegrable Ising model which exhibits ballistic entanglement growth for times up to L/vk, where vk is the maximal dispersion as- sociated with quasiparticle momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This timescale is upper-bounded by L/(2J) which provides the charac- teristic timescale of nearest-neighbor interactions multi- plied by the length of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The factor of two comes from periodic boundary conditions and simply captures the longest path between spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the λ = 0 curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3(c) we see that for increasing system size, entangle- ment saturation occurs on a timescale that scales linearly with L and the maximal quasiparticle velocity remains independent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entanglement saturates like Llog(2) as the effective infinite temperature state ex- plores the full Hilbert space of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' With the struc- ture of correlations reaching the scale of the Hilbert space O(2L) we then expect entanglement to be similar to the log of full operator space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As we slowly increase the nonlocal coupling to the cen- tral c-qubit, we expect that nonintegrability is maintained and now the shortest path between sites is next-nearest neighbor, mediated by the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The timescale for the c-qubit mediated entanglement spread is 2/λ, where a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 hc/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 hc/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 hc/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 hc/L =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 hc/L =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 5 10 15 20 25 30 35 b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 = 10 decoherent =1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4 regime L = 20 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' )V coherent L = 30 regime L = 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 L = 50 L = 60 10-5 10-4 10-3 10-2 10-1 100 101 h/L c) 100 Eo 入= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 E1 入= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-1 22 入= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 入 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-2 10-2 10-1 100 101 h/ La) b) 6 8 3 6 4 入= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 4 入 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='28 入 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='56 α logt 1 2 2 入 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='84 入 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='34 0 入 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='28 0 100 101 102 c) d) tJ = 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 = 13 入=0 = 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 =17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 = 19 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='8 vN vL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 Su(t) α tα 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4 入= VL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-1 100 入 0 5 10 15 20 tJ8 the factor of 2 comes from the second order interaction with the central c-qubit that allows correlations to de- velop between sites i, j ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the extremely weak regime λ << J,h, g, the c-qubit qubit can similarly be thought of as a cavity, where the finite size Hilbert space can be dis- regarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Including the c-qubit modifies the early-time entangle- ment growth, as operators grow ballistically due to lo- cal transport and super-ballistically with an additional nearest-neighbor c-qubit mediated contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3(a) that for L + 1 = 20 as λ → h the linear coeffi- cient of entanglement growth grows continuously and the saturation value is modified by the addition of a single qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In this enhanced rapid scrambling regime, if we reparameterize the central qubit coupling λ → λ/ √ L such that the effective spin-spin interaction is not extensive Jeff ∼ λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0/L, we find that the coefficient for linear growth is roughly independent of system size (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3(b) λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 √ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entanglement entropy growth rate that increases with system size is indicative of fast-scrambling behavior [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λ > h, we see a surprising entanglement transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The early time growth defined as t < 2 1 λ exhibits rapid entanglement growth but becomes sub-ballistic at inter- mediate times 2 1 λ < t < tsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3(b) the width of this intermediate timescale grows exponentially with increas- ing λ as exhibited by the logarithmic scale in time, while SvN(t → ∞) remains largely unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We perform a polynomial fit of the entanglement growth in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3(d) over the region log[2] < SvN(t) < SvN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='sat and find that in the local and fast-scrambling regime, entanglement contin- ues to grow ballistically with exponent α ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λ > λc, α monotonically decreases with λ as expected if SvN ∝ 1 λ log[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In order to determine the location of the phase transi- tion and relevant scalings, we analyze the entanglement entropy at a fixed time t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We choose a time such that the SvN(t∗,λ = 0) is roughly 1 2 SvN(t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4(a) we plot SvN(t∗) as a function of two magnetic fields h depicted by solid/dashed curves and as a function of system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We extract the maxima of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4(a) and perform a size and field scaling and find that the fast-to-slow scrambling transition value occurs like λc ∝ L(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05)h(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='02) c , (b,c) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the high hc limit, the transition be- comes nearly independent of system size (hc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4(c)) and in the low field limit, is dominated by system size: hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='75 curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Greater details on identifying the entanglement transition are included in the Supplemental Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This contrasts the dy- namical transition observed in the star Ising model with a transition that occurs at λc = hc/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The novel entanglement transition mediated by the c- qubit dynamics is extremely surprising in that not only does the nonlocal coupling provide a secondary channel for distributing entanglement, but in the strong coupling regime it inhibits growth of even the local Ising interac- tions with which it commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The problem is similarly interesting in that it is completely disorder-free, so the slow information growth in the system can be attributed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Identifying entanglement dynamics crossover: (a) Entan- glement entropy calculated at time t⋆ ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5tJ plotted as a func- tion of c-qubit coupling λ (SvN(t⋆)/t⋆ ∼ vb) for three transverse magnetic fields and various system sizes (hc ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='L ∈ [11−21,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For nonzero λ, entanglement growth rate varies as a function of hc,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Crossover point λc determined as maxima of the curves in (a) with λc(L) and λc(hc) plotted in (b,c), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the respective hc dominant and L dominant regimes, the critical point find λc ∝ Lγhκ c with γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='02(1) and κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' to purely coherent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In order to shed more light on how this central coupling serves to rapidly/slowly scram- ble quantum information we examine the transverse and longitudinal OTOCs (Cxx,Czz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator Spreading Starting from an initial state selected from the Haar measure, as to approximate infinite temperature state, we examine how operators initially prepared on site i = 0 spread under the influence of Ising interactions, where site c represents the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' When calculating the OTOC we simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='1, as the local ˆX, ˆZ operators are Hermitian and initially commute on all sites at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We then explicitly evaluate CVW = 1−⟨ ˆW(j,t) ˆV(i,0) ˆW(j,t) ˆV(i,0)⟩ (17) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5(a), when considering only local interactions we see that operator weight develops according to a ballistic a) h= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 h= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='25 4 × 100 h= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 3 × 100 (*7) 2 × 100 100 10-1 100 入 b) 2 × 100 L-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='3 2 L-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 hz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 100 h, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content="50 101 2 × 101 L c) 2 × 100 00'6 = 7 L = 19." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 100 6 × 10-1 100 2 × 1009 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Space-time OTOC spreading: (a-c) OTOC for longitu- dinal spin component Czz for λ = 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Ini- tial linear light cone spreads on top of c-qubit mediated op- erator weight that spreads super-ballistically on all sites af- ter an initial wait time that scales with min[1/hx,1/λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For (λ/ √ L)/h ≥ 1, Czz is suppressed on all sites beyond a weak ini- tial light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) In contrast, for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6, transverse OTOC Cxx rapidly fluctuates on the initial site-i and spreads to nearest neighbors i+1 and the central site L on the order of t = 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The light cone profile is no longer visible and Cxx rapidly oscillates with frequency 2λ for sites ∣i − j∣ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' System size L +1 = 14 and J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' light cone with a bubble-like profile that saturates on all sites following the wavefront: Czz(r,t) ∝ t2r r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' With small, nonzero λ (b) the light cone remains apparent but now spreads on top of growing operator weight distributed by the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The central qubit super-ballistically distributes operator weight to all sites on a timescale like 2 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The growth timescale on the c-qubit is half that of the bulk spins, as the σz operator must decohere on the initial site and again on the central site and hence no longer com- mutes with the Ising interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λ << hc, 1/λ sets the timescale for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the strong coupling regime (c), a weak light cone remains but transfers a small frac- tion of the original operator weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator spreading becomes increasingly restricted where the lightcone pro- file is only visible for few tJ and Czz(j,t) grows increas- ingly slowly on sites far from the initialized operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This behavior is puzzling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' the highly nonlocal coupling c-qubit with increasing interaction leads to unintuitively slow lo- cal operator spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we can draw insight from the analytic results on the star-Ising model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The L−body interaction on the central qubit acts to coherently project operators into σz, while rapidly aliasing orthogonal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once σz grows on the central qubit, it develops a coherent life- time with operator weight decaying like ∼ e−1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Czz characterizes how operators no longer commute with σz, and in the same vain as the star-Ising model, we ex- pect operators on the central site to be strongly projected into the z−subspace and coherently oscillate like 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Cxx(∣i − j∣, t) spreading in the ring-star Ising model: Real-time evolution of Cxx(i − j, t) for varying c-qubit coupling λ, plotted on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' λ = 0 exhibits lightcone spreading Cxx ∝ t2(i−j) (i−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the fast-scrambling regime (λ < λc) (b,c), oper- ator weight grows homogeneously across all sites at half the rate as the on the c-qubit site (Cxx(c, t), blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) As λ ∼ O(1), Cxx(c, t) becomes ∼ O(1) at t ∝ λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once Cxx(c, t) becomes O(1), the rapid growth on spin sites ∣i − j∣ > 1 slows and coher- ent oscillations develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 and focusing on distant sites (purple) and tJ ∈ [1,5], the operator growth after c-qubit satura- tion becomes slower than that observed for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' System size L+1 = 15 and J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' the central c-qubit is highly/fully entangled with the re- maining chain, the coherent projection on the central site then similarly restricts how rapidly the many-body state of the L−spins decoheres from the z−subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We ex- pect Czz(t) to exhibit slow behavior as z−operators are strongly driven on all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We then examine Cxx(t), which captures the rapid growth of ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5(d) we see that Cxx(t) continues to grow like O(1) across all sites, with strong, oscillatory behavior on the central qubit (i − j = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Cxx(j,t) on all sites similarly oscillates at frequency 2λ but shifted by half a period compared to Cxx(c,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' On the initial site i, as σz i decoheres under the trans- verse magnetic field it no longer commutes with itself nor the Ising interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This leads to a nonzero Czz(t) on site−i and subsequent operator weight growing on near- est neighbors and the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once operator weight grows on the c-qubit, the associated decoherence time of σz c is exponential in λ and as weak operator weight leaks onto bulk spins, the same coherent oscillations and slow decay is imparted on the local spin-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We gain further insight by examining the individual curves that make up the density plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We see the polynomial growth associated with the light-cone spreading on top of the exponential c-qubit mediated growth (a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='In the slow scrambling regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6(d)), we find that for λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 that Cxx(c,t) becomes O(1) at tc = π/2λ (dashed, red) and af- ter time tλ = π/λ the rate of change of Cxx(i − j,t) de- creases dramatically (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once σz exists on the central Czz(i,j, t) Czz(i,j, t) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 a)=0 b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 5 10 C 0 5 10 C Czz(i,j, t) Cxx(i,j, t) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 c) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 d) α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 5 10 0 5 10 C C i-j i-ja) b 100 i=0 i=2 10-1 ^=0 i=4 i-i=6 10-2 ^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 i-i=8 i-i=10 10-3 i-j=12 10-4 c) 100 10-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 t 10-2 ^= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 ^ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 1 2 3 10-4 4 tJ 10-1 100 10-1 100 tJ t)10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Czz(t) spreading in the ring-star Ising model: Longitu- dinal OTOC Czz(t) on (a) site j = 10 and (c) c-qubit j = c with real-time evolution plotted on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (b, d) Polynomial fit at intermediate times Czz(t) ∝ αtβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As λ increases, coef- ficient of longitudinal operator growth becomes exponentially suppressed (top to bottom curve on log scale) logα ∝ λ/ √ Lh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator growth on distant spin sites is approximately linear β ∼ 1 in the slow scrambling regime and coefficient α = e − 3 2 λ √ Lh2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) Czz the c-qubit site grows with exponent βc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 and coeffi- cient αc = e− λ h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator growth on bulk spin sites is expected to behave like Czz(j, t) = Czz(L, t)2, as the effective c-qubit me- diated spin-spin interaction is second order process in Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' qubit, orthogonal operators on the c-qubit and bulk sites rapidly fluctuate and lead to an effective wait time, slow- ing the amount of operator growth under the action of the Ising−zz interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' A complementary perspective for operators orthogonal to ˆZ is observed by Czz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Looking at the OTOC in the slow scrambling regime, operator weight on the c- qubit and bulk sites grows like h2t6, as captured by early- time expansion, up to time tλ = π/2λ [see Supplemental for greater details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For t > tλ, operators become essen- tially projected onto z−operator subspace due to the co- herent lifetime of operators on the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Fitting the OTOC to αtβ shows that operator weight decoheres on the c-qubit with exponent βc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45, while on bulk sites it is nearly linear with βj =∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' On all sites, the coef- ficient of growth becomes exponentially suppressed in λ with log[α] ∝ −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This can be understood heuristically: once operator weight lies on the central site, it coherently oscillates between sites like 2λ and leads to a waiting time during which little to no operator weight has deco- hered from the z−subspace as it oscillates between z − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The amount of operator weight that decoheres in-between these waiting periods is ∫ tc2 tc1 e−λ so the rate of operator growth into this orthogonal subspace is independent of time and leads generically to Cxx(j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' = c,t) ∝ e−λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This slow growth for simple two-body body operators outside of the z−subspace continues to be exponentially suppressed for greater complexity many-body operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The sublin- ear growth of OTOCs into the bulk of operator Hilbert space then generically provides log[t] entanglement en- tropy growth, as we have observed for quantum quenches from the ∣+ y⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' DISCUSSION Here we have shown how the collective interactions between many constituent spins and a central auxiliary qubit is both able to rapidly scramble information across all degrees of freedom and restrict state/operator growth from exploring the full Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' From an entangle- ment entropy picture, the central bit entangles rapidly with its surrounding environment, the state space of the qubit simply being q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Treating it as a 2-level sys- tem in an effective bath, the extensive bath interactions and on-site magnetic field frustrate the central qubit and lead to a rapidly fluctuating spin-moment when Nλ ∼ hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the operator picture this leads to rapidly decohering operators that quickly lie in a superposition of states { ˆX, ˆY , ˆI, ˆZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For λ >> hc, { ˆZ} operators are strongly driven to the qubit and therefore do not fully decohere while { ˆX, ˆY} become essentially echoed-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This picture is more clearly developed when we again turn-off local interactions, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In this case only the cen- tral bit is able to mediate entanglement throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As a function of λ/hc the half-chain entangle- ment entropy grows with a velocity like 1/λ until λ hc and surprisingly saturates with an entropy that is in- dependent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The entropy growth rate and saturation value do not change when crossing over into the strong-coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though the half-system en- tropy value is identical, we know from an operator pic- ture that scrambling occurs slowly on the leaves of the star graph, leaving the central qubit to scramble rapidly while other sites scramble slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' From both the opera- tor and entropy picture, we know that the central bit is highly entangled with the system and the saturation en- tropy remains fixed regardless of coupling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' therefore once the central bit becomes highly entangled, it similarly can- not scramble quantum information throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The sat- uration in information capacity of the star-graph remains fixed as a function of λ, so by tuning λ we tune whether entanglement is frozen in a highly entangled auxiliary bit or distributed globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Understanding the information dynamics of the star-graph then informs the surprising behavior observed in the fully interacting star-local Ising system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once the central qubit becomes highly entangled with it’s environment, it similarly acts to project operators into the z−subspace, inhibiting the local, nonintegrable Ising interactions from fully scrambling the system and reaching full thermalization quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the commutivity graph that translates operators from site−i to site−j requires operation of the Ising−zz interactions, which similarly commutes with the central spin coupling, local operator growth spreading is inhib- ited on the same decoherence timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The local channel is effectively turned off by the slow operator decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' If the local interactions were orthogonal to the central a) 100 (q Czz(L, t) α αth t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='4 B α入 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='2 4 10-2 10-1 100 101 0 2 4 6 tI 入 c) d) 100 : 10, 入 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='28 0 入= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='57 [0]60] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-1 入= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='85 B 入= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='13 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 5 ZZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 C 10-1 100 101 0 2 4 6 t/ 入11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Information spreading in the transverse ring-star model (λ∑L−1 i σx i σx c, J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0): (a) Early-time entanglement entropy growth crosses over from ballistic to sub-ballistic with increas- ing λ, plotted on a semi-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entropy depicts a prether- mal phase with entanglement that saturates around t, then exhibits characteristic MBL, logarithmic growth ∝ log(t) out to late times tJ = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Early-time entropy growth increases smoothly with λ until reaching a discontinuity at λ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (c, d, e, f) OTOC for transverse spin component for λ = 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='84,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In contrast to the parallel ring-star Ising model, the light cone becomes increasingly visible as λ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (c) In the nonintegrable, purely local case, operator scrambling saturates following the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As c-qubit coupling grows, operators orthogonal to σx rapidly fluctuate, so that following the light cone, σx operator weight dominates each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (e) The light cone boundary eventually becomes the only region where operators are orthogonal to the c-qubit coupling and prop- agate under local Ising interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' System size L = 13 and J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0,h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' coupling, we expect different operator spreading results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we take a simple extension of our model and change the form of the auxiliary coupling to be Ising−xx which we term a compass, ring-star Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Now the the auxil- iary coupling projects the system into ˆX subspace, which no longer commutes with the local −zz interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This should allow many-body operators to propagate through- out but inhibit the growth of operator strings contain- ing ˆY , ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Translating the operator picture to the entropy picture, we are allowing higher order many-body opera- tors to develop through local channels, so the intermedi- ate time saturation value should be significantly larger and scale extensively with system size, the logarithmi- cally slow growth of orthogonal operator strings should then extend to long times as operators can decohere to larger regions of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 8 we observe exactly this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Noting that the magnitude of the orthogonal field on the c-qubit changes from h → g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45, the inhibited operator growth goes as λx/g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The entanglement entropy (a) following a quench from ∣+ y⟩ transitions from local spreading, fast- spreading, to confined for λx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For times tJ < 10, the system exhibits ballistic operator growth regardless of coupling but reaches an intermediate saturation value SvN ≈ 3 with slow growth extending out to late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The OTOC agrees identically, where we have the same ballis- tic spreading lightcone with fully scrambled operators fol- lowing the wavefront (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' With increasing coupling (c) we observe a small amount of operator weight that spreads super-ballistically across all sites, a well defined wave- front, and a suppression in the OTOC behind the wave- front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This becomes more dramatic deeper into the con- fined regime (d,e), where the wavefront becomes the only region where operators orthogonal to ˆX may be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the local Ising interaction is orthogonal to the operator subspace in which the c-qubit protects, the wavefront may freely propagate but operators behind the wavefront are aliased to predominantly span strings of ˆX and ˆI opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' CONCLUSION We have interrogated the quantum entanglement and operator dynamics in a central spin-bath design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The local Ising chain acts as a structured, thermalizing bath that when coupled to a nonlocal central qubit, is able to rapidly scramble quantum information across the system up un- til the information capacity of the c-qubit is quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' When the central bit is rapidly quenched in the strong coupling regime, it leads to operator confinement in sec- tors of the Hilbert space spanned by ˆI and operators par- allel to the central qubit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Entanglement entropy grows slowly out to late times as observed in effective in- finite temperature states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This model admits unique limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' where, for weak cou- pling it exhibits super-ballistic OTOC spreading and fast- scrambling as supported by previous works examining a similar star-graph structure, and a confined quantum Zeno-like limit in which the complexity of operator growth is inhibited by extensive, coherent interaction with the two level central mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This confined regime mirrors a larger class of models which admit prethermal or local- ization physics: disorder driven many-body localization, projective or weak quantum measurement, floquet-driven prethermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Our star-local work is similar in that the c-qubit drives operator fluctuations on the leaves on a timescale like 1/λ once λ is greater than noncommut- ting fields on the central site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In contrast, the qubit re- tains its infinite-range information sharing capacity com- pared to the external drive scenario in floquet physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the measurement theory sense, once the central qubit be- comes nearly maximally entangled with the Ising spin chain bath, its as if the bath projects the central spin a) 10 4 8 3 6 2 4 2 1 0 100 101 102 tJ Cxx(i, j) Cxx(i,j) 15 b) ^=0 c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='6 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 15 d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='7 e) ^ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 0 5 10 C 0 5 10 c i-j i-j12 which in turn freezes the information sharing dynamics with which the bath is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' And finally in the MBL case, instead of local integrals of motion that are nearly integrable with exponentially small overlap, here we have global operators orthogonal to the auxiliary coupling that have a logarthimically slow thermalization timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This work provides a novel mechanism for exploring a host of quantum information dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here we outline the existence of a dynamical transition that occurs under static Hamiltonian dynamics, and in future work it would be fruitful to investigate the nature of the transition and how similar physics can be observed in RUCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Tracing out the central qubit to produce non-Hermitian physics may similarly illuminate how this model relates to strong peri- odic driving prethermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' As the inverse problem, it would be interesting if exotic non-Hermitian physics has a hidden unitary quantum system analog, where multiple ancillary qubit degrees of freedom produce the observed non-Hermitian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Future work should also seek to understand how this transition persists with larger c- qudit towards an infinite bosonic fields or multiple dissi- pative modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' It would be an interesting engineering ap- plication if uniform coupling to a bosonic mode or central- qubit can protect against dephasing errors in the environ- mental qubit platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ACKNOWLEDGEMENTS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' would like to thank Sumilan Banerjee, Chandrasekhar Ramanathan, Brian Skinner, Xiaozhou Feng, Shi Feng, and Sayantan Roy for useful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 82 Joseph C Szabo and Nandini Trivedi, “Entanglement dynam- ics between ising spins and a central ancilla,” Physical Re- view A 105, 052431 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 83 Tibor Rakovszky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' von Keyserlingk, and Frank Pollmann, “Entanglement growth after inhomogenous quenches,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' B 100, 125139 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 84 Phillip Weinberg and Marin Bukov, “QuSpin: a Python Pack- age for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains,” SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 2, 003 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 15 Supplemental Material for "Entanglement Dynamics between Ising Spins and a Central Ancilla" I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' OPERATOR SPREADING ON THE STAR GRAPH In addition to the exactly solvable star-Ising model presented in the main text, we present how the dynamics extend with a magnetic field applied to the c-qubit and all other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' H = λ∑ i σz cσz i + hσx i + hσx c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (18) The early-time behavior using the BCH expansion of the commutator and for general (h/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' [σx j(t),σx i ] = ∞ ∑ m (it)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' [[H,σx i ]m,σx j] [A,B]m = [A,[A,B]m−1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='[A,B]0 = B [H,σx i ]1 = it[σz i ,σx i ] = itλσz 0σy i [H,σx i ]2 = (it2) 2 [H,λσz 0σy i ] = −t2 2 (−λ2σx i + hλσy 0σy i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' [H,σx i ]3 = (it3) 6 [H,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='] = −it3 6 (−λ3σz 0σy i + hλ2(∑ j≠i σx 0σz jσy i +σx 0σx i )− h2λσz 0σy i ) In the limit h/λ = 0,λ = 1 we have: [σx j(t),σx i ] = [∑ m (it)2m 2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' σx i + (it)2m+1 (2m+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='σz 0σy i ,σx j] = [cos(2t)σx i − isin(2t)σz 0σy i ,σx j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Then the two-time correlator behaves as: ⟨σx i (t)∣σx i ⟩ = cos2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (19) And the OTOC goes as: Cxx(i,1,t) = 2sin2(2t), (20) strictly for i ∈ c,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Under the operation of the central Ising interaction, the operator weight is completely preserved to live in the projective space as we have seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' When adding a non-commuting term that rotates states away from this space, how does this then modify growth on the remaining sites?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Looking back, we see that ⟨σx j(t)∣σx i ⟩ ∼ O(1) ∝ t3 6 h2 (21) Cxx(j, i,t) ∝ h2t6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (22) As exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S1, we see that for fixed λ = 1, that h2t6 provides an excellent approximation to the early-time growth behavior of OTOC on next nearest neighbors to the initial site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This rapid growth on the central site and corresponding neighbors at early-times in essence captures the extreme early-time entanglement growth that is observed to grow steadily with growing h/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The interesting phenomena occurs when t ∼ π 2λ, where in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S1(a) the operator space q is completely preserved, allowing C(0, i,t) to decay to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' On this same timescale we see in (b, c) and most clearly in (d) that this leads to an effective wait time that is proportional λ and similarly corresponds to the amount of operator weight outside of the space q on the c-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Also, any weight operator weight transferred to edge spins then undergoes similar oscillations seen in (a) during this wait time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the weakly decoherent regime h << λ, the wait time leads to essentially linear OTOC growth with oscillations that similarly grow in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' OTOC Cxx(i, j, t) for (a) h = 0, (b) hc > 0, (c, d) h > 0 with L + 1 = 13, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (a) OTOC is only nonzero on the edge and central c-qubit as coherent evolution under the star-coupling does not allow operators to propagate beyond the c-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (b,c) early- time growth behavior is accurately captured by BCH expansion of the two-time correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) At intermediate times, t, as operator weight transitions back to the initial site, there is then essentially a waiting time that occurs before more operator weight may grow on sites j ≠ [i,0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This leads to a polynomial growth in time of the two-time correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' ENTANGLEMENT SPREADING ON THE STAR GRAPH Here we examine the corresponding entanglement entropy growth on the Ising star graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Starting from the same polarized product state used in the main results (∣ + y⟩) we plot the half-system entanglement entropy following a quantum quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We treat λ as the variable and set h,hc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05 and g, gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The entanglement entropy saturates to a length independent value Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S2(a), constant for system sizes L = 11 − 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Though the mixture of onsite fields allows local operators to explore the operator space of single sites, the collective magnetic field and collective c-qubit interaction still limit the system to behave semiclassically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S2(b) we note that the entanglement entropy growth rate and saturation value become independent for λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' This is an interesting finding as we know in the large λ >> h limit that operators simply scramble between the initial site and the cqubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The corresponding entanglement entropy between halves is dominated by the system-qubit entanglement as entanglement generated between spins grows slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Once the system rapidly entangles with the cqubit, the entanglement entropy of the system remains fixed but becomes uniformly distributed between all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In the OTOC picture operators are scrambled with the cqubit and then leak into the remaining sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' OTOC GROWTH VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' SYSTEM SIZE SCALING In the main text we examined the longitudinal OTOC and found that it goes like Czz(t) ≈ e−λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S3 we perform the same analysis and vary the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We see that close to the transition λ ≤ 1 that the coefficient on OTOC growth is roughly independent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' For increasing λ we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S3(top) that there is weak decrease in log(α) with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Looking at the 1/α to highlight the slight difference in system size, we see that into the localized operator phase λ > 3−4 that larger system sizes exhibit increasingly slow operator growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' a) b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-1 α h2t6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 t 10-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 hc/^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='26 li-j=2 10-5 hc/^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='53 li-jlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 hc/△ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='79 li-jl=c 10-7 hc/△ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='68 sin2(2t) hc/^ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 10-9 5 10 10-2 10-1 100 101 t t c) d) 10-1 10-1 t h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='26 hj^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='38 hj^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='13 10-3 10-3 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='51 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='26 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='79 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='38 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='^ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='44 cQubit:hi/^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='13 10-5 10-5 10-1 100 101 10-1 100 101 t t17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Information spreading star model (J = 0): Real-time evolution of the 1/2−chain entanglement entropy initialized in ∣+ y⟩ (a, b), and OTOC following quench from a Haar random initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Information spreading in the star-Ising model (J = 0): Real-time evolution of the the 1/2−chain entanglement entropy initialized in ∣+ y⟩ (a,b) and Czz(t, i− j) with initial state chosen randomly from the Haar distribution (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (a) Entanglement entropy plotted as a function of length for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Growth is independent of system size for fixed λ and saturates to a value independent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (b) Entanglement growth rate and steady state value for λ > h saturate and become independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (c, d) OTOC Czz for λ/ √ L = {0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='78}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (c) Initial operator simply fluctuates under mixed-fields and does not propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (d) OTOC grows on the c-qubit at a rate that grows with λ after an initial wait time that scales with min[1/h,1/λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Operator weight then spreads to the remaining leaves after a time 2 min[1/h,1/λ] corresponding to two processes of operator transfer that occur on the c-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' SPECTRAL STATISTICS OF THE STAR-ISING MODEL We calculate the spectral statistics for the central qubit Ising model studied in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' These statistics provide insight into the random nature like behavior of the density matrix or whether there are conservation laws present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Anything beyond random statistics manifests as prethermalization, MBL, or integrable behavior when a system is quench from a highly energetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We calculate the spectral statistics using 100 states surrounding the middle of the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' The ratio is between two levels ri is calculated as ri = min[Ei − Ei−1 Ei + Ei−1 , Ei+1 − Ei Ei + Ei+1 ] (23) We then take the average spacing between all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We work in the k = 0 sector as well as the paritiy conserving sector, as the translational invariance leads to degeneracies between k−sectors when only one particular sector is meaningful under dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We take the tensor product of this Ising spin-chain Hilbert space with the central qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='S4 we observe that the only point in phase space that does not exhibit GOE random matrix statistics ⟨r⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='53 as for λ = 0, g = 0, where we have the integrable TFIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Here the system exhibits sub-Poissonian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 a) b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='00 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 L = 11 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='75 L = 13 入=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 L = 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 入=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='34 L = 17 入=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='28 L = 19 入= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 5 10 15 20 0 5 10 15 20 t t Czz(i, j) Czz(i, j) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 c) = 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 d) ^ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 5 10 15 0 5 10 15 i-j i-i18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Finite size scaling of the coefficient of linear growth αtβ (see Main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='7) in Czz(∣i − j∣ = 6, t) for system sizes L = [8,10,12,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (Top) log[α] decreases linearly with λ as exponentially less operator weight decoheres on all sites and spreads throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (Bottom) Plot of 1/α for better resolution of the system size dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We observe a small decrease in α with increasing system size, corresponding to the weak coefficient in λc ∼ √ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' Average adjacency level spacing of energy eigenvalues r(Ei,Ei+1) for the ring-star Ising Hamiltonian for L +1 = 16 with periodic boundary conditions for fixed momentum and Z-reflection symmetry blocks of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' (a) Spacing as a function of λ, g for fixed h, (b) h, lambda, g = 0, and (c) h, g,λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' We find strong evidence of the integrable case of h ≠ 0, g = 0,λ = 0 in (b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 log[α] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 B L=8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 L=10 L=12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 L=14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0 2 4 L=8 L=10 L=12 200 L=14 1/α 0 0 2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='550 h =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='525 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdA0T4oBgHgl3EQfJf-W/content/2301.02091v1.pdf'} +page_content='5 0.' 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b/n9E1T4oBgHgl3EQfOgPu/content/tmp_files/2301.03017v1.pdf.txt @@ -0,0 +1,2684 @@ + + +1 +Development of a novel nonlinear dynamic cavitation model +and its numerical validations + +Haidong Yu12, Xiaobo Quan3, Haipeng Wei1, Matevž Dular4 and Song Fu2* +1Beijing Institute of Astronautical System Engineering, Beijing, 100076, China +2School of Aerospace Engineering, Tsinghua University, Beijing 100084, China +3China Academy of Launch Vehicle Technology, Beijing, 100076, China +4Laboratory for Water and Turbine Machines, University of Ljubljana, Aškerčeva 6, +1000 Ljubljana, Slovenia +Abstract +Aiming at modeling the cavitation bubble cluster, we propose a novel nonlinear +dynamic cavitation model (NDCM) considering the second derivative term in +Rayleigh-Plesset equation through strict mathematical derivation. There are two +improvements of the new model: i) the empirical coefficients are eliminated by +introduction of the nonuniform potential functions of 𝜓𝑣 and 𝜓𝑐 for growth and +collapse processes respectively, and ii) only two model parameters are required, which +both base on physical quantities – the Blake critical radius 𝑅𝑏 and the average +maximum growth radius 𝑅𝑚. The corresponding cavitation solver was developed by +using OpenFOAM in which we implemented the modified momentum interpolation +(MMI) method to ensure that the calculated results are independent of time step size. +Three validation cases, namely numerical bubble cluster collapse, ultrasonic horn +experiment, and hydrodynamic cavitation around slender body are employed. The +results indicate that 𝜓𝑣 and 𝜓𝑐 can reveal the nonlinear characteristics for cavity +accurately, and 𝑅𝑏 and 𝑅𝑚 can reflect the relevance between cavitation model and +actual physical quantities. Moreover, it is discussed the potentiality of NDCM that is +generally applied on the cavitating flow possessing with dispersed bubbly cloud. + +keywords: Cavitation model, Rayleigh-Plesset equation, Bubble cluster collapse, +OpenFOAM + + + +* Corresponding author, email: fs-dem@tsinghua.edu.cn + + + +2 +Highlights +➢ A nonlinear dynamic cavitation model (NDCM) is established through strict +mathematical derivation. +➢ The formula of NDCM employs the potential functions of 𝜓𝑣 and 𝜓𝑐 instead on +empirical coefficients to describe the nonlinear effects. +➢ The model parameters of 𝑅𝑏 and 𝑅𝑚 represent the physical characteristics of +bubble cluster. +➢ The NDCM is best valid for the cavitating flows with dispersed bubbles. + +1. Introduction +Cavitation often occurs in liquid flows when the ambient pressure drops below a +certain threshold. The cavitating bubbles will emerge gradually from “cracked” liquid +medium at its weak points [1]. Individual bubbles cluster and form a complex two- +phase mixture cloud, which shape depends strongly on the structure of the flow field. +The cavitation bubble cluster exhibits many unique characteristics, such as strong +collapse accompanied by a shockwave, or the natural frequency far lower than single +bubble’s etc. [2, 3] There are many researches on single bubble dynamics which can be +described by Rayleigh-Plesset type equation [4, 5]. However, an approach to build +cavitation model based on Rayleigh-Plesset equation to investigate the dynamics of +bubble cluster can be considered questionable. +Numerical simulation of cavitating flows and specifically the development of +transport equation model (TEM) has received enormous attention from investigators in +recently years. Instead of potential flow theory implemented in early engineering +applications, the Eulerian's one field formulation (OFM) of the two-phase Navier– +Stokes equation [6-9], which combines the properties of each phase as a single mixed +one, is popularly applied as the methodology of multiphase model. The cavitation +model for bubble cluster is embedded into the convective phase equation as source +terms. There is also an alternative approach based on Eulerian–Lagrangian method [10, +11] but is not within the present framework. +The prototype of TEM was introduced by Kubota [12] who assumed that the bubble +nuclei are uniformly distributed in the flow, and the simplified Rayleigh-Plesset +equation, which considered the SGS bubble interaction was used to determine the +change of radius and consequently the mixture density in each computational cell. The +advantage of this approach is that the dynamic response of local equilibrium bubbles +can be estimated precisely. However, solving the nonlinear transport equation often +faces the difficulty of convergence and its application was merely limited in studying +steady flow. +In order to develop the Kubota’s method, the modeling strategy was focused on larger +scale of bubble cloud rather than the tiny scale of individual bubbles. A different + + + +3 +approach with the same form of transport equation models had already been proposed +in literature [6-9], in which the nonlinear ODE is replaced by a convective equation of +void fraction. The mass transfer rate of bubbles is contained implicitly into source terms. +It has the advantage that the convective character of equation is more appropriate for +describing the topological evolution of bubble cluster particularly in unsteady situations. +In the literature, there are two main approaches to build the empirical phase-transition +laws associating with surrounding pressure. Merkel et al. [13-15] proposed simplest +formulation based on dimensional analysis that defined the characteristic velocity 𝑢∞ +or temporal scale 𝑡∞. However, these parameters are prescribed as a constant during +the whole dynamic period, especially in collapsing process, which would cause large +deviation. In order to make the model more physical, the void fraction was regarded +equivalently as a group of identical bubbles. Schnerr et al. [16-19] simplified the +Rayleigh-Plesset equation that only the linear part was used to redefine the velocity +scale as √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 +⁄ + for both growth and collapse situations. The advantage of +this formulation is that it follows, to some part, a physical law and reduces the +complexity. Nevertheless, the empirical coefficients are still needed to regulate the +order of magnitude between evaporation and condensation terms. +Another way of modelling is to establish the barotropic relation that couples the +mixture density and local pressure as called equation of state (EoS). The model was +first introduced by Delannoy and Kueny [20] and later widely used by others [21-23]. +Some results obtained by EoS model show good agreement with benchmarks in term +of strong compressible multiphase flows. However, the mechanisms inside cavity are +complicated more than barotropic assumption that the gradient parallel between +pressure and density can hardly be guaranteed. For weak compressible issues, +numerical algorithms lost robustness that would induce numerical instability and +sometimes poor convergence. +All the mentioned cavitation models were developed and compared with a variety of +experiments. Despite some satisfactory results have been achieved by existing models, +some questions still remain to be discussed. One of the basic principles for cavitation +model application is that the flow pattern of vapor-liquid mixture should be +homogeneous dispersed bubbly flow. It will exceed the model capability if the +frequency of bubble coalescence is too high, for instance, in the case of supercavitating +flow. Though few models are still workable on the flow out of their application scope, +one has to select weird values for parameters in order to match the results. Furthermore, +it can be inferred from several simulation comparisons [24, 25] that the model +parameters cannot reflect the physical characteristics sufficiently because the empirical +coefficients in specified cavitation model sometimes need to be recalibrated for +different flow conditions. Also, the results calculated by different cavitation models +present respective tendencies under the same experimental condition [26, 27]. We +speculate the main reasons causing these deviations come from two sides. One is that +the underlying characteristic scales produce large errors when the second derivative +term in Rayleigh-Plesset equation becomes predominant. As previously mentioned, the +characteristic velocity in current models is given as an immutable value 𝑢∞ or linear +form √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 +⁄ + so that it is necessary to introduce empirical coefficients to + + + +4 +compensate the information mismatch. On the other hand, the typical configuration of +existing models with two empirical coefficients and several parameters have good +adaptability but wide range of proper values to guess. Defining the parameters which +can reveal the physical relevance between model and objective phenomenon would +narrow the search regions. Thus, we suggest three improvements to account for the +effect of nonlinear term, eliminating the empirical coefficients, and parameters +reflected more intrinsic law to remedy the cavitation model with wider applicability. +In this paper, a novel TEM-kind nonlinear dynamic cavitation model is proposed +with rigorous mathematical derivation. There arise two functions 𝜓𝑣 and 𝜓𝑐 instead +of empirical coefficients to represent the nonlinear effects for growth and collapse +periods respectively. Only two meaningful parameters, the Blake critical radius 𝑅𝑏 +and the average maximum growth radius 𝑅𝑚 , are adopted. The new model is +implemented in the open source C++ package OpenFOAM based on the +interPhaseChangeFoam solver [28]. To illustrate the superiorities of the new model +clearly, three typical validations cases, sequentially from simple to complex, were +adopted: numerical bubble cluster collapse [29], ultrasonic horn experiment [30, 31], +and hydrodynamic cavitation around a slender body [32]. In these cases, the similar +dispersed structures of bubble cloud satisfy the homogeneous assumption. +In the following sections, basic assumptions and simplifications for bubble cluster +are stated firstly. The model derivation processes are then elaborated in detail. The +algorithms, including VoF-based interface capture method MULES [33, 34] in +OpenFOAM and modified momentum interpolation (MMI) method [35, 36] for +multiphase flows, are implemented correspondingly. Finally, qualitative and +quantitative validations of the improved model performance are presented. +2. Modeling methodology +2.1 Assumptions +The cavitating cluster composed by multi-radius bubbles makes the overall problem +difficult to model. In order to reduce the complexity of modeling the bubble cloud, we +propose to introduce four assumptions to describe its main mechanisms approximately. +I. No bubbles’ coalescence or breakage. +II. Local homogeneous assumption: +As is shown in Fig. 1, the bubble cloud is divided into lattices by CFD grid in +physical space. The real bubbles included in each grid cell are equivalent as +uniform radius according to the local vapor volume so that the free degrees are +degenerated into one. +III. Neglecting the local bubble interactions: + The intensity of bubble interactions has positive correlations with the bubble +population 𝑁 [37]. Noting that only partial region of bubble cloud is filled in +local grid cell. If the mesh resolution is fine enough to contain few bubbles in + + + +5 +single element that the interactions between these localized bubbles could be +ignored. +IV. The law of mean bubble dynamics: + Given a pressure field, bubbles with different scales experience similar +dynamics which can be governed by Rayleigh-Plesset-type equations. We +suppose the mean bubble dynamic processes existed which could be obtained +from statistical operations on the whole cloud bubbles. +In local point of view, the assumptions II and III are indicated that the dynamics of +average bubbles, including growth and collapse, insides each grid cell can be treated as +synchronized processes. It should be noted that the assumption III doesn’t mean nothing +interaction existed inside bubble cloud. Actually, these interactions are implied in +numerical fluxes between adjacent grid cells, which will be illustrated in section 4.1. It +can also be inferred that there is a lower limit of mesh resolution that fulfills the +requirements in assumption III. We recommend to generate a proper mesh by using grid +independence principle. +In overall view of bubble cloud, the structure of void fraction which exhibits +concentrated interior and sparse border has been observed through several experiments +[38, 39]. This characteristic can be reflected as radius distribution of the average +bubbles mentioned above in computational space. The assumption IV is indicated that +such distribution can be decided by the mean bubble dynamics which is considered as +internal law that every bubble obeys to manifest the main dynamic character of bubble +cloud. Therefore, the modeling strategy can be simplified as to investigate the mean +dynamics of local average bubbles. + +Fig. 1 Diagram of bubble cluster in physical/computational space +2.2 Simplifications +With the above assumptions, the model expressions can be derived from the +individual bubble with Rayleigh–Plesset equation: +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +⏟ +1 ++ 𝑝𝑔0 +𝜌𝑙 +(𝑅0 +𝑅 ) +3𝛾 +⏟ +2 += 𝑅 𝑑2𝑅 +𝑑𝑡2 +⏟ +3 ++ 3 +2 (𝑑𝑅 +𝑑𝑡 ) +2 +⏟ +4 ++ 2𝜎 +𝜌𝑙𝑅 +⏟ +5 +− 4𝜇 +𝜌𝑙𝑅 +𝑑𝑅 +𝑑𝑡 +⏟ +6 +. +(1) +Here the first term represents the difference of vapor saturation pressure 𝑝𝑠𝑎𝑡 and +ambient pressure 𝑝̅. In Euler mixture model, we should point out that 𝑝̅ is a phase- + +Four +assumptions +Physical space +Computationalspace + +6 +weighted mixing pressure [6] which can be considered as ambient pressure for those +under-resolved bubbles. The second terms describe the influence of gases inside the +bubble. The inertial effects are given by the third and the fourth terms, while terms 5 +and 6 depict the influence of surface tension and liquid viscosity, respectively. +Schnerr [17] has introduced the bubble density 𝑛 which definition represents the +bubble population per unit liquid volume is adopted uniformly to build the relation +between liquid volume fraction 𝛼𝑙 and bubble radius: +𝛼𝑙 = +𝑉𝑙 +𝑉𝑙 + 4 +3 𝜋𝑅3𝑛𝑉𝑙 += +1 +1 + 4 +3 𝜋𝑅3𝑛 +. +(2) +Due to the cavitating bubble is commonly treated as pure vapor, the expression for +mass transfer rate per unit volume can be built as follows: +𝑚̇ = 1 +𝑉 +𝑑 +𝑑𝑡 (𝜌𝑣 ∗ 4 +3 𝜋𝑅3 ∗ 𝑛𝛼𝑙𝑉) = 𝜌𝑣𝛼𝑙 +2 𝑑 +𝑑𝑡 (4 +3 𝜋𝑛𝑅3) . +(3) +According to the meaning of 𝑛 , the non-dimensional term 4𝜋𝑛𝑅3 3 +⁄ inside time +derivative indicates the volume ratio of vapor to liquid. The original way to handle this +term is to expand differentially into 4𝜋𝑅2 𝑑𝑅 𝑑𝑡 +⁄ +, then specify the velocity 𝑑𝑅 𝑑𝑡 +⁄ + by +using the linear part √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 +⁄ + in Rayleigh-Plesset equation for both growth and +collapse processes. Except for the oversimplification of linearized bubble velocity as +previously discussed, such instantaneous form would underestimate the intensity of +source terms during time marching. Thus, we suggest to utilize the average value within +the evolution time 𝜏 (Eqn.4) during which the bubble radius would vary from present +𝑅𝑃 to ultimate 𝑅𝑈: +𝑑 +𝑑𝑡 (4 +3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ +4 +3 𝜋𝑅𝑈 +3 − 4 +3 𝜋𝑅𝑃 +3 +𝜏 +. +(4) +It is illustrated the implication of 𝜏 through the numerical results of Rayleigh- +Plesset equation. As is shown in Fig. 2, neglecting the bubble rebound, the typical +dynamic period can be divided into four stages, namely inception, growth, slow-down +and collapse, also several critical moments are marked - including the Blake radius 𝑅𝑏, +zero acceleration point 𝑅𝑑, maximum radius 𝑅𝑚, and the final collapse radius 𝑅𝑐0. +The dynamics before 𝑅𝑏 is gas dominated oscillation, after that the bubble content is +occupied by vapor gradually, then unstable cavitation took place. The expansion +process from 𝑅𝑏 to 𝑅𝑚 is split by inflection point 𝑅𝑑 into growth and slow-down, +where the ratio of 𝑅𝑑 and 𝑅𝑚 is approximately 0.8. Besides, we emphasize that the +nonlinear effect should be considered for the accelerated collapse. The cavitation model +is only devoted to describing growth and collapse so the evolution time for these two +periods, 𝜏𝑣 and 𝜏𝑐 , are determined from the present point 𝑅 to 𝑅𝑑 and 𝑅𝑐0 , +respectively. + + + + +7 + +Fig. 2 Typical bubble dynamic period + +Therefore, the main task is to calculate the values of 𝜏𝑣 and 𝜏𝑐 in term of +Rayleigh–Plesset equation (Eqn.1) where terms including surface tension, viscosity and +gas had been neglected, +𝑅𝑅̈ + 3 +2 𝑅̇ 2 = 𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +. +(5) +Here, the saturated vapor pressure 𝑝𝑠𝑎𝑡 and liquid density 𝜌𝑙 are treated as constants, +and the LHS in Eqn.5 can also be reformulated as the following equation. +𝑅𝑅̈ + 3 +2 𝑅̇ 2 = +1 +2𝑅2𝑅̇ +𝑑 +𝑑𝑡 (𝑅̇ 2𝑅3). +(6) +It can be expected that the exact solutions for these evolution time can be derived +from integrating the bubble velocity 𝑑𝑅 𝑑𝑡 +⁄ +, i.e., to integrate the Eqn.5 twice under +compatible initial conditions. The deductions are discussed in next section. +2.3 Modeling +2.3.1 Growth potential function 𝑚̇ 𝑣 +Based on the interval of growth period shown in Fig. 2, the integration and +corresponding initial condition for obtaining the velocity at 𝑅 are given in Eqn.7.1 and +7.2, where the velocity at 𝑅𝑏 is treated approximately zero: +𝑡 = 0, 𝑅 = 𝑅𝑏 𝑎𝑛𝑑 𝑅̇ ≈ 0, +(7.1) +∫ 𝑑 +𝑑𝑡 (𝑅̇ 2𝑅3) +𝑡𝑅 +0 +𝑑𝑡 = 𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +∗ ∫ 2𝑅2 𝑑𝑅 +𝑑𝑡 +𝑡𝑅 +0 +𝑑𝑡. +(7.2) +Assuming that the ambient pressure 𝑝̅ is constant during short period of time, the +integral result of growing velocity is then given in Eqn.8: +𝑑𝑅 +𝑑𝑡 = √2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +(1 − (𝑅𝑏 +𝑅 ) +3 +) . +(8) +Obviously, it approaches to the inertial velocity √2|𝑝̅ − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 +⁄ + when the bubble +expands significantly greater than 𝑅𝑏. Consequently, the evolution time of 𝜏𝑣 can be + +175 +- +- +1 +R +1 +- +- +1 +m +- +1 +- +140 +- +R +1 +R +- +- +1 +- +- +1 +- +lius +- +1 +- +- +- +- +IRadi +- +105- +- +1 +1 +1 +- +- +1 +- +- +1 +1 +- +Nondimensional +- +- +1 +- +70- +- +- +1 +- +- +- +1 +- +- +R +1 +1 +1 +1 +1 +1 +- +1 +1 +35- +- +1 +1 +- +1 +1 +- +1 +R +1 +1 +1 +之 +R +- +1 +co +0 +- +1 +- +growth +1 +collapse +- +inception +(t) +islow-down! +Irebound +- + +8 +integrated from 𝑅 to 𝑅𝑑 using Eqn.8: +𝜏𝑣 = ∫ +𝑑𝑅 +√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +(1 − (𝑅𝑏 +𝑅 ) +3 +) +𝑅𝑑 +𝑅 += +1 +√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +∫ +𝑑𝑅 +√1 − (𝑅𝑏 +𝑅 ) +3 +𝑅𝑑 +𝑅 +. +(9) + +Letting 𝑥 = (𝑅𝑏 𝑅 +⁄ )3, so we have: +𝑑𝑅 = − 𝑅𝑏 +3 𝑥−4 +3𝑑𝑥. +(10) +Thus, Eqn.9 can be reformulated as follows: +𝜏𝑣 = +𝑅𝑏 +3√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +∫ +𝑥−4 +3 +(𝑅𝑏 +𝑅 ) +3 +(𝑅𝑏 +𝑅𝑑) +3 +(1 − 𝑥)−1 +2𝑑𝑥, +(11) +where the definite integral in Eqn.11 can be expanded by applying the Gauss +hypergeometric function, the result becomes: +𝜏𝑣 = +2𝑅𝑏 +3√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +[ + + + + + √1 − (𝑅𝑏 +𝑅𝑑 +) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅𝑑 +) +3 +) +−√1 − (𝑅𝑏 +𝑅 ) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅 ) +3 +) +] + + + + + +. (12) +Note that there is an ultimate limit under the large ratio of 𝑅𝑑 an 𝑅𝑏, then: +lim +𝑅𝑏 +𝑅𝑑→0 +(𝑅𝑏 +𝑅𝑑 +) √1 − (𝑅𝑏 +𝑅𝑑 +) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅𝑑 +) +3 +) = 3 +2 . +(13) +The Eqn.12 is simplified as follow: +𝜏𝑣 = +2 +3√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +[3 +2 𝑅𝑑 − 𝑅𝑏√1 − (𝑅𝑏 +𝑅 ) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅 ) +3 +)]. +(14) +Thus, combing with the Eqn.2 and Eqn.14, we can get the approximation of derivative +in Eqn.4 for growth situation: +𝑑 +𝑑𝑡 (4 +3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ +4 +3 𝜋𝑅𝑑 +3 − 4 +3 𝜋𝑅3 +𝜏𝑣 += 1 − 𝛼𝑙 +𝛼𝑙 +1 +𝑅 √2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 +∗ 𝜓𝑣, +(15.1) +𝜓𝑣 = +(𝑅𝑑 +𝑅 ) +3 +− 1 +(𝑅𝑑 +𝑅 ) − 2 +3 (𝑅𝑏 +𝑅 ) √1 − (𝑅𝑏 +𝑅 ) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅 ) +3 +) +. +(15.2) +There, it emerges the formulation 𝜓𝑣 which we call growth potential function that +represents the capacity of continuous growth until 𝑅𝑑 being reached. Recalling the +ratio of 𝑅𝑑 𝑅𝑚 +⁄ +≈ 𝜂 = 0.8, it can be expressed by 𝑅𝑚 alternatively: + + + +9 +𝜓𝑣 ≅ +(𝜂 𝑅𝑚 +𝑅 ) +3 +− 1 +(𝜂 𝑅𝑚 +𝑅 ) − 2 +3 (𝑅𝑏 +𝑅 )√1 − (𝑅𝑏 +𝑅 ) +3 +𝐹 +2 1 (1 +2 , 4 +3 ; 3 +2 ; 1 − (𝑅𝑏 +𝑅 ) +3 +) +. +(16) +2.3.2 Collapse potential function 𝑚̇ 𝑐 +Analogously, with the same integration in Eqn.7.2, the initial condition at max radius +𝑅𝑚 for obtaining the collapse velocity is given as follows: +𝑡 = 0, 𝑅 = 𝑅𝑚 𝑎𝑛𝑑 𝑅̇ = 0, +(17) +which yields +𝑑𝑅 +𝑑𝑡 = −√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +((𝑅𝑚 +𝑅 ) +3 +− 1) . +(18) +The evolution time 𝜏𝑐 can be integrated from 𝑅 to 𝑅𝑐0 by Eqn.18: +𝜏𝑐 = − ∫ +𝑑𝑅 +√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +((𝑅𝑚 +𝑅 ) +3 +− 1) +𝑅𝑐0 +𝑅 += +1 +√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +∫ +𝑑𝑅 +√(𝑅𝑚 +𝑅 ) +3 +− 1 +𝑅 +𝑅𝑐0 +. +(19) + +Letting 𝑥 = (𝑅 𝑅𝑚 +⁄ +)3, so we have: +𝑑𝑅 = 𝑅𝑚 +3 𝑥−2 +3𝑑𝑥. +(20) +Eqn.19 can be reformulated as follows: +𝜏𝑐 = 𝑅𝑚 +3 +1 +√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +∫ +𝑥−1 +6(1 − 𝑥)−1 +2𝑑𝑥. +( 𝑅 +𝑅𝑚) +3 +(𝑅𝑐0 +𝑅𝑚) +3 +(21) +According to the expression of incomplete beta function, we obtain: +𝜏𝑐 = 𝑅𝑚 +3 +1 +√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +[𝛽 (5 +6 , 1 +2 , ( 𝑅 +𝑅𝑚 +) +3 +) − 𝛽 (5 +6 , 1 +2 , (𝑅𝑐0 +𝑅𝑚 +) +3 +)] . +(22) +Employing Eqn.2 and Eqn.22, we can get the approximation of derivative in Eqn.4 for +collapse situation: +𝑑 +𝑑𝑡 (4 +3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ +4 +3 𝜋𝑅𝑐0 +3 − 4 +3 𝜋𝑅3 +𝜏𝑐 += − 1 − 𝛼𝑙 +𝛼𝑙 +1 +𝑅 √2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 +∗ 𝜓𝑐, +(23.1) +𝜓𝑐 = +3 ( 𝑅 +𝑅𝑚) [1 − (𝑅𝑐0 +𝑅 ) +3 +] +𝛽 (5 +6 , 1 +2 , ( 𝑅 +𝑅𝑚) +3 +) − 𝛽 (5 +6 , 1 +2 , (𝑅𝑐0 +𝑅𝑚) +3 +) +. +(23.2) +Emerged 𝜓𝑐 is called collapse potential function considering the historical effect that +reflects the magnitude of bubble velocity applied by continuously accelerated. +It is further proposed that the function 𝜓𝑐 can be simplified by constraining the +bubble radius 𝑅 above the initial radius, where the last moment at 𝑅𝑐0 is actually at +least one magnitude less than the initial state. This yield: + + + +10 +𝜓𝑐 ≅ +3 ( 𝑅 +𝑅𝑚) +𝛽 (5 +6 , 1 +2 , ( 𝑅 +𝑅𝑚) +3 +) +. +(24) +Based on analysis above, the final form of NDCM is given below by replacing +Eqn.15.1 and Eqn.23.1 into Eqn.3: +𝑚̇ 𝑣 = 𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 𝜓𝑣√2 +3 +𝑝𝑠𝑎𝑡 − 𝑝̅ +𝜌𝑙 + (𝑝𝑠𝑎𝑡 > 𝑝̅), +(25.1) +𝑚̇ 𝑐 = −𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 𝜓𝑐√2 +3 +𝑝̅ − 𝑝𝑠𝑎𝑡 +𝜌𝑙 + (𝑝𝑠𝑎𝑡 < 𝑝̅), +(25.2) +Here, the dimensionless functions of growth 𝜓𝑣 and collapse 𝜓𝑐 are given by +Eqns.16 and 24, respectively, their variations against bubble radius are plotted in Fig. +3. Noting that the smaller bubble has larger value during both dynamic processes. For +growth period, tiny bubble possesses long expansion time ( 𝜏𝑣 ) which is cubic +proportional to volume change that leads to strong growth intensity. As for collapsing, +the bubble speed at the direction of inward radial is accelerated gradually so that the +intensity increases. + +Fig. 3 The graphs of potential functions, 𝜓𝑣 and 𝜓𝑐 + +Apparently, only two parameters with significant physical meaning of cavitating +bubbles, namely the Blake radius 𝑅𝑏 and the average maximum radius 𝑅𝑚 , are +employed to regulate the model application. The empirical coefficients are substituted +by the nonuniform potential functions, 𝜓𝑣 and 𝜓𝑐, considered the nonlinear effects +for both growth and collapse periods. Comparing with the Schnerr-Sauer model given +in the following equations, +𝑚̇ 𝑣𝑆 = 𝐶𝑣 ∗ 3𝜌𝑙𝜌𝑣 +𝜌 +𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 √2(𝑝𝑠𝑎𝑡 − 𝑝̅) +3𝜌𝑙 + (𝑝̅ < 𝑝𝑠𝑎𝑡), +(26.1) +𝑚̇ 𝑐𝑆 = −𝐶𝑐 ∗ 3𝜌𝑙𝜌𝑣 +𝜌 +𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 √2(𝑝̅ − 𝑝𝑠𝑎𝑡) +3𝜌𝑙 + (𝑝̅ > 𝑝𝑠𝑎𝑡). +(26.2) + +10000 +1000 +Potential Function +100 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +NondimensionalRadius/(R/RR/R.) + +11 +the difference is, except the additional potential functions, that the density factor in +front can be approximated as +𝜌𝑙𝜌𝑣 +𝜌 𝛼𝑙 ≈ 𝜌𝑣 based on 𝜌𝑙 ≫ 𝜌𝑣. The contribution of 𝛼𝑙 +vanishes that indicates both source terms are only proportional to (1 − 𝛼𝑙) but not +𝛼𝑙(1 − 𝛼𝑙). +Moreover, the Schnerr-Sauer model can be reformulated as the same form of NDCM. +In view of the symmetrical feature, we only discuss the collapse term as follow: +𝑚̇ 𝑐𝑆 = −𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 𝜓𝑐𝑆√2(𝑝̅ − 𝑝𝑠𝑎𝑡) +3𝜌𝑙 + (𝑝̅ > 𝑝𝑠𝑎𝑡). +(27.1) +where 𝜓𝑐 +𝑆 ≅ 3𝐶𝑐 𝛼𝑙 +⁄ + . Using the Eqn. 2 to substitute 𝛼𝑙 , we can get the equivalent +collapse potential function of Schnerr model: +𝜓𝑐𝑆 = 3𝐶𝑐 (1 + 4 +3 𝜋𝑛𝑅3) +(27.2) +In contrast to 𝜓𝑐 which is inversely proportional to 𝑅 , 𝜓𝑐 +𝑆 presents positive +relation that will conclude a confusing interpretation that larger bubbles have stronger +intensity during collapse. The comparison of 𝜓𝑐 and 𝜓𝑐 +𝑆 will be discussed in section +4.1. +3. Mathematical formulations and numerical method +3.1 Governing equations +The methodology in OpenFOAM for describing the two-phase system of cavitating +flow is based on Eulerian’s homogenous mixture approach, in which both phases are +treated as incompressible, isothermal, and immiscible. It is simple and efficient to +employ the one-field formulation (OFF) of Navier-Stokes equations that the properties +of two phases, including density and viscosity, are hybrid as equivalent single-phase +flow. The filtered governing equations including transport equation of liquid volume +fraction and momentum conservation of effective fluid are given in Eqn. 28 and 29 +respectively, and the velocity divergence in Eqn. 30 comes from summing over the +volume fraction equations of liquid and vapor that enables to build the pressure +equation to update the flux field. More details can be referred to Fleckenstein [6] for a +rigorous derivation: +𝜕𝛼𝑙 +𝜕𝑡 + 𝑼 ∙ ∇𝛼𝑙 = +𝜌 +𝜌𝑙𝜌𝑣 +(|𝑚̇ 𝑐| − |𝑚̇ 𝑣|), +(28) +𝜕 +𝜕𝑡 (𝜌𝑼) + ∇ ∙ (𝜌𝑼𝑼) = −∇𝑝𝑟𝑔ℎ + ∇ ∙ [𝜇(∇𝑼 + ∇𝑇𝑼) − 𝝉𝑇] − 𝒈 ∙ 𝒉∇𝜌, +(29) +∇ ∙ 𝑼 = ( 1 +𝜌𝑙 +− 1 +𝜌𝑣 +) (|𝑚̇ 𝑐| − |𝑚̇ 𝑣|). +(30) +Here, two phases are assumed to share the same velocity denoted as 𝑼. The hybrid +density 𝜌 and viscosity 𝜇 are weighted based on volume fraction 𝛼𝑙 linearly with +the constant properties of each phase, namely 𝜌𝑙, 𝜌𝑣, 𝜇𝑙, 𝜇𝑣, which are: + + + +12 +𝜌 = 𝜌𝑙𝛼𝑙 + 𝜌𝑣𝛼𝑣, +(31.1) +𝜇 = 𝜇𝑙𝛼𝑙 + 𝜇𝑣𝛼𝑣. +(31.2) +Note that the pressure 𝑝𝑟𝑔ℎ is relative to hydrostatic pressure 𝜌𝒈 ∙ 𝒉 to avoid the +algorithmic trouble of artificial diffusion induced by height difference. The non-linear +stress 𝝉𝑇 is closed by RANS or LES turbulence models. In the present work, we +employed the k-omega-SST two equation model [40]. Thus, the complete framework +is composed of three parts which are multiphase model (Eqn.28, 29 and 30), turbulence +model (𝝉𝑇 in Eqn.29) and cavitation model (𝑚̇ 𝑐 and 𝑚̇ 𝑐 in Eqn.28). +3.2 Algorithms and discretization +The above governing equations are implemented into CFD code based on the solver +of interPhaseChangeFoam, in which two important algorithms are developed for +capturing the topological changes of bubble cluster and coupling the velocity-pressure +to prevent checkerboard distribution. +One of the mature algorithms to solve a convective-only transport equation is VoF- +based interface capturing method that a new high-resolution algebraic reconstruction +proposed by Weller [34] based on flux-corrected transport (FCT) is implemented in +OpenFOAM. For further sharpness, comparing to the traditional approaches of +compressive schemes like HRIC or CICSAM, the “counter-gradient” diffusion term is +joined into transport equation to compress the interface in the reverse direction of +volume fraction gradient that has good performance for 2D and 3D complex flows. It +should be noted that the interface of cavitation cavity refers to the boundary of vapor- +liquid mixture (0 < 𝛼𝑙 < 1) and pure liquid (𝛼𝑙 = 1) differently from the one between +water (𝛼𝑙 = 1) and vapor (𝛼𝑙 = 0) where the ambiguous region (0 < 𝛼𝑙 < 1) should +be contracted as sharper as possible. Therefore, it is improper to employ the +compression term which would bring unreasonable diffusion inside cavity. We +recommend to use the FCT-based numerical method of semi-implicit multi- +dimensional limiter for explicit solution (MULES) for better boundedness and +consistency. A detailed description of this algorithm can be found in [33]. +The SIMPLE/PISO algorithm on collocated grid is realized via the technique of +momentum interpolation proposed by Rhie-Chow [41] that the serrated pressure would +be eliminated by introducing the third pressure derivative in correction equation. +However, only an incomplete method is implemented in OpenFOAM for robustness. +The simulation results would be depending on time step size so that we suggest a +modified momentum interpolation (MMI) method referring to the work by Cubero [35, +36] to remove the drawback. The original momentum interpolation (OMI) in +OpenFOAM is given by: +𝑼𝑓 +∗ = 1 +𝑎𝑃 +𝑯∗ +̅̅̅̅̅̅̅ +− 1 +𝑎𝑃 +̅̅̅ +∇𝑝𝑟𝑔ℎ,𝑓 +∗ ++ 𝜖 ∗ 1 +𝑎𝑃 +̅̅̅ +𝑎𝑃 +𝑡 +̅̅̅(𝑼𝑓 +𝑛 − 𝑼𝑃 +𝑛 +̅̅̅̅), +(32) +that the operator of hat bar is linear interpolation from cell to face center, and the +superscript ∗ means the mid-iteration till convergence to 𝑛 + 1 . The discrete +coefficient 𝑎𝑃 can be decomposed into temporal 𝑎𝑃 +𝑡 and spacial 𝑎𝑃 +𝑠 dependent items, +and the vector 𝑯 represents the collection of neighbor points summation and other + + + +13 +sources. Noting that the last term, called Choi correction, is emerged in unsteady +problems that the flux difference of previous time step is used to correct the interpolated +velocity, where 𝜖 employed in OpenFOAM is the empirical factor less than unity to +prevent the correction value inducing instability. Though, theoretically, the Choi +correction would be vanished through several iterations, the time step is contained in +coefficient 𝑎𝑃 that leads to the convergence result still associating with time. To +remedy the problem, the specific value is defined as 𝑑𝑃 = 𝑎𝑃 +𝑡 𝑎𝑃 +𝑠 +⁄ +, thereby the Rhie- +Choi interpolation can be reformulated as: +𝑼𝑓 +∗ = +1 +1 + 𝑑𝑃 +1 +𝑎𝑃 +𝑠 𝑯∗ +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ +− +1 +1 + 𝑑𝑓 +1 +𝑎𝑓 +𝑠 ∇𝑝𝑟𝑔ℎ,𝑓 +∗ ++ ( +1 +1 + 𝑑𝑓 +𝑎𝑓 +𝑡 +𝑎𝑓 +𝑠 𝑼𝑓 +𝑛 − +1 +1 + 𝑑𝑃 +𝑎𝑃 +𝑡 +𝑎𝑃 +𝑠 𝑼𝑃 +𝑛 +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ +) . +(33) +Here, it is assumed that all the coefficients on face center are interpolated linearly by +cell values, and the approximate relations are introduced: +1 +1 + 𝑑𝑃 +1 +𝑎𝑃 +𝑠 𝑯∗ +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ +≈ +1 +1 + 𝑑𝑃 +̅̅̅̅ +1 +𝑎𝑃 +𝑠 𝑯∗ +̅̅̅̅̅̅̅ +, +(34.1) +1 +1 + 𝑑𝑃 +𝑎𝑃 +𝑡 +𝑎𝑃 +𝑠 𝑼𝑃 +𝑛 +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ +≈ +1 +1 + 𝑑𝑃 +̅̅̅̅ 𝑑𝑃𝑼𝑃 +𝑛 +̅̅̅̅̅̅̅. +(34.2) +Thus, the formulation of MMI can be derived by substituting Eqn34.1 and 34.2 into +Eqn.33: +𝑼𝑓 +∗ ≅ +1 +1 + 𝑑𝑃 +̅̅̅̅ ( 1 +𝑎𝑃 +𝑠 𝑯∗ +̅̅̅̅̅̅̅ +− 1 +𝑎𝑃 +𝑠 +̅̅̅ +∇𝑝𝑟𝑔ℎ,𝑓 +∗ +) + +1 +1 + 𝑑𝑃 +̅̅̅̅[𝑑𝑃 +̅̅̅̅𝑼𝑓 +𝑛 − 𝑑𝑃𝑼𝑃 +𝑛 +̅̅̅̅̅̅̅]. +(35) +To ensure stability of solving pressure equation, the cavitation source term in Eqn.30 +is handled as semi-implicit form based on the principal diagonal dominant as follows: +∇ ∙ [ +1 +1 + 𝑑𝑃 +̅̅̅̅ ( 1 +𝑎𝑃 +𝑠 𝑯∗ +̅̅̅̅̅̅̅ +− 1 +𝑎𝑃 +𝑠 +̅̅̅ +∇𝑝𝑟𝑔ℎ,𝑓 +∗ +) + +1 +1 + 𝑑𝑃 +̅̅̅̅ [𝑑𝑃 +̅̅̅̅𝑼𝑓 +𝑛 − 𝑑𝑃𝑼𝑃 +𝑛 +̅̅̅̅̅̅̅]] += ( 1 +𝜌𝑙 +− 1 +𝜌𝑣 +) 𝐹(𝛼𝑙)√ +2 +3𝜌𝑙(|𝑝𝑠𝑎𝑡 − 𝑝̅𝑛| + 0.001 ∗ 𝑝𝑠𝑎𝑡) [𝑝𝑟𝑔ℎ +∗ +− (𝑝𝑠𝑎𝑡 − 𝜌𝒈 ∙ 𝒉)], +(36.1) + +where 𝐹(𝛼𝑙) is 𝛼𝑙 dependent function: +𝐹(𝛼𝑙) = 𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 +𝑅 [𝜓𝑣 ∗ 𝑝𝑜𝑠(𝑝𝑠𝑎𝑡 − 𝑝̅) + 𝜓𝑐 ∗ 𝑛𝑒𝑔(𝑝𝑠𝑎𝑡 − 𝑝̅)]. +(36.2) +The Choi correction will disappear during PISO/SIMPLE loops, and the time step is +excluded out of 𝑎𝑃 +𝑠. We employ two cases of lid-driven cavity (single phase flow) and +2D bubble rising [42] (multiphase flow) to test the performance of different methods +shown in Fig. 4. It is seen that the results obtained by MMI are overlapped completely +to indicate the independence of time step (Fig. 4 (b) and (d)) for both situations. + + + +14 + + +(a) Lid-driven cavity (OMI) +(b) Lid-driven cavity (MMI) + + +(c) Rising bubble (OMI) +(d) Rising bubble (MMI) +Fig. 4 Comparison of the velocity profiles between OMI and MMI against different +time steps +3.3 Numerical configuration +The strategy of velocity-pressure coupling is designed dual loops as called PIMPLE +which have inner PISO and outer SIMPLE for accommodating large time step size [28]. +The TVD type high order resolution schemes are used for the convective terms, and the +second order central difference scheme is used for the diffusion terms. The first-order +implicit Euler scheme is used for the transient terms. +4. Validation cases +In this section, the main purpose is to validate the cavitation model (NDCM) +developed in this work through three fundamental cases. For lack of exact solution +about cavitating flows, the first case is to simulate the collapse process of vapor bubble +cluster to illustrate differences between linear and nonlinear models. The real bubble +cluster excited by ultrasonic field is then investigated to reveal physical connotation of +model parameters through a series of simulations for different experimental conditions. +The new model is finally applied to the convective bubble cloud in hydrodynamic + +1.0 +dT=0.1 +dT=0.01 +0.5- +dT=0.001 +dT=0.0001 +0.0- +3 +-0.5 +-1.0- +-1.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x/L1.0 +dT=0.1 +dT=0.01 +0.5- +dT=0.001 +-dT=0.0001 +0.0- +3 +-0.5- +-1.0- +-1.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +X/LdT=5e-3 +dT=5e-4 +0.20- +dT=5e-5 +Rising velocity / [m/s] +0.15 +0.185- +0.180- +0.10 +8.175- +6.175- +0.185 +0.160 +0.05- +8.155 +0.00- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time / [s]dT=5e-3 +dT=5e-4 +0.20- +dT=5e-5 +Rising velocity / [m/s] +0.15 +0.10 +0.05- +0.00- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time / [s] + +15 +cavitation of slender bodies with conical or blunt nose to further highlight the capability +of NDCM. +4.1 Bubble cluster collapse +The necessity of considering the nonlinear effects represented by the potential +function of 𝜓𝑐 during bubble collapse can be demonstrated clearly by this validation +case. Here, we recommend to use the simulation case by Schmidt [29] who developed +a thermodynamic equilibrium model that the interface can be resolved implicitly when +the grid resolution is sufficiently fine so as comparable to DNS. As is shown in Fig. +5(a), the bubble cluster above the wall covers a spherical domain with a diameter of +𝑟𝑏 = 30mm within which 𝑁 = 150 spherical bubbles of equal radii 𝑟0 distributed +from dense center to sparse border are randomly generated. All bubbles are filled with +water vapor while the surrounding domain contains liquid water at an initial pressure +of 𝑝∞ = 100𝑏𝑎𝑟. The initial pressure inside the bubbles is equal to the vapor pressure +𝑝𝑠𝑎𝑡 = 2340𝑃𝑎. The velocity field is initially at rest. + + +(a) +(b) +Fig. 5 Distribution of bubble cluster (a), lateral section of computational mesh (b) + +The simulation is carried out for bubbles with the radius 𝑟0 of 1mm and 1.5mm by +NDCM and Schnerr-Sauer models. The lateral section of 3D hexahedral grid is shown +in Fig. 5(b), three mesh resolutions for the spherical domain containing under-resolved +bubbles are tested by the nodes of 𝑎 × 𝑏 × 𝑐. We introduce a factor 𝜆 that is defined +as the ratio between bubble radius (𝑟0) and the numerical resolution scale (√𝑉̅ +3 +), where +(√𝑉̅ +3 +) is the cubic root of cell-weighted average volume in bubble cluster. The model +parameters and initial volume fraction 𝛼𝑙 can be calculated from: +𝛼𝑙 = 1 − 𝑁 ∗ (𝑟0 +𝑟𝑏 +) +3 +, +(37.1) +𝑛 = 1 − 𝛼𝑙 +𝛼𝑙 +1 +4 +3 𝜋𝑟0 +3 , +(37.2) +that the setup details are shown in the Table 1. According to the conclusions from the +reference paper, the bubble distribution influences the pressure field significantly but +trivial for the collapse period, so we initialize the uniform field for 𝛼𝑙 to quantitatively + +35 +20N +15 +20 +25 +25 +20 +30 +30 +X +3535alpha.water +8.5e-010.880.90.920.940.96 +1.0e+00 + +16 +investigate the main collapse process whereas only compare the pressure results +qualitatively. The numerical pressure transducer is used consistently with [29] that the +sampling frequency is 49.3MHz on the wall of 15 × 15𝑚𝑚2 directly beneath the +bubble cluster. + +Table 1 Initializations for the vapor bubble cluster + +𝑟0 = 1.0 [𝑚𝑚] +𝑟0 = 1.5 [𝑚𝑚] +Liquid volume fraction +𝛼𝑙 [−] +0.95556 +0.85 +Bubble number density +𝑛 [1/𝑚3] +11103833 +12482740 +Resolution +ratio + 𝜆 = 𝑟0 +√𝑉̅ +3 + [−] +𝑀𝑒𝑠ℎ 1 (6.5k cells) +0.386 +0.579 +𝑀𝑒𝑠ℎ 2 (16k cells) +0.521 +0.782 +𝑀𝑒𝑠ℎ 3 (26k cells) +0.613 +0.919 + +The validation for grid independence is shown in Fig. 6, that all the variables are +nondimensionalized by the equivalent radius 𝑅𝑒𝑞𝑛 and its Rayleigh time 𝜏. It can be +seen that the collapse time calculated by NDCM (solid lines) is converged on Mesh 2 +and 3 for both radii. Recalling the assumption III which is indicated that the bubble +interactions could be ignored above the resolution of Mesh 2. Differently, the results by +Schnerr model (symbol-solid lines) given the value of 𝐶𝑐 as unity are overlapped +together which duration is longer than the former. + +𝑟0 = 1.0 [𝑚𝑚] + +𝑟0 = 1.5 [𝑚𝑚] +Fig. 6 The grid independence verifications for different meshes + +Fig. 7 shows model comparisons on Mesh 2. The collapse time in reference paper is +taken as benchmarks which accounts for 60% and 65% of Rayleigh time 𝜏 of the +radius of 1.0mm and 1.5mm respectively. The collapse time interval by NDCM agrees +well with the benchmarks although the rate of change has discrepancy which is likely +due to employ the uniform distribution of 𝛼𝑙 inconsistently with the original method. +However, it is deviated widely by Schnerr model with 𝐶𝑐 = 1, renamed as Schnerr-Cc + +1.0 +NDCM-Mesh1 +NDCM-Mesh2 +0.8- +NDCM-Mesh3 +Schnerr-Mesh1 +Schnerr-Mesh2 +0.6 +Schnerr-Mesh3 +uba +R/R +0.4 - +0.2- +0.0- +0.0 +0.5 +1.0 +1.5 +t/t1.0 +NDCM-Mesh1 +NDCM-Mesh2 +0.8- +NDCM-Mesh3 +.Schnerr-Mesh1 +Schnerr-Mesh2 +0.6 +---Schnerr-Mesh3 +uba +R/R. +0.4 - +0.2- +0.0- +0.0 +0.5 +1.0 +1.5 +t/t + +17 +as follows, which has insufficient intensity of source term that induces bubbles hard to +collapse especially in the last stage. In our opinion, the problem is caused by model +error that different potential functions of Eqns. 24 and 27.2 depict contradictory trend +against bubble radius. It can be considered that there is a collapsing shell at the border +of bubble cloud based on the fact that its character of collapse processes is layer-by- +layer. The internal bubbles are normally larger than the external along the shell radial. +The potential function 𝜓𝑐 +𝑆 in Schnerr model gives a positive correlation with bubble +radius that may induce the high intensity of source term to be emerged inside bubble +cloud. Thus, the external tiny bubbles are constrained by weak sources that leads to +difficult collapse. Even though 𝜓𝑐 +𝑆 is incompatible with physical actuality, the Schnerr +model can still be used by changing the coefficient 𝐶𝑐. Ghahramani et al. [43] studied +the Schnerr approach to simulate bubble cluster collapse, and they employed a high 𝐶𝑐 +to obtain better results of collapse period but emerged huge numerical pressure wiggles +unexpectedly. We also tried larger values 𝐶𝑐 of 800 and 1200 for both cases that make +the collapse period matched well with the benchmarks. As is shown in Fig. 8, highest +pressure pulse occurs at the last process of collapse. However, only the result by +Schnerr-1 has smooth curve whereas others do not. Comparison of the results by +NDCM (red line) and Schnerr with large 𝐶𝑐 (blue line), we can see that the new model +can suppress most extents of spurious pressure wiggles. Noting that the only difference +in the contrast is the potential functions that indicate the physics implied by 𝜓𝑐 is more +reasonable than 𝜓𝑐 +𝑆. Comparison of the influence by different meshes (red and green +lines) of NDCM results show that finer resolution contributes to control unphysical +oscillations, particularly in 𝑟0 = 1.5. It can be found that the pressure is more sensitive +to mesh resolution although the collapse period has been converged. + + +(a) 𝑟0 = 1.0 [𝑚𝑚] + +(b) 𝑟0 = 1.5 [𝑚𝑚] +Fig. 7 Comparison of time history of bubble cluster collapse + + +1.0 +RefbySchmidt +NDCM,Mesh2 +0.8 +Schnerr,Cc=1,Mesh2 +Schnerr.Cc=800,Mesh2 +0.6- +uba, +R/R +0.4- +0.2- +0.0- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t/t1.0 +RefbySchmidt +NDCM,Mesh2 +0.8- +Schnerr,Cc=1,Mesh2 +Schnerr,Cc=1200,Mesh2 +0.6- +eqn +R/R +0.4- +0.2- +-0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t/t + +18 + +(a) 𝑟0 = 1.0 [𝑚𝑚] + +(b) 𝑟0 = 1.5 [𝑚𝑚] +Fig. 8 Comparison of time history of pressure transducer on the wall + +For further comparison and understanding the differences between two models, the +bubble cloud structure, pressure contours and streamlines for the case of 𝑟0 = 1𝑚𝑚 +at different time instances are shown in Fig. 9. The behaviour of bubble cluster collapse +is seen in the left sides that the variation of cloud radius is tardy from 𝑡 = 0 to 𝑡 = +0.618𝑇 , subsequently dramatic collapse occurs during the rest of times. As is +mentioned in assumption III, interactions inside bubble cloud can be reflected in the +flux transfer between neighbor cells even if the local interactions are ignored. The +motivation is that the velocity doesn’t be divergence free (Eqn.30) that indicating the +velocity field will be affected by cavitation source terms which inverse gradient will +generate streamlines in the same direction. As is seen from left sides in Fig. 9(b)~(d), +the structures of velocity field of NDCM and Schnerr-1 are quite similar. The high- +speed collapsing shell is driven by appropriate source term where large values distribute +at the outside, but the low-speed internal region almost immunes to collapse until the +last period at 𝑡 = 0.927𝑇. However, the situation in Schnerr-800 is that unreasonable +inverse flow is formed due to the higher source values emerge inside that sometimes +induces the internal bubbles collapse priorly instead of the external (red dash circle in +Fig. 9(b. II)). The temporal evolution of bubble radius distribution in the direction of +𝜌𝑟 is shown in Fig. 10. The serrated line marked by the dash circle at 𝑡 = 0.618𝑇 +illustrates the bubble collapsing sequence is from inside-out. Moreover, the innermost +bubbles are affected prematurely from the boundary (dash arrow). +Apparently, the property of potential functions (𝜓𝑐 and 𝜓𝑐 +𝑆) is the main reason to +influence the source term distribution. Besides, the local pressure 𝑝̅ (in Eqns. 26.2 and +27.1) is another key factor that exhibits positive correlation to source terms, so that high +pressure around outer bubble cluster enhances local sources. It should be mentioned +that 𝑝̅ is compatible with 𝜓𝑐 that will further strengthen outer source values, but 𝜓𝑐 +𝑆 +weakens the effect by 𝑝̅. For the case of Schnerr-1, it can be inferred that 𝜓𝑐 +𝑆 is trivial +relative to pressure 𝑝̅ which dominates the source term over the whole collapse period, +thus large source values only appear externally to avoid inverse flows. However, such +distribution of source term is destroyed by employing large 𝐶𝑐 which is augmented +800 times. +It is seen from right sides in Fig. 9 that there are some local pressure pulses around +bubble cloud for contours of NDCM and Schnerr-800. These pulses are the reasons for + +2000 +NDCM, Mesh2 +NDCM,Mesh3 +Schnerr,Cc=1.Mesh2 +1500- +Schnerr,Cc=800,Mesh2 +Pressure / [bar] +1000- +500- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +tt2000 +NDCM,Mesh2 +NDCM.Mesh3 +Schnerr,Cc=1,Mesh2 +1500- +Schnerr,Cc=1200,Mesh2 +Pressure / [bar] +1000 +500- +0- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t/t + +19 +wiggles detected by the wall pressure transducer. Rossinelli et al. [44] has implemented +a two-phase flow DNS to simulate 15000 bubbles collapse, which found that the +pressure wiggles should be existed during temporal evolution. Thus, the smooth +pressure profile predicted by Schnerr-1 is unreasonable. In Fig. 9(b.III)~(e.III), the +bubble cloud is dispersed excessively due to insufficient intensity of cavitation sources +that the dissipated pressure gradient prevents local high pressure to happen. +Comparison of pressure field by two models in Fig. 9(d. I) and (d. II), the positions of +high pressure predicted by NDCM are located at the external shell from where to +infinity the values decrease monotonously to ambient pressure. However, the +corresponding points in Schnerr-800 invade into the bubble cloud where pressure pulse +is surrounded by an additional low-pressure band (red dash arrow). We suppose that the +misplacing pressure distribution is one of the reasons to cause the spurious pressure +pulses which can be eliminate effectively by using the derived potential function 𝜓𝑐 +in NDCM. + + + +(𝑎) 𝑡 = 0 +(𝑓) 𝑡 = 𝑇 + + + +(I) +(II) +(III) +(𝑏) 𝑡 = 0.515𝑇 + + + +(I) +(II) +(III) +(𝑐) 𝑡 = 0.618𝑇 + + + +20 + + + +(I) +(II) +(III) +(𝑑) 𝑡 = 0.755𝑇 + + + +(I) +(II) +(III) +(𝑒) 𝑡 = 0.927𝑇 + + + +Fig. 9 Comparison of NDCM (I), Schnerr with 𝐶𝑐 = 800 (II), and Schnerr with +𝐶𝑐 = 1 (III) in prediction of cloud structure (left side), pressure contours (right side) +and streamlines at different time instances + + +(a) NDCM + +(b) Schnerr (𝐶𝑐 = 800) +Fig. 10 Comparison of distribution of bubble radius in radial direction of bubble +cluster + + +5.0e+05 +5e+6 +1e+7 +1.5e+7 +2e+7 +2.5e+7 +3.0e+07 +Pressure: p [Pa]0.0e+000.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +4.0e-02 +Void fraction: αy0.0e+00 +2 +3 +4 +5 +6 +7 +8 +9 1.0e+01 +1 +Velocity: U [m/s]t=0.515T, +t=0.618T +t=0.755T, +t=0.927T +1.0 +LO +0.5- +L +0.0- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P, / rpt=0.515T, +t=0.618T +t=0.755T, +t=0.927T +1.0 +LO +0.5- +0.0- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p,/ rp + +21 +4.2 Ultrasonic horn +After exhibiting the basic expectation of NDCM, the discussion will be concentrated +on the real bubble cloud generated by the high frequency oscillating ultrasonic horn. It +has been observed that if the horn tip is sufficiently small and driven at high amplitude, +cavitation is very strong and the tip can be covered entirely by the gas/vapor phase for +longer time intervals. We employ the experiment designed by Žnidarčič et al. [30] who +investigated a systematic study in water at a 20 kHz with the horn diameter of 3 mm +under variation of driving power, air saturation, viscosity, surface tension and +temperature, that the attached cavity emerged peculiar dynamics with a self-generated +frequency of expansion and collapse periodically. After that, they [31] carried out the +correspondingly simulation studies and obtained poor predictions of flow features with +the original TEM method, i.e., Schnerr-Sauer model. In their opinion, the Schnerr- +Sauer-like model cannot adapt to the rapidly changing driving pressures, they presented +an improved approach which also considered the second derivative term of Rayleigh- +Plesset equation but in differential form. Good agreements comparing with +measurements were then revealed for cavity shape and its frequency. However, the +evolutionary tendency of the bubble cloud does not match well with experiment +especially in expansion process. In this section, the tasks are not only to demonstrate +better predictive results by applying the NDCM method but also the rules of parameter +regulation. Table 2 lists five experimental conditions for water at room temperature by +which comparison of simulations can validate the physical meaning of model +parameters. +Table 2 Four experimental conditions for ultrasonic horn +Case +Percentage of +max power [%] +Vibrating amplitude +𝐴ℎ [𝜇𝑚] +Saturation +[%] +A +70 +164 +100 +B +50 +C +20 +D +30 +100 +100 +4.2.1 Model parameters determination +Recalling that the model parameters, 𝑅𝑏 and 𝑅𝑚, are the particular points living on +the bubble dynamics curve, it is therefore demanded to determine the initial bubble +state to estimate their reasonable values. The theory of rectified mass diffusion which +is the mechanism of cavitation inception in acoustic field is adopted to describe the +formation of microbubbles from dissolved gas. Here, the bubble nuclei are formed from +microbubbles by gradually mass transfer, between which the difference of magnitude +order is commonly at one. The analytical model for air-water systems proposed by +Crum [45] that there exists a certain critical amplitude of acoustic pressure 𝑝̃𝑐 above +which the microbubbles will begin to grow by rectified diffusion. The expression is +given as follow: + + + +22 +𝑝̃𝑐 = 𝜌𝑅0 +2𝜔𝑁 +2 √ +[(1 − 𝜔2 +𝜔𝑁 +2) +2 ++ (𝑏𝜔 +𝜔𝑁) +2 +] (1 + 2𝜎 +𝑅0𝑝∞ − 𝑐𝑖 +𝑐0) +(3 + 4𝐾) (𝑐𝑖 +𝑐0) − [3(𝜂 − 1)(3𝜂 − 4) +4 ++ (4 − 3𝜂)𝐾](1 + 2𝜎 +𝑅0𝑝∞) +, +(38) +where the 𝑅0 is the microbubble radius, the 𝑐𝑖 and 𝑐0 are the concentration at +bubble interface and ambient liquid respectively, which ratio represents the gas +saturation, whereas other variables can be seen in [45]. Shown in Fig. 11 are the values +of Eqn. 38 for the driving frequency at 20kHz. It is indicated that large microbubbles +tend to grow easily on the same saturation curve, besides, it is difficult to form nuclei +for the same size of microbubble in degassed water. + +Fig. 11 The graphs of critical pressure 𝑝̃𝑐 against microbubble radius 𝑅0 + +As is obtained the one-to-one relation between acoustic pressure 𝑝̃ and microbubble +𝑅0 so we can get the size ranging of activated microbubbles under a given acoustic +field. Referring to the work by Mellow [46, 47], the approximate analytic solution of +acoustic field 𝑝̃ for the vibrating horn is available using the Green function method +which formulas indicate that the distribution of 𝑝̃ relates to three variables, the horn +radius 𝑎 , the vibrating frequency 𝑓 , and the vibrating amplitude 𝐴ℎ , but is only +proportional to 𝐴ℎ since the other two are fixed conditions in experiment. It is seen +that the acoustic fields in Case A, B and C are identical, and become weaker in Case C +and D sequentially for the vibrating amplitude reduced. +The diagrams in Fig. 12 illustrate the nondimensionalized analytic solution of +acoustic field +𝑝̃ +𝜌𝑙𝑐𝑢0 where the maximum value is located at the center of bottom wall +and the semi-ellipsoidal iso-surfaces monotonically decrease down to the far field. It +can be inferred from the background experimental pictures which represent the peak +bulk of cavity that the region surrounded by the red iso-surface enables to provide +sufficient driving force to develop the microbubbles evolving as cavitation nuclei. Thus, +in term of the theoretical values of acoustic pressure 𝑝̃ inside the region, the span of +qualified microbubbles, shown in Table 3, can be evaluated by Eqn. 38. According to +the conclusions in Fig. 11, the ranges of 𝑅0 display that the larger nuclei are appeared +in the more degassed water (Case A, B and C), likewise situations embodied in weaker +acoustic field (Case A and D). + +32 +c/c。= 1 +16 +c/c = 0.5 +Pressure amplitude / [atm] +8 +c/c。= 0.2 +4 +2 +1 +0.5 +0.25 +0.125 +0.0625 +- +0 +20 +40 +60 +80 +100 +Microbubbleradius/ [um] + +23 + + +Case A +Case B + + +Case C +Case D + +Fig. 12 The estimation of cavitation bubble production region + +Since the value scope of microbubbles 𝑅0 inside the effective region of acoustic +field has been acquired, we suggest the multiples about 6~8 of 𝑅0 to define the value +range for model parameter 𝑅𝑏. And the range of maximum radius 𝑅𝑀 can exploit the +numerical solution of Rayleigh-Plesset equation with the initial conditions of endpoint +values of 𝑝̃ and 𝑅0 that can estimate the parameter 𝑅𝑚 . For clarity, all these +specified values of five cases are listed in Table 3 as well. + +Table 3 Estimation of model parameters + +Case A +Case B +Case C +Case D +Pressure amplitude in +effective region 𝑝̃ [𝑏𝑎𝑟] +6.18~15.8 7.72~15.8 9.27~15.8 6.12~9.63 +Activated microbubbles +𝑅0 [𝜇𝑚] +0.36~0.60 +0.75~1.0 +4.5~6.0 +0.46~0.61 +Model parameters +of 𝑅𝑏 [𝜇𝑚] +Range +2.2~4.8 +4.5~8.0 +27~48 +2.8~4.9 +Value +3.5 +5.2 +35 +4.6 +Maximum growth radius +𝑅𝑀 [𝑚𝑚] +0.41~0.76 0.49~0.77 0.56~0.77 0.41~0.56 +Model parameters of +𝑅𝑚 [𝑚𝑚] +0.58 +0.62 +0.66 +0.48 +4.2.2 Simulation setup +As is shown in Fig. 13, we employ a 2D axisymmetric computational domain +consistently with the experiment with the dimensions of 40 mm height and 25 mm +radius, in which the horn tip of 3 mm diameter is placed vertically from top 30 mm +above the bottom. It is note that the near-wall grid is densified to ensure the y+ is lower + +0.0e+00 +0.02 +0.04 +0.06 +0.08 +1.0e-01 +Nondimensional acoustic pressure +picuo + +24 +than one. + +Fig. 13 Computational domain for ultrasonic horn +All the walls are used the no-slip velocity boundary condition and zero gradient for +pressure. The top of atmosphere is defined as fixed pressure at 1 atm. The horn vibration +in a sinusoidal manner at a frequency of 20 kHz, at various amplitudes, depending on +the power. To capture the movement a dynamic mesh approach was used that the mesh +must constantly be updated by laplacian smoothing and local remeshing. It was +determined that the mesh, due to very small deformation of the domain, preserves an +extremely low value of cell skew. +Three mesh densities were tested and it is found that it does not influence the outcome +of the calculation of cavitation dynamics, but the model parameters have to change +slightly for different resolutions. Consequently, the following results are calculated on +a medium grid with 23550 cells. +4.2.3 Results +A series of simulations by the new cavitation model are compared with experiments +including the volume evolution of the bubble cloud and the acoustic pressure probed +by hydrophone. Considering that the physics of these cavitating flow do not differ +significantly between Case A-D, we emphatically analyze the results of Case A whereas +others are given more briefly in data charts. +A sequence of the spatial structures of cavity beneath the tip of the horn are shown +in Fig. 14 graphically displayed as left simulation and right experiment. It is seen that +the mushroom-like shape cloud is formed rapidly during the interval of 0 to 40𝜇𝑠. +Then, the generated bubbles keep the dynamic balance from 60𝜇𝑠 to 100𝜇𝑠 in +which the maximum volume is achieved about 80𝜇𝑠. After that the cavity contracts at +the outer rim and the violent collapse happens at the end of cavitation period. A +comparison between the measured and predicted cavity volume is shown in Fig. 15(a). +It is evident that the simulation by NDCM accurately predicts the dynamics of the +cavity volume which cycle (5010Hz) is about a quarter of the driving frequency +(20kHz). Noting that the cavity frequency calculated by Schnerr-Sauer model can agree +with the experiment, the produced vapor volume is insufficient though. The typical + +atmosphere +horn +40mm +axis +30mm +-walls +25mm + +25 +period inside pink dash line is magnified shown in Fig. 15(b), and attached the results +obtained by Znidarcic’s model additionally (dash line). We can see that the tendency is +almost the same before 30𝜇𝑠, however the shrink of cavity in growth period predicted +by Znidarcic model mismatches with the measurements, moreover, the variation rate of +cavity volume shifts much faster than the experiment during collapse process. + + + + + + + + + + +Fig. 14 Typical cycle of the oscillation of a large cavity between simulation and +experiment at the driving frequency of 20kHz + +Some discrepancies should be pointed out that a pinch of bubbles at the tip of cavity + +O μs +Simulation +Experiment20μs +Simulation +Experiment40 μs +Simulation +Experiment60 μs +Simulation +Experiment80μus +Simulation +Experiment100μs +Simulation +Experiment120 μs +Simulation +Experiment140 μs +Simulation +Experiment180μs +Simulation +Experiment200μs +Simulation +Experiment + +26 +cannot capture perfectly possibly due to the 2D simulation method since the bubble +cloud has non-axisymmetric 3D structure shown in experiment. Besides, the convection +of corner bubbles which is likely induced by Bjerknes [48] force is unable to take into +account here limited by the model capability. + + + +(a) +(b) +Fig. 15 Comparison between the predicted and measured attached cavity volumes for +Case A + +A comparison of measured and simulated pressure evolutions at a distance of 7 mm +from the tip of the horn are shown in Fig. 16(a). It is indicated that high pressure pulse +is emitted at the last stage of cavity collapse. The peak pressure amplitude seems to be +slightly overpredicted which could be caused by the assumption of incompressibility +for both phases, and furthermore missing the isolated bubble structures (see the right +side in Fig. 14) where the cavitation model can only capture the main cavity bulks. Also +note that the high frequency components are smoothed out because of the insufficient +mesh resolution at the vicinity of probe and low order of temporal scheme under the +RANS simulation scenarios. Nevertheless, the periodicity of pressure peaks is correctly +predicted shown in Fig. 16(b) of power spectrum density (PSD). It is evident that the +primary and second frequency in PSD are identical with the cavity and the driving horn +respectively, which reflect main dynamic characteristics in the system. + + + +(a) +(b) +Fig. 16 Comparison between the predicted and measured acoustic pressure (a), PSD + +8 +Case A-NDCM +Schnerr-Sauer +Experiment +vaporvolume/ [mm' +o +4 +3 +N +0.0 +0.5 +1.0 +1.5 +2.0 +Time / [ms]9.0 +Experiment +CaseA-NDCM +7.5 - +- +RefbyAnton +6.0- +4.5 +3.0 +1.5 +0.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +Time / [ms]3.0 +CaseA-NDCM +2.5 +Experiment +Acoustic pressure/ [bar] +2.0 +1.5 +1.0 +0.5 +0.0 +-0.5- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time / [ms]30 +5010Hz +25- +20000Hz +20 +D +15 +PSD +10- +5 +0 +0 +100000 +200000 +300000 +400000 +500000 +frequency/ [Hz] + +27 +analysis for the obtained acoustic pressure in Case A (b) +As unsuccessful application of Schnerr-Sauer model is well illustrated, we will +discuss the rest of Cases B-D which model parameters are orderly defined in Table 3. +The typical cavity volume evolution for degassed water of Case B and C are shown in +Fig. 17(a) and Fig. 18(a) respectively, we can see that the peak volume are declining as +the gas content decreased (about 7~8𝑚𝑚3 for Case A, 6~7𝑚𝑚3 for Case B, and +4~5𝑚𝑚3 for Case C), meanwhile accompany with increasing unsteadiness of the +cavitation dynamics. This character can also be observed in acoustic pressure variations +(Fig. 17 (b) and Fig. 18(b)) which amplitude wiggles violently by degassed extent. +There exist complex bubble interactions inferred from Fig. 18(b) (Case C) that the +pressure pulse radiates more frequently than the previous two where the calculated +pressure reflects these frequency components qualitatively as well. However, the +periodicity of the cavity does not damage which is slightly increased (5010Hz for Case +A, 5079Hz for Case B, and 5259Hz for Case C) for more additional high frequency +emerged. Comparing with Cases A, the peak volume is significantly reduced in Case D +shown in Fig. 19(a) which is caused by the weak acoustic field by the low driving power. +Differently the cyclic evolution period of cavity (6517Hz) is raised to about one-third +of driving frequency. +It is seen that the NDCM is capable to predict the cavity dynamics accurately. +Moreover, the proposed model parameters, 𝑅𝑏 and 𝑅𝑚, are physically based where +good agreements can be obtained by setting reasonable values. + + +(a) + + + +8 +Experiment +CaseB-NDCM +6 +vaporvolume/ [mm +5 +3 +2 +0 +0.0 +0.5 +1.0 +1.5 +2.0 +Time/ [ms]3.0 +CaseB-NDCM +2.5 +Experiment +Acoustic pressure / [bar] +2.0 +1.5 +1.0 +0.5 +0.0 +0.5- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time / [ms]35 +5079Hz +30- +25 +20000Hz +20 - +8 +P15- +10- +5- +0- +0 +100000 +200000 +300000 +400000 +500000 +frequency/ [Hz] + +28 +(b) +(c) +Fig. 17 Comparison between the predicted results and measured data for Case B (70% +max power, 50% saturated) + + +(a) + + +(b) +(c) +Fig. 18 Comparison between the predicted results and measured data for Case C (70% +max power, 20% saturated) + + +(a) + +Experiment +6 +CaseC-NDCM +5 +vaporvolume/[mm +3- +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +Time/ [ms]3.0 +CaseC-NDCM +2.5- +Experiment +2.0- +[bar] +1.5 +Acousticpressure / +1.0 +0.5 +0.0 +-0.5 +1 +-1.0- +-1.5 +0.0 +0.6 +1.2 +1.8 +2.4 +3.0 +Time/ [ms]25 +20000Hz +20 - +15- +S +P +10- +5259Hz +5- +0- +0 +100000 +200000 +300000 +400000 +500000 +freguency/[Hz]Experiment +CaseD-NDCM +3 +2 +- +1 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Time/[ms] + +29 + + +(b) +(c) +Fig. 19 Comparison between the predicted results and measured data for Case D (30% +max power, 100% saturated) +4.3 Slender body +In this section, we extend the application for NDCM on the hydrodynamic cavitation +flow which contains the structures of dispersed bubbly cloud. The natural cavitation +experiments [32] for axisymmetric bodies with blunt and conical heads are employed +for elementary investigation, also to compare with the results by Merkle’s model in the +article. +The 2D axisymmetric computational domains are adopted shown in Fig. 20. The +diameter of the slender body is 20 mm and the length 7.5d. The domain extension has +15d upstream and 20d downstream. The inlet velocity is 6.8m/s and fixed outlet +pressure based on the cavitation number 𝜎 with 0.5 and 0.3. The slender walls are +specified as no-slip boundary and the outer ring is set as slip wall. The boundary layer +grid is generated to ensure y+ less than 1. + +Fig. 20 Computational domain for slender bodies with conical and blunt heads + +In order to determine the model parameters, it only needs to select the middle mesh +with the quantity of 37700 cells and 45400 cells for conical and blunt heads respectively +between three resolutions which the simulation results are close. For the lack of +information to estimate the size of bubble nuclei, the approximate values of 𝑅𝑏 and + +2.0 +CaseD-NDCM +Experiment +1.6- +1.2 +0.8 +0.4 +0.0 +0.4 +-0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Time / [ms]15 +6517Hz +12- +20000Hz +9- +PSD +-9 +3- +0 +0 +100000 +200000 +300000 +400000 +500000 +frequency/ [Hz]42.5d +slipwall +outlet +15d +inlet +7.5d +axis +axis +10mm +10mm + +30 +𝑅𝑚 for both heads listed in Table 4 are guessed from several trial calculations. We +suppose that these values are reasonable because the cavitation region has higher +negative pressure and wider area as dropping the cavitation number, which enables +smaller nuclei to grow (𝑅𝑏) and more residence time leads to bubbles expand larger +(𝑅𝑚). +Table 4 Model parameters of NDCM model + +𝜎 = 0.5 +𝜎 = 0.3 +Blunt +𝑅𝑏 = 32 𝜇𝑚 +𝑅𝑚 = 0.53 𝑚𝑚 +𝑅𝑏 = 25 𝜇𝑚 +𝑅𝑚 = 0.62 𝑚𝑚 +Conical +𝑅𝑏 = 28𝜇𝑚 +𝑅𝑚 = 0.43𝑚𝑚 +𝑅𝑏 = 18𝜇𝑚 +𝑅𝑚 = 0.52𝑚𝑚 + +The pressure distributions for blunt head with cavitation number of 0.3 and 0.5 are +shown in Fig. 21(a). We can see that the results by NDCM achieves good agreements +especially for 𝜎 = 0.5 that the cavity length predicted by new model (solid line) is +almost identical with the experiment but underestimated in Merkle’s (dash line). The +insufficient length of cavity predicted by Merkle’s model is embodied notably in +conical head shown in Fig. 21(b) whereas the NDCM presents a reasonable prediction +matching with experiments. +However, some discrepancies still exist limited by 2D geometry that the recovery +pressure at the cavity tail is a bit lower than experiment at 𝜎 = 0.5 and the cavity +length is little overpredicted for blunt at 𝜎 = 0.3 . It is more likely to use the 3D +simulation to capture the asymmetric structures of bubble cloud which are shedding +periodically that can obtain better results. + + +(a) blunt +(b) conical + +Fig. 21 Comparison of pressure distribution for blunt and conical head body +5. Conclusions +In this study, a novel nonlinear dynamic cavitation model (NDCM) is proposed +against the bubble cluster structure through strictly mathematical derivation. Firstly, the +four thorough assumptions for TFM-type models are explicated that the filtered bubbles +are mapped from physical to computational space. Then, we introduced the integral + +1.2 +Sim,NDCM,=0.3 +1.0 +Sim,NDCM,=0.5 +0.8 +Ref,Merkle,=0.3 +Ref, Merkle, =0.5 +0.6 +Exp.=0.3 +0.4 +Exp,α=0.5 +8 +0.2 +0.0 +-0.2 +-0.4 +0.6 +0 +2 +3 +4 +5 +p/X1.2 +Sim.NDCM.g-0.3 +1.0 +Sim,NDCM,=0.5 +0.8 +Ref,Merkle,=0.3 +Ref, Merkle, α=0.5 +0.6 +Exp,=0.3 +0.4 +Exp,α=0.5 +8 +0.2- +0.0 +-0.2 +-0.4 +0.6 +0 +2 +3 +4 +5 +p/X + +31 +average method to calculate the time derivative term +𝑑 +𝑑𝑡 ( +4 +3 𝜋𝑛𝑅3) that the second +derivative in Rayleigh-Plesset equation can be considered in the characteristic time +during growth and collapse, namely 𝜏𝑣 and 𝜏𝑐, solved analytically. Consequently, two +additional potential functions 𝜓𝑣 and 𝜓𝑐 emerge in model formula which represent +the nonlinear effects in cavity dynamics. In addition, without any empirical coefficients, +there are merely two parameters with definitude physical meaning in which 𝑅𝑏 and +𝑅𝑚 indicate the Blake critical radius and the average maximum growth radius, +respectively. +In order to validate the performance of the new model, three simulation cases, from +simple to complex, were employed including the collapse of numerical bubble cluster, +periodic generation and collapse of real bubble cloud in ultrasonic horn experiment, +and hydrodynamic cavitation of slender body. +For the first case, the results showed that the collapse time of NDCM and benchmark +agreed well except the speed rate which may cause by different initialization method. +The layer-by-layer collapse character and pressure shock at last stage were revealed +correctly. On the contrary, the Schnerr-Sauer model with parameter 𝐶𝑐 = 1 +overpredicted the collapse time because of insufficient intensity of source term. More +importantly, the potential function 𝜓𝑐 +𝑆 implied in Schnerr model gives a positive +relation with bubble radius 𝑅𝑏 that contradicts with 𝜓𝑐 in NDCM. Although the +collapse time can be remedied by employing large coefficient 𝐶𝑐, the model errors were +also augmented that brought great numerical pressure wiggles and incorrect collapse +processes. Most of spurious pressure can be suppressed by applying NDCM, but the +remaining components should be further studied in future. +The NDCM was then applied to the real bubble cloud generated in acoustic field. +The main purpose was to confirm the physical relevance of 𝑅𝑏 and 𝑅𝑚 . Four +experimental conditions were adopted that the theoretical value ranges of model +parameters were determined based on the rectified diffusion theory. It was found that +the variation of those well-matched simulation results was in accord with the laws of +specified model parameters, and more sensitive to 𝑅𝑏 which should be given larger +values in more degassed water or weaker acoustic field. A detailed comparison of Case +A against the results from Znidarcic showed that the second derivative term in +Rayleigh-Plesset equation considered in integral form rather than differential can +provide better predictions. +Finally, the new model was extended to hydrodynamic cavitation of convective +dominated flow that the slender bodies with two heads of conical and blunt were +simulated under the cavitation number of 0.3 and 0.5. The good agreements of cavity +length and pressure distribution further indicated that the NDCM is applicable for the +cavitation cavity with dispersed bubble structures. + + + +32 +Acknowledgement +This work is supported by National Key Project GJXM92579 and National Sci. & +Tech. 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Bensow, A comparative study between +numerical methods in simulation of cavitating bubbles. International Journal of +Multiphase Flow, 2019. 111: p. 339-359. +44. +D. Rossinelli, B. Hejazialhosseini, P. Hadjidoukas, C. Bekas, A. Curioni, A. +Bertsch, S. Futral, S.J. Schmidt, N.A. Adams, P. Koumoutsakos, 11 PFLOP/s +simulations of cloud cavitation collapse, in Proceedings of the International +Conference on High Performance Computing, Networking, Storage and +Analysis. 2013, Association for Computing Machinery: Denver, Colorado. p. +Article 3. +45. +L.A. Crum, Acoustic cavitation series: Part five rectified diffusion. Ultrasonics, +1984. 22(5): p. 215-223. + + + +35 +46. +T. Mellow, On the sound field of a resilient disk in free space. Journal of the +Acoustical Society of America, 2008. 123(4): p. 1880-1891. +47. +T. Mellow, L. Kärkkäinen, On the sound field of an oscillating disk in a finite +open and closed circular baffle. 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The Journal of the +Acoustical Society of America, 1949. 21(5): p. 551-551. + + diff --git a/n9E1T4oBgHgl3EQfOgPu/content/tmp_files/load_file.txt b/n9E1T4oBgHgl3EQfOgPu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..826b3449ad45f96192bc06f01c4b6c5ec241ef6c --- /dev/null +++ b/n9E1T4oBgHgl3EQfOgPu/content/tmp_files/load_file.txt @@ -0,0 +1,1406 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf,len=1405 +page_content='1 Development of a novel nonlinear dynamic cavitation model and its numerical validations Haidong Yu12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Xiaobo Quan3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Haipeng Wei1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Matevž Dular4 and Song Fu2* 1Beijing Institute of Astronautical System Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 100076,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' China 2School of Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' China 3China Academy of Launch Vehicle Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 100076,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' China 4Laboratory for Water and Turbine Machines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' University of Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Aškerčeva 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1000 Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Slovenia Abstract Aiming at modeling the cavitation bubble cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' we propose a novel nonlinear dynamic cavitation model (NDCM) considering the second derivative term in Rayleigh-Plesset equation through strict mathematical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' There are two improvements of the new model: i) the empirical coefficients are eliminated by introduction of the nonuniform potential functions of 𝜓𝑣 and 𝜓𝑐 for growth and collapse processes respectively, and ii) only two model parameters are required, which both base on physical quantities – the Blake critical radius 𝑅𝑏 and the average maximum growth radius 𝑅𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The corresponding cavitation solver was developed by using OpenFOAM in which we implemented the modified momentum interpolation (MMI) method to ensure that the calculated results are independent of time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Three validation cases, namely numerical bubble cluster collapse, ultrasonic horn experiment, and hydrodynamic cavitation around slender body are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The results indicate that 𝜓𝑣 and 𝜓𝑐 can reveal the nonlinear characteristics for cavity accurately, and 𝑅𝑏 and 𝑅𝑚 can reflect the relevance between cavitation model and actual physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Moreover, it is discussed the potentiality of NDCM that is generally applied on the cavitating flow possessing with dispersed bubbly cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' keywords: Cavitation model, Rayleigh-Plesset equation, Bubble cluster collapse, OpenFOAM Corresponding author, email: fs-dem@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='cn 2 Highlights ➢ A nonlinear dynamic cavitation model (NDCM) is established through strict mathematical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' ➢ The formula of NDCM employs the potential functions of 𝜓𝑣 and 𝜓𝑐 instead on empirical coefficients to describe the nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' ➢ The model parameters of 𝑅𝑏 and 𝑅𝑚 represent the physical characteristics of bubble cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' ➢ The NDCM is best valid for the cavitating flows with dispersed bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Introduction Cavitation often occurs in liquid flows when the ambient pressure drops below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The cavitating bubbles will emerge gradually from “cracked” liquid medium at its weak points [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Individual bubbles cluster and form a complex two- phase mixture cloud, which shape depends strongly on the structure of the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The cavitation bubble cluster exhibits many unique characteristics, such as strong collapse accompanied by a shockwave, or the natural frequency far lower than single bubble’s etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [2, 3] There are many researches on single bubble dynamics which can be described by Rayleigh-Plesset type equation [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, an approach to build cavitation model based on Rayleigh-Plesset equation to investigate the dynamics of bubble cluster can be considered questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Numerical simulation of cavitating flows and specifically the development of transport equation model (TEM) has received enormous attention from investigators in recently years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=" Instead of potential flow theory implemented in early engineering applications, the Eulerian's one field formulation (OFM) of the two-phase Navier– Stokes equation [6-9], which combines the properties of each phase as a single mixed one, is popularly applied as the methodology of multiphase model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The cavitation model for bubble cluster is embedded into the convective phase equation as source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' There is also an alternative approach based on Eulerian–Lagrangian method [10, 11] but is not within the present framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The prototype of TEM was introduced by Kubota [12] who assumed that the bubble nuclei are uniformly distributed in the flow, and the simplified Rayleigh-Plesset equation, which considered the SGS bubble interaction was used to determine the change of radius and consequently the mixture density in each computational cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The advantage of this approach is that the dynamic response of local equilibrium bubbles can be estimated precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, solving the nonlinear transport equation often faces the difficulty of convergence and its application was merely limited in studying steady flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In order to develop the Kubota’s method, the modeling strategy was focused on larger scale of bubble cloud rather than the tiny scale of individual bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' A different 3 approach with the same form of transport equation models had already been proposed in literature [6-9], in which the nonlinear ODE is replaced by a convective equation of void fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The mass transfer rate of bubbles is contained implicitly into source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It has the advantage that the convective character of equation is more appropriate for describing the topological evolution of bubble cluster particularly in unsteady situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In the literature, there are two main approaches to build the empirical phase-transition laws associating with surrounding pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Merkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [13-15] proposed simplest formulation based on dimensional analysis that defined the characteristic velocity 𝑢∞ or temporal scale 𝑡∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, these parameters are prescribed as a constant during the whole dynamic period, especially in collapsing process, which would cause large deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In order to make the model more physical, the void fraction was regarded equivalently as a group of identical bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Schnerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [16-19] simplified the Rayleigh-Plesset equation that only the linear part was used to redefine the velocity scale as √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 ⁄ for both growth and collapse situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The advantage of this formulation is that it follows, to some part, a physical law and reduces the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Nevertheless, the empirical coefficients are still needed to regulate the order of magnitude between evaporation and condensation terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Another way of modelling is to establish the barotropic relation that couples the mixture density and local pressure as called equation of state (EoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The model was first introduced by Delannoy and Kueny [20] and later widely used by others [21-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Some results obtained by EoS model show good agreement with benchmarks in term of strong compressible multiphase flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, the mechanisms inside cavity are complicated more than barotropic assumption that the gradient parallel between pressure and density can hardly be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For weak compressible issues, numerical algorithms lost robustness that would induce numerical instability and sometimes poor convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' All the mentioned cavitation models were developed and compared with a variety of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Despite some satisfactory results have been achieved by existing models, some questions still remain to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' One of the basic principles for cavitation model application is that the flow pattern of vapor-liquid mixture should be homogeneous dispersed bubbly flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It will exceed the model capability if the frequency of bubble coalescence is too high, for instance, in the case of supercavitating flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Though few models are still workable on the flow out of their application scope, one has to select weird values for parameters in order to match the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Furthermore, it can be inferred from several simulation comparisons [24, 25] that the model parameters cannot reflect the physical characteristics sufficiently because the empirical coefficients in specified cavitation model sometimes need to be recalibrated for different flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Also, the results calculated by different cavitation models present respective tendencies under the same experimental condition [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We speculate the main reasons causing these deviations come from two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' One is that the underlying characteristic scales produce large errors when the second derivative term in Rayleigh-Plesset equation becomes predominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As previously mentioned, the characteristic velocity in current models is given as an immutable value 𝑢∞ or linear form √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 ⁄ so that it is necessary to introduce empirical coefficients to 4 compensate the information mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' On the other hand, the typical configuration of existing models with two empirical coefficients and several parameters have good adaptability but wide range of proper values to guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Defining the parameters which can reveal the physical relevance between model and objective phenomenon would narrow the search regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, we suggest three improvements to account for the effect of nonlinear term, eliminating the empirical coefficients, and parameters reflected more intrinsic law to remedy the cavitation model with wider applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In this paper, a novel TEM-kind nonlinear dynamic cavitation model is proposed with rigorous mathematical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' There arise two functions 𝜓𝑣 and 𝜓𝑐 instead of empirical coefficients to represent the nonlinear effects for growth and collapse periods respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Only two meaningful parameters, the Blake critical radius 𝑅𝑏 and the average maximum growth radius 𝑅𝑚 , are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The new model is implemented in the open source C++ package OpenFOAM based on the interPhaseChangeFoam solver [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' To illustrate the superiorities of the new model clearly, three typical validations cases, sequentially from simple to complex, were adopted: numerical bubble cluster collapse [29], ultrasonic horn experiment [30, 31], and hydrodynamic cavitation around a slender body [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In these cases, the similar dispersed structures of bubble cloud satisfy the homogeneous assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In the following sections, basic assumptions and simplifications for bubble cluster are stated firstly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The model derivation processes are then elaborated in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The algorithms, including VoF-based interface capture method MULES [33, 34] in OpenFOAM and modified momentum interpolation (MMI) method [35, 36] for multiphase flows, are implemented correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Finally, qualitative and quantitative validations of the improved model performance are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Modeling methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 Assumptions The cavitating cluster composed by multi-radius bubbles makes the overall problem difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In order to reduce the complexity of modeling the bubble cloud, we propose to introduce four assumptions to describe its main mechanisms approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' No bubbles’ coalescence or breakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Local homogeneous assumption: As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1, the bubble cloud is divided into lattices by CFD grid in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The real bubbles included in each grid cell are equivalent as uniform radius according to the local vapor volume so that the free degrees are degenerated into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Neglecting the local bubble interactions: The intensity of bubble interactions has positive correlations with the bubble population 𝑁 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Noting that only partial region of bubble cloud is filled in local grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' If the mesh resolution is fine enough to contain few bubbles in 5 single element that the interactions between these localized bubbles could be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The law of mean bubble dynamics: Given a pressure field, bubbles with different scales experience similar dynamics which can be governed by Rayleigh-Plesset-type equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We suppose the mean bubble dynamic processes existed which could be obtained from statistical operations on the whole cloud bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In local point of view, the assumptions II and III are indicated that the dynamics of average bubbles, including growth and collapse, insides each grid cell can be treated as synchronized processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It should be noted that the assumption III doesn’t mean nothing interaction existed inside bubble cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Actually, these interactions are implied in numerical fluxes between adjacent grid cells, which will be illustrated in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It can also be inferred that there is a lower limit of mesh resolution that fulfills the requirements in assumption III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We recommend to generate a proper mesh by using grid independence principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In overall view of bubble cloud, the structure of void fraction which exhibits concentrated interior and sparse border has been observed through several experiments [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' This characteristic can be reflected as radius distribution of the average bubbles mentioned above in computational space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The assumption IV is indicated that such distribution can be decided by the mean bubble dynamics which is considered as internal law that every bubble obeys to manifest the main dynamic character of bubble cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Therefore, the modeling strategy can be simplified as to investigate the mean dynamics of local average bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 Diagram of bubble cluster in physical/computational space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Simplifications With the above assumptions, the model expressions can be derived from the individual bubble with Rayleigh–Plesset equation: 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 ⏟ 1 + 𝑝𝑔0 𝜌𝑙 (𝑅0 𝑅 ) 3𝛾 ⏟ 2 = 𝑅 𝑑2𝑅 𝑑𝑡2 ⏟ 3 + 3 2 (𝑑𝑅 𝑑𝑡 ) 2 ⏟ 4 + 2𝜎 𝜌𝑙𝑅 ⏟ 5 − 4𝜇 𝜌𝑙𝑅 𝑑𝑅 𝑑𝑡 ⏟ 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (1) Here the first term represents the difference of vapor saturation pressure 𝑝𝑠𝑎𝑡 and ambient pressure 𝑝̅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In Euler mixture model, we should point out that 𝑝̅ is a phase- Four assumptions Physical space Computationalspace 6 weighted mixing pressure [6] which can be considered as ambient pressure for those under-resolved bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The second terms describe the influence of gases inside the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The inertial effects are given by the third and the fourth terms, while terms 5 and 6 depict the influence of surface tension and liquid viscosity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Schnerr [17] has introduced the bubble density 𝑛 which definition represents the bubble population per unit liquid volume is adopted uniformly to build the relation between liquid volume fraction 𝛼𝑙 and bubble radius: 𝛼𝑙 = 𝑉𝑙 𝑉𝑙 + 4 3 𝜋𝑅3𝑛𝑉𝑙 = 1 1 + 4 3 𝜋𝑅3𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (2) Due to the cavitating bubble is commonly treated as pure vapor, the expression for mass transfer rate per unit volume can be built as follows: 𝑚̇ = 1 𝑉 𝑑 𝑑𝑡 (𝜌𝑣 ∗ 4 3 𝜋𝑅3 ∗ 𝑛𝛼𝑙𝑉) = 𝜌𝑣𝛼𝑙 2 𝑑 𝑑𝑡 (4 3 𝜋𝑛𝑅3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (3) According to the meaning of 𝑛 , the non-dimensional term 4𝜋𝑛𝑅3 3 ⁄ inside time derivative indicates the volume ratio of vapor to liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The original way to handle this term is to expand differentially into 4𝜋𝑅2 𝑑𝑅 𝑑𝑡 ⁄ , then specify the velocity 𝑑𝑅 𝑑𝑡 ⁄ by using the linear part √2|𝑝 − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 ⁄ in Rayleigh-Plesset equation for both growth and collapse processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Except for the oversimplification of linearized bubble velocity as previously discussed, such instantaneous form would underestimate the intensity of source terms during time marching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, we suggest to utilize the average value within the evolution time 𝜏 (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4) during which the bubble radius would vary from present 𝑅𝑃 to ultimate 𝑅𝑈: 𝑑 𝑑𝑡 (4 3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ 4 3 𝜋𝑅𝑈 3 − 4 3 𝜋𝑅𝑃 3 𝜏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (4) It is illustrated the implication of 𝜏 through the numerical results of Rayleigh- Plesset equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2, neglecting the bubble rebound, the typical dynamic period can be divided into four stages, namely inception, growth, slow-down and collapse, also several critical moments are marked - including the Blake radius 𝑅𝑏, zero acceleration point 𝑅𝑑, maximum radius 𝑅𝑚, and the final collapse radius 𝑅𝑐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The dynamics before 𝑅𝑏 is gas dominated oscillation, after that the bubble content is occupied by vapor gradually, then unstable cavitation took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The expansion process from 𝑅𝑏 to 𝑅𝑚 is split by inflection point 𝑅𝑑 into growth and slow-down, where the ratio of 𝑅𝑑 and 𝑅𝑚 is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Besides, we emphasize that the nonlinear effect should be considered for the accelerated collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The cavitation model is only devoted to describing growth and collapse so the evolution time for these two periods, 𝜏𝑣 and 𝜏𝑐 , are determined from the present point 𝑅 to 𝑅𝑑 and 𝑅𝑐0 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2 Typical bubble dynamic period Therefore, the main task is to calculate the values of 𝜏𝑣 and 𝜏𝑐 in term of Rayleigh–Plesset equation (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) where terms including surface tension, viscosity and gas had been neglected, 𝑅𝑅̈ + 3 2 𝑅̇ 2 = 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (5) Here, the saturated vapor pressure 𝑝𝑠𝑎𝑡 and liquid density 𝜌𝑙 are treated as constants, and the LHS in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 can also be reformulated as the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 𝑅𝑅̈ + 3 2 𝑅̇ 2 = 1 2𝑅2𝑅̇ 𝑑 𝑑𝑡 (𝑅̇ 2𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (6) It can be expected that the exact solutions for these evolution time can be derived from integrating the bubble velocity 𝑑𝑅 𝑑𝑡 ⁄ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=', to integrate the Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 twice under compatible initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The deductions are discussed in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Modeling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 Growth potential function 𝑚̇ 𝑣 Based on the interval of growth period shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2, the integration and corresponding initial condition for obtaining the velocity at 𝑅 are given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2, where the velocity at 𝑅𝑏 is treated approximately zero: 𝑡 = 0, 𝑅 = 𝑅𝑏 𝑎𝑛𝑑 𝑅̇ ≈ 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) ∫ 𝑑 𝑑𝑡 (𝑅̇ 2𝑅3) 𝑡𝑅 0 𝑑𝑡 = 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 ∗ ∫ 2𝑅2 𝑑𝑅 𝑑𝑡 𝑡𝑅 0 𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) Assuming that the ambient pressure 𝑝̅ is constant during short period of time, the integral result of growing velocity is then given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8: 𝑑𝑅 𝑑𝑡 = √2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 (1 − (𝑅𝑏 𝑅 ) 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (8) Obviously, it approaches to the inertial velocity √2|𝑝̅ − 𝑝𝑠𝑎𝑡| 3𝜌𝑙 ⁄ when the bubble expands significantly greater than 𝑅𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Consequently, the evolution time of 𝜏𝑣 can be 175 1 R 1 1 m 1 140 R 1 R 1 1 lius 1 IRadi 105- 1 1 1 1 1 1 Nondimensional 1 70- 1 1 R 1 1 1 1 1 1 1 1 35- 1 1 1 1 1 R 1 1 1 之 R 1 co 0 1 growth 1 collapse inception (t) islow-down!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Irebound 8 integrated from 𝑅 to 𝑅𝑑 using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8: 𝜏𝑣 = ∫ 𝑑𝑅 √2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 (1 − (𝑅𝑏 𝑅 ) 3 ) 𝑅𝑑 𝑅 = 1 √2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 ∫ 𝑑𝑅 √1 − (𝑅𝑏 𝑅 ) 3 𝑅𝑑 𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (9) Letting 𝑥 = (𝑅𝑏 𝑅 ⁄ )3, so we have: 𝑑𝑅 = − 𝑅𝑏 3 𝑥−4 3𝑑𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (10) Thus, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='9 can be reformulated as follows: 𝜏𝑣 = 𝑅𝑏 3√2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 ∫ 𝑥−4 3 (𝑅𝑏 𝑅 ) 3 (𝑅𝑏 𝑅𝑑) 3 (1 − 𝑥)−1 2𝑑𝑥, (11) where the definite integral in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='11 can be expanded by applying the Gauss hypergeometric function, the result becomes: 𝜏𝑣 = 2𝑅𝑏 3√2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 [ √1 − (𝑅𝑏 𝑅𝑑 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅𝑑 ) 3 ) −√1 − (𝑅𝑏 𝑅 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅 ) 3 ) ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (12) Note that there is an ultimate limit under the large ratio of 𝑅𝑑 an 𝑅𝑏, then: lim 𝑅𝑏 𝑅𝑑→0 (𝑅𝑏 𝑅𝑑 ) √1 − (𝑅𝑏 𝑅𝑑 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅𝑑 ) 3 ) = 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (13) The Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='12 is simplified as follow: 𝜏𝑣 = 2 3√2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 [3 2 𝑅𝑑 − 𝑅𝑏√1 − (𝑅𝑏 𝑅 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅 ) 3 )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (14) Thus, combing with the Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='14, we can get the approximation of derivative in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 for growth situation: 𝑑 𝑑𝑡 (4 3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ 4 3 𝜋𝑅𝑑 3 − 4 3 𝜋𝑅3 𝜏𝑣 = 1 − 𝛼𝑙 𝛼𝑙 1 𝑅 √2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 ∗ 𝜓𝑣, (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝜓𝑣 = (𝑅𝑑 𝑅 ) 3 − 1 (𝑅𝑑 𝑅 ) − 2 3 (𝑅𝑏 𝑅 ) √1 − (𝑅𝑏 𝑅 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅 ) 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) There, it emerges the formulation 𝜓𝑣 which we call growth potential function that represents the capacity of continuous growth until 𝑅𝑑 being reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Recalling the ratio of 𝑅𝑑 𝑅𝑚 ⁄ ≈ 𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8, it can be expressed by 𝑅𝑚 alternatively: 9 𝜓𝑣 ≅ (𝜂 𝑅𝑚 𝑅 ) 3 − 1 (𝜂 𝑅𝑚 𝑅 ) − 2 3 (𝑅𝑏 𝑅 )√1 − (𝑅𝑏 𝑅 ) 3 𝐹 2 1 (1 2 , 4 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1 − (𝑅𝑏 𝑅 ) 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Collapse potential function 𝑚̇ 𝑐 Analogously, with the same integration in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2, the initial condition at max radius 𝑅𝑚 for obtaining the collapse velocity is given as follows: 𝑡 = 0, 𝑅 = 𝑅𝑚 𝑎𝑛𝑑 𝑅̇ = 0, (17) which yields 𝑑𝑅 𝑑𝑡 = −√2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 ((𝑅𝑚 𝑅 ) 3 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (18) The evolution time 𝜏𝑐 can be integrated from 𝑅 to 𝑅𝑐0 by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='18: 𝜏𝑐 = − ∫ 𝑑𝑅 √2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 ((𝑅𝑚 𝑅 ) 3 − 1) 𝑅𝑐0 𝑅 = 1 √2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 ∫ 𝑑𝑅 √(𝑅𝑚 𝑅 ) 3 − 1 𝑅 𝑅𝑐0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (19) Letting 𝑥 = (𝑅 𝑅𝑚 ⁄ )3, so we have: 𝑑𝑅 = 𝑅𝑚 3 𝑥−2 3𝑑𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (20) Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='19 can be reformulated as follows: 𝜏𝑐 = 𝑅𝑚 3 1 √2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 ∫ 𝑥−1 6(1 − 𝑥)−1 2𝑑𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' ( 𝑅 𝑅𝑚) 3 (𝑅𝑐0 𝑅𝑚) 3 (21) According to the expression of incomplete beta function, we obtain: 𝜏𝑐 = 𝑅𝑚 3 1 √2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 [𝛽 (5 6 , 1 2 , ( 𝑅 𝑅𝑚 ) 3 ) − 𝛽 (5 6 , 1 2 , (𝑅𝑐0 𝑅𝑚 ) 3 )] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (22) Employing Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='22, we can get the approximation of derivative in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 for collapse situation: 𝑑 𝑑𝑡 (4 3 𝜋𝑛𝑅3) ≈ 𝑛 ∗ 4 3 𝜋𝑅𝑐0 3 − 4 3 𝜋𝑅3 𝜏𝑐 = − 1 − 𝛼𝑙 𝛼𝑙 1 𝑅 √2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 ∗ 𝜓𝑐, (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝜓𝑐 = 3 ( 𝑅 𝑅𝑚) [1 − (𝑅𝑐0 𝑅 ) 3 ] 𝛽 (5 6 , 1 2 , ( 𝑅 𝑅𝑚) 3 ) − 𝛽 (5 6 , 1 2 , (𝑅𝑐0 𝑅𝑚) 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) Emerged 𝜓𝑐 is called collapse potential function considering the historical effect that reflects the magnitude of bubble velocity applied by continuously accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is further proposed that the function 𝜓𝑐 can be simplified by constraining the bubble radius 𝑅 above the initial radius, where the last moment at 𝑅𝑐0 is actually at least one magnitude less than the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' This yield: 10 𝜓𝑐 ≅ 3 ( 𝑅 𝑅𝑚) 𝛽 (5 6 , 1 2 , ( 𝑅 𝑅𝑚) 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (24) Based on analysis above, the final form of NDCM is given below by replacing Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3: 𝑚̇ 𝑣 = 𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 𝜓𝑣√2 3 𝑝𝑠𝑎𝑡 − 𝑝̅ 𝜌𝑙 (𝑝𝑠𝑎𝑡 > 𝑝̅), (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝑚̇ 𝑐 = −𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 𝜓𝑐√2 3 𝑝̅ − 𝑝𝑠𝑎𝑡 𝜌𝑙 (𝑝𝑠𝑎𝑡 < 𝑝̅), (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) Here, the dimensionless functions of growth 𝜓𝑣 and collapse 𝜓𝑐 are given by Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='16 and 24, respectively, their variations against bubble radius are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Noting that the smaller bubble has larger value during both dynamic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For growth period, tiny bubble possesses long expansion time ( 𝜏𝑣 ) which is cubic proportional to volume change that leads to strong growth intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As for collapsing, the bubble speed at the direction of inward radial is accelerated gradually so that the intensity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3 The graphs of potential functions, 𝜓𝑣 and 𝜓𝑐 Apparently, only two parameters with significant physical meaning of cavitating bubbles, namely the Blake radius 𝑅𝑏 and the average maximum radius 𝑅𝑚 , are employed to regulate the model application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The empirical coefficients are substituted by the nonuniform potential functions, 𝜓𝑣 and 𝜓𝑐, considered the nonlinear effects for both growth and collapse periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Comparing with the Schnerr-Sauer model given in the following equations, 𝑚̇ 𝑣𝑆 = 𝐶𝑣 ∗ 3𝜌𝑙𝜌𝑣 𝜌 𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 √2(𝑝𝑠𝑎𝑡 − 𝑝̅) 3𝜌𝑙 (𝑝̅ < 𝑝𝑠𝑎𝑡), (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝑚̇ 𝑐𝑆 = −𝐶𝑐 ∗ 3𝜌𝑙𝜌𝑣 𝜌 𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 √2(𝑝̅ − 𝑝𝑠𝑎𝑡) 3𝜌𝑙 (𝑝̅ > 𝑝𝑠𝑎𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) 10000 1000 Potential Function 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 NondimensionalRadius/(R/RR/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=') 11 the difference is, except the additional potential functions, that the density factor in front can be approximated as 𝜌𝑙𝜌𝑣 𝜌 𝛼𝑙 ≈ 𝜌𝑣 based on 𝜌𝑙 ≫ 𝜌𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The contribution of 𝛼𝑙 vanishes that indicates both source terms are only proportional to (1 − 𝛼𝑙) but not 𝛼𝑙(1 − 𝛼𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Moreover, the Schnerr-Sauer model can be reformulated as the same form of NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In view of the symmetrical feature, we only discuss the collapse term as follow: 𝑚̇ 𝑐𝑆 = −𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 𝜓𝑐𝑆√2(𝑝̅ − 𝑝𝑠𝑎𝑡) 3𝜌𝑙 (𝑝̅ > 𝑝𝑠𝑎𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) where 𝜓𝑐 𝑆 ≅ 3𝐶𝑐 𝛼𝑙 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Using the Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2 to substitute 𝛼𝑙 , we can get the equivalent collapse potential function of Schnerr model: 𝜓𝑐𝑆 = 3𝐶𝑐 (1 + 4 3 𝜋𝑛𝑅3) (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) In contrast to 𝜓𝑐 which is inversely proportional to 𝑅 , 𝜓𝑐 𝑆 presents positive relation that will conclude a confusing interpretation that larger bubbles have stronger intensity during collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The comparison of 𝜓𝑐 and 𝜓𝑐 𝑆 will be discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Mathematical formulations and numerical method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 Governing equations The methodology in OpenFOAM for describing the two-phase system of cavitating flow is based on Eulerian’s homogenous mixture approach, in which both phases are treated as incompressible, isothermal, and immiscible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is simple and efficient to employ the one-field formulation (OFF) of Navier-Stokes equations that the properties of two phases, including density and viscosity, are hybrid as equivalent single-phase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The filtered governing equations including transport equation of liquid volume fraction and momentum conservation of effective fluid are given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 28 and 29 respectively, and the velocity divergence in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 30 comes from summing over the volume fraction equations of liquid and vapor that enables to build the pressure equation to update the flux field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' More details can be referred to Fleckenstein [6] for a rigorous derivation: 𝜕𝛼𝑙 𝜕𝑡 + 𝑼 ∙ ∇𝛼𝑙 = 𝜌 𝜌𝑙𝜌𝑣 (|𝑚̇ 𝑐| − |𝑚̇ 𝑣|), (28) 𝜕 𝜕𝑡 (𝜌𝑼) + ∇ ∙ (𝜌𝑼𝑼) = −∇𝑝𝑟𝑔ℎ + ∇ ∙ [𝜇(∇𝑼 + ∇𝑇𝑼) − 𝝉𝑇] − 𝒈 ∙ 𝒉∇𝜌, (29) ∇ ∙ 𝑼 = ( 1 𝜌𝑙 − 1 𝜌𝑣 ) (|𝑚̇ 𝑐| − |𝑚̇ 𝑣|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (30) Here, two phases are assumed to share the same velocity denoted as 𝑼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The hybrid density 𝜌 and viscosity 𝜇 are weighted based on volume fraction 𝛼𝑙 linearly with the constant properties of each phase, namely 𝜌𝑙, 𝜌𝑣, 𝜇𝑙, 𝜇𝑣, which are: 12 𝜌 = 𝜌𝑙𝛼𝑙 + 𝜌𝑣𝛼𝑣, (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝜇 = 𝜇𝑙𝛼𝑙 + 𝜇𝑣𝛼𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) Note that the pressure 𝑝𝑟𝑔ℎ is relative to hydrostatic pressure 𝜌𝒈 ∙ 𝒉 to avoid the algorithmic trouble of artificial diffusion induced by height difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The non-linear stress 𝝉𝑇 is closed by RANS or LES turbulence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In the present work, we employed the k-omega-SST two equation model [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, the complete framework is composed of three parts which are multiphase model (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='28, 29 and 30), turbulence model (𝝉𝑇 in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='29) and cavitation model (𝑚̇ 𝑐 and 𝑚̇ 𝑐 in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Algorithms and discretization The above governing equations are implemented into CFD code based on the solver of interPhaseChangeFoam, in which two important algorithms are developed for capturing the topological changes of bubble cluster and coupling the velocity-pressure to prevent checkerboard distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' One of the mature algorithms to solve a convective-only transport equation is VoF- based interface capturing method that a new high-resolution algebraic reconstruction proposed by Weller [34] based on flux-corrected transport (FCT) is implemented in OpenFOAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For further sharpness, comparing to the traditional approaches of compressive schemes like HRIC or CICSAM, the “counter-gradient” diffusion term is joined into transport equation to compress the interface in the reverse direction of volume fraction gradient that has good performance for 2D and 3D complex flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It should be noted that the interface of cavitation cavity refers to the boundary of vapor- liquid mixture (0 < 𝛼𝑙 < 1) and pure liquid (𝛼𝑙 = 1) differently from the one between water (𝛼𝑙 = 1) and vapor (𝛼𝑙 = 0) where the ambiguous region (0 < 𝛼𝑙 < 1) should be contracted as sharper as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Therefore, it is improper to employ the compression term which would bring unreasonable diffusion inside cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We recommend to use the FCT-based numerical method of semi-implicit multi- dimensional limiter for explicit solution (MULES) for better boundedness and consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' A detailed description of this algorithm can be found in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The SIMPLE/PISO algorithm on collocated grid is realized via the technique of momentum interpolation proposed by Rhie-Chow [41] that the serrated pressure would be eliminated by introducing the third pressure derivative in correction equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, only an incomplete method is implemented in OpenFOAM for robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The simulation results would be depending on time step size so that we suggest a modified momentum interpolation (MMI) method referring to the work by Cubero [35, 36] to remove the drawback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The original momentum interpolation (OMI) in OpenFOAM is given by: 𝑼𝑓 ∗ = 1 𝑎𝑃 𝑯∗ ̅̅̅̅̅̅̅ − 1 𝑎𝑃 ̅̅̅ ∇𝑝𝑟𝑔ℎ,𝑓 ∗ + 𝜖 ∗ 1 𝑎𝑃 ̅̅̅ 𝑎𝑃 𝑡 ̅̅̅(𝑼𝑓 𝑛 − 𝑼𝑃 𝑛 ̅̅̅̅), (32) that the operator of hat bar is linear interpolation from cell to face center, and the superscript ∗ means the mid-iteration till convergence to 𝑛 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The discrete coefficient 𝑎𝑃 can be decomposed into temporal 𝑎𝑃 𝑡 and spacial 𝑎𝑃 𝑠 dependent items, and the vector 𝑯 represents the collection of neighbor points summation and other 13 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Noting that the last term, called Choi correction, is emerged in unsteady problems that the flux difference of previous time step is used to correct the interpolated velocity, where 𝜖 employed in OpenFOAM is the empirical factor less than unity to prevent the correction value inducing instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Though, theoretically, the Choi correction would be vanished through several iterations, the time step is contained in coefficient 𝑎𝑃 that leads to the convergence result still associating with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' To remedy the problem, the specific value is defined as 𝑑𝑃 = 𝑎𝑃 𝑡 𝑎𝑃 𝑠 ⁄ , thereby the Rhie- Choi interpolation can be reformulated as: 𝑼𝑓 ∗ = 1 1 + 𝑑𝑃 1 𝑎𝑃 𝑠 𝑯∗ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ − 1 1 + 𝑑𝑓 1 𝑎𝑓 𝑠 ∇𝑝𝑟𝑔ℎ,𝑓 ∗ + ( 1 1 + 𝑑𝑓 𝑎𝑓 𝑡 𝑎𝑓 𝑠 𝑼𝑓 𝑛 − 1 1 + 𝑑𝑃 𝑎𝑃 𝑡 𝑎𝑃 𝑠 𝑼𝑃 𝑛 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (33) Here, it is assumed that all the coefficients on face center are interpolated linearly by cell values, and the approximate relations are introduced: 1 1 + 𝑑𝑃 1 𝑎𝑃 𝑠 𝑯∗ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ≈ 1 1 + 𝑑𝑃 ̅̅̅̅ 1 𝑎𝑃 𝑠 𝑯∗ ̅̅̅̅̅̅̅ , (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 1 1 + 𝑑𝑃 𝑎𝑃 𝑡 𝑎𝑃 𝑠 𝑼𝑃 𝑛 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ≈ 1 1 + 𝑑𝑃 ̅̅̅̅ 𝑑𝑃𝑼𝑃 𝑛 ̅̅̅̅̅̅̅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) Thus, the formulation of MMI can be derived by substituting Eqn34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='33: 𝑼𝑓 ∗ ≅ 1 1 + 𝑑𝑃 ̅̅̅̅ ( 1 𝑎𝑃 𝑠 𝑯∗ ̅̅̅̅̅̅̅ − 1 𝑎𝑃 𝑠 ̅̅̅ ∇𝑝𝑟𝑔ℎ,𝑓 ∗ ) + 1 1 + 𝑑𝑃 ̅̅̅̅[𝑑𝑃 ̅̅̅̅𝑼𝑓 𝑛 − 𝑑𝑃𝑼𝑃 𝑛 ̅̅̅̅̅̅̅].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (35) To ensure stability of solving pressure equation, the cavitation source term in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='30 is handled as semi-implicit form based on the principal diagonal dominant as follows: ∇ ∙ [ 1 1 + 𝑑𝑃 ̅̅̅̅ ( 1 𝑎𝑃 𝑠 𝑯∗ ̅̅̅̅̅̅̅ − 1 𝑎𝑃 𝑠 ̅̅̅ ∇𝑝𝑟𝑔ℎ,𝑓 ∗ ) + 1 1 + 𝑑𝑃 ̅̅̅̅ [𝑑𝑃 ̅̅̅̅𝑼𝑓 𝑛 − 𝑑𝑃𝑼𝑃 𝑛 ̅̅̅̅̅̅̅]] = ( 1 𝜌𝑙 − 1 𝜌𝑣 ) 𝐹(𝛼𝑙)√ 2 3𝜌𝑙(|𝑝𝑠𝑎𝑡 − 𝑝̅𝑛| + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='001 ∗ 𝑝𝑠𝑎𝑡) [𝑝𝑟𝑔ℎ ∗ − (𝑝𝑠𝑎𝑡 − 𝜌𝒈 ∙ 𝒉)], (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) where 𝐹(𝛼𝑙) is 𝛼𝑙 dependent function: 𝐹(𝛼𝑙) = 𝜌𝑣𝛼𝑙(1 − 𝛼𝑙) 1 𝑅 [𝜓𝑣 ∗ 𝑝𝑜𝑠(𝑝𝑠𝑎𝑡 − 𝑝̅) + 𝜓𝑐 ∗ 𝑛𝑒𝑔(𝑝𝑠𝑎𝑡 − 𝑝̅)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) The Choi correction will disappear during PISO/SIMPLE loops, and the time step is excluded out of 𝑎𝑃 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We employ two cases of lid-driven cavity (single phase flow) and 2D bubble rising [42] (multiphase flow) to test the performance of different methods shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is seen that the results obtained by MMI are overlapped completely to indicate the independence of time step (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4 (b) and (d)) for both situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 14 (a) Lid-driven cavity (OMI) (b) Lid-driven cavity (MMI) (c) Rising bubble (OMI) (d) Rising bubble (MMI) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4 Comparison of the velocity profiles between OMI and MMI against different time steps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Numerical configuration The strategy of velocity-pressure coupling is designed dual loops as called PIMPLE which have inner PISO and outer SIMPLE for accommodating large time step size [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The TVD type high order resolution schemes are used for the convective terms, and the second order central difference scheme is used for the diffusion terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The first-order implicit Euler scheme is used for the transient terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Validation cases In this section, the main purpose is to validate the cavitation model (NDCM) developed in this work through three fundamental cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For lack of exact solution about cavitating flows, the first case is to simulate the collapse process of vapor bubble cluster to illustrate differences between linear and nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The real bubble cluster excited by ultrasonic field is then investigated to reveal physical connotation of model parameters through a series of simulations for different experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The new model is finally applied to the convective bubble cloud in hydrodynamic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 dT=0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time / [s]dT=5e-3 dT=5e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='20- dT=5e-5 Rising velocity / [m/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='05- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time / [s] 15 cavitation of slender bodies with conical or blunt nose to further highlight the capability of NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 Bubble cluster collapse The necessity of considering the nonlinear effects represented by the potential function of 𝜓𝑐 during bubble collapse can be demonstrated clearly by this validation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Here, we recommend to use the simulation case by Schmidt [29] who developed a thermodynamic equilibrium model that the interface can be resolved implicitly when the grid resolution is sufficiently fine so as comparable to DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 5(a), the bubble cluster above the wall covers a spherical domain with a diameter of 𝑟𝑏 = 30mm within which 𝑁 = 150 spherical bubbles of equal radii 𝑟0 distributed from dense center to sparse border are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' All bubbles are filled with water vapor while the surrounding domain contains liquid water at an initial pressure of 𝑝∞ = 100𝑏𝑎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The initial pressure inside the bubbles is equal to the vapor pressure 𝑝𝑠𝑎𝑡 = 2340𝑃𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The velocity field is initially at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 5 Distribution of bubble cluster (a), lateral section of computational mesh (b) The simulation is carried out for bubbles with the radius 𝑟0 of 1mm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5mm by NDCM and Schnerr-Sauer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The lateral section of 3D hexahedral grid is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 5(b), three mesh resolutions for the spherical domain containing under-resolved bubbles are tested by the nodes of 𝑎 × 𝑏 × 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We introduce a factor 𝜆 that is defined as the ratio between bubble radius (𝑟0) and the numerical resolution scale (√𝑉̅ 3 ), where (√𝑉̅ 3 ) is the cubic root of cell-weighted average volume in bubble cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The model parameters and initial volume fraction 𝛼𝑙 can be calculated from: 𝛼𝑙 = 1 − 𝑁 ∗ (𝑟0 𝑟𝑏 ) 3 , (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) 𝑛 = 1 − 𝛼𝑙 𝛼𝑙 1 4 3 𝜋𝑟0 3 , (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2) that the setup details are shown in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' According to the conclusions from the reference paper, the bubble distribution influences the pressure field significantly but trivial for the collapse period, so we initialize the uniform field for 𝛼𝑙 to quantitatively 35 20N 15 20 25 25 20 30 30 X 3535alpha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='water 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5e-010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+00 16 investigate the main collapse process whereas only compare the pressure results qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The numerical pressure transducer is used consistently with [29] that the sampling frequency is 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3MHz on the wall of 15 × 15𝑚𝑚2 directly beneath the bubble cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Table 1 Initializations for the vapor bubble cluster 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 [𝑚𝑚] 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 [𝑚𝑚] Liquid volume fraction 𝛼𝑙 [−] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='95556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='85 Bubble number density 𝑛 [1/𝑚3] 11103833 12482740 Resolution ratio 𝜆 = 𝑟0 √𝑉̅ 3 [−] 𝑀𝑒𝑠ℎ 1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5k cells) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='579 𝑀𝑒𝑠ℎ 2 (16k cells) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='782 𝑀𝑒𝑠ℎ 3 (26k cells) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='919 The validation for grid independence is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 6, that all the variables are nondimensionalized by the equivalent radius 𝑅𝑒𝑞𝑛 and its Rayleigh time 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It can be seen that the collapse time calculated by NDCM (solid lines) is converged on Mesh 2 and 3 for both radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Recalling the assumption III which is indicated that the bubble interactions could be ignored above the resolution of Mesh 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Differently, the results by Schnerr model (symbol-solid lines) given the value of 𝐶𝑐 as unity are overlapped together which duration is longer than the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 [𝑚𝑚] 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 [𝑚𝑚] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 6 The grid independence verifications for different meshes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 7 shows model comparisons on Mesh 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The collapse time in reference paper is taken as benchmarks which accounts for 60% and 65% of Rayleigh time 𝜏 of the radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0mm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5mm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The collapse time interval by NDCM agrees well with the benchmarks although the rate of change has discrepancy which is likely due to employ the uniform distribution of 𝛼𝑙 inconsistently with the original method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, it is deviated widely by Schnerr model with 𝐶𝑐 = 1, renamed as Schnerr-Cc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 NDCM-Mesh1 NDCM-Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8- NDCM-Mesh3 Schnerr-Mesh1 Schnerr-Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 Schnerr-Mesh3 uba R/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 t/t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 NDCM-Mesh1 NDCM-Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8- NDCM-Mesh3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Schnerr-Mesh1 Schnerr-Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 ---Schnerr-Mesh3 uba R/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 t/t 17 as follows, which has insufficient intensity of source term that induces bubbles hard to collapse especially in the last stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In our opinion, the problem is caused by model error that different potential functions of Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 24 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 depict contradictory trend against bubble radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It can be considered that there is a collapsing shell at the border of bubble cloud based on the fact that its character of collapse processes is layer-by- layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The internal bubbles are normally larger than the external along the shell radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The potential function 𝜓𝑐 𝑆 in Schnerr model gives a positive correlation with bubble radius that may induce the high intensity of source term to be emerged inside bubble cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, the external tiny bubbles are constrained by weak sources that leads to difficult collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Even though 𝜓𝑐 𝑆 is incompatible with physical actuality, the Schnerr model can still be used by changing the coefficient 𝐶𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Ghahramani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [43] studied the Schnerr approach to simulate bubble cluster collapse, and they employed a high 𝐶𝑐 to obtain better results of collapse period but emerged huge numerical pressure wiggles unexpectedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We also tried larger values 𝐶𝑐 of 800 and 1200 for both cases that make the collapse period matched well with the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 8, highest pressure pulse occurs at the last process of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, only the result by Schnerr-1 has smooth curve whereas others do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Comparison of the results by NDCM (red line) and Schnerr with large 𝐶𝑐 (blue line), we can see that the new model can suppress most extents of spurious pressure wiggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Noting that the only difference in the contrast is the potential functions that indicate the physics implied by 𝜓𝑐 is more reasonable than 𝜓𝑐 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Comparison of the influence by different meshes (red and green lines) of NDCM results show that finer resolution contributes to control unphysical oscillations, particularly in 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It can be found that the pressure is more sensitive to mesh resolution although the collapse period has been converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 [𝑚𝑚] (b) 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 [𝑚𝑚] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 7 Comparison of time history of bubble cluster collapse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 RefbySchmidt NDCM,Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 Schnerr,Cc=1,Mesh2 Schnerr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Cc=800,Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6- uba, R/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 t/t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 RefbySchmidt NDCM,Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8- Schnerr,Cc=1,Mesh2 Schnerr,Cc=1200,Mesh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6- eqn R/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content="2- 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 t/t 18 (a) 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 [𝑚𝑚] (b) 𝑟0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 [𝑚𝑚] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 8 Comparison of time history of pressure transducer on the wall For further comparison and understanding the differences between two models, the bubble cloud structure, pressure contours and streamlines for the case of 𝑟0 = 1𝑚𝑚 at different time instances are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The behaviour of bubble cluster collapse is seen in the left sides that the variation of cloud radius is tardy from 𝑡 = 0 to 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='618𝑇 , subsequently dramatic collapse occurs during the rest of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As is mentioned in assumption III, interactions inside bubble cloud can be reflected in the flux transfer between neighbor cells even if the local interactions are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The motivation is that the velocity doesn’t be divergence free (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='30) that indicating the velocity field will be affected by cavitation source terms which inverse gradient will generate streamlines in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' As is seen from left sides in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9(b)~(d), the structures of velocity field of NDCM and Schnerr-1 are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The high- speed collapsing shell is driven by appropriate source term where large values distribute at the outside, but the low-speed internal region almost immunes to collapse until the last period at 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='927𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, the situation in Schnerr-800 is that unreasonable inverse flow is formed due to the higher source values emerge inside that sometimes induces the internal bubbles collapse priorly instead of the external (red dash circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' II)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The temporal evolution of bubble radius distribution in the direction of 𝜌𝑟 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The serrated line marked by the dash circle at 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='618𝑇 illustrates the bubble collapsing sequence is from inside-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Moreover, the innermost bubbles are affected prematurely from the boundary (dash arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Apparently, the property of potential functions (𝜓𝑐 and 𝜓𝑐 𝑆) is the main reason to influence the source term distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Besides, the local pressure 𝑝̅ (in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1) is another key factor that exhibits positive correlation to source terms, so that high pressure around outer bubble cluster enhances local sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It should be mentioned that 𝑝̅ is compatible with 𝜓𝑐 that will further strengthen outer source values, but 𝜓𝑐 𝑆 weakens the effect by 𝑝̅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For the case of Schnerr-1, it can be inferred that 𝜓𝑐 𝑆 is trivial relative to pressure 𝑝̅ which dominates the source term over the whole collapse period, thus large source values only appear externally to avoid inverse flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, such distribution of source term is destroyed by employing large 𝐶𝑐 which is augmented 800 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is seen from right sides in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9 that there are some local pressure pulses around bubble cloud for contours of NDCM and Schnerr-800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' These pulses are the reasons for 2000 NDCM, Mesh2 NDCM,Mesh3 Schnerr,Cc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Mesh2 1500- Schnerr,Cc=800,Mesh2 Pressure / [bar] 1000- 500- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 tt2000 NDCM,Mesh2 NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Mesh3 Schnerr,Cc=1,Mesh2 1500- Schnerr,Cc=1200,Mesh2 Pressure / [bar] 1000 500- 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 t/t 19 wiggles detected by the wall pressure transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Rossinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [44] has implemented a two-phase flow DNS to simulate 15000 bubbles collapse, which found that the pressure wiggles should be existed during temporal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, the smooth pressure profile predicted by Schnerr-1 is unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='III)~(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='III), the bubble cloud is dispersed excessively due to insufficient intensity of cavitation sources that the dissipated pressure gradient prevents local high pressure to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Comparison of pressure field by two models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9(d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' I) and (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' II), the positions of high pressure predicted by NDCM are located at the external shell from where to infinity the values decrease monotonously to ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, the corresponding points in Schnerr-800 invade into the bubble cloud where pressure pulse is surrounded by an additional low-pressure band (red dash arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We suppose that the misplacing pressure distribution is one of the reasons to cause the spurious pressure pulses which can be eliminate effectively by using the derived potential function 𝜓𝑐 in NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (𝑎) 𝑡 = 0 (𝑓) 𝑡 = 𝑇 (I) (II) (III) (𝑏) 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='515𝑇 (I) (II) (III) (𝑐) 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='618𝑇 20 (I) (II) (III) (𝑑) 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='755𝑇 (I) (II) (III) (𝑒) 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='927𝑇 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 9 Comparison of NDCM (I), Schnerr with 𝐶𝑐 = 800 (II), and Schnerr with 𝐶𝑐 = 1 (III) in prediction of cloud structure (left side), pressure contours (right side) and streamlines at different time instances (a) NDCM (b) Schnerr (𝐶𝑐 = 800) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 10 Comparison of distribution of bubble radius in radial direction of bubble cluster 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+05 5e+6 1e+7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5e+7 2e+7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5e+7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+07 Pressure: p [Pa]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='035 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e-02 Void fraction: αy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+00 2 3 4 5 6 7 8 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+01 1 Velocity: U [m/s]t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='515T, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='618T t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='755T, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='927T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 LO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5- L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 P, / rpt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='515T, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='618T t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='755T, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='927T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 LO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 p,/ rp 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Ultrasonic horn After exhibiting the basic expectation of NDCM, the discussion will be concentrated on the real bubble cloud generated by the high frequency oscillating ultrasonic horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It has been observed that if the horn tip is sufficiently small and driven at high amplitude, cavitation is very strong and the tip can be covered entirely by the gas/vapor phase for longer time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We employ the experiment designed by Žnidarčič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' [30] who investigated a systematic study in water at a 20 kHz with the horn diameter of 3 mm under variation of driving power, air saturation, viscosity, surface tension and temperature, that the attached cavity emerged peculiar dynamics with a self-generated frequency of expansion and collapse periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' After that, they [31] carried out the correspondingly simulation studies and obtained poor predictions of flow features with the original TEM method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=', Schnerr-Sauer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In their opinion, the Schnerr- Sauer-like model cannot adapt to the rapidly changing driving pressures, they presented an improved approach which also considered the second derivative term of Rayleigh- Plesset equation but in differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Good agreements comparing with measurements were then revealed for cavity shape and its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, the evolutionary tendency of the bubble cloud does not match well with experiment especially in expansion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In this section, the tasks are not only to demonstrate better predictive results by applying the NDCM method but also the rules of parameter regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Table 2 lists five experimental conditions for water at room temperature by which comparison of simulations can validate the physical meaning of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Table 2 Four experimental conditions for ultrasonic horn Case Percentage of max power [%] Vibrating amplitude 𝐴ℎ [𝜇𝑚] Saturation [%] A 70 164 100 B 50 C 20 D 30 100 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='1 Model parameters determination Recalling that the model parameters, 𝑅𝑏 and 𝑅𝑚, are the particular points living on the bubble dynamics curve, it is therefore demanded to determine the initial bubble state to estimate their reasonable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The theory of rectified mass diffusion which is the mechanism of cavitation inception in acoustic field is adopted to describe the formation of microbubbles from dissolved gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Here, the bubble nuclei are formed from microbubbles by gradually mass transfer, between which the difference of magnitude order is commonly at one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The analytical model for air-water systems proposed by Crum [45] that there exists a certain critical amplitude of acoustic pressure 𝑝̃𝑐 above which the microbubbles will begin to grow by rectified diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The expression is given as follow: 22 𝑝̃𝑐 = 𝜌𝑅0 2𝜔𝑁 2 √ [(1 − 𝜔2 𝜔𝑁 2) 2 + (𝑏𝜔 𝜔𝑁) 2 ] (1 + 2𝜎 𝑅0𝑝∞ − 𝑐𝑖 𝑐0) (3 + 4𝐾) (𝑐𝑖 𝑐0) − [3(𝜂 − 1)(3𝜂 − 4) 4 + (4 − 3𝜂)𝐾](1 + 2𝜎 𝑅0𝑝∞) , (38) where the 𝑅0 is the microbubble radius, the 𝑐𝑖 and 𝑐0 are the concentration at bubble interface and ambient liquid respectively, which ratio represents the gas saturation, whereas other variables can be seen in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 11 are the values of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 38 for the driving frequency at 20kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is indicated that large microbubbles tend to grow easily on the same saturation curve, besides, it is difficult to form nuclei for the same size of microbubble in degassed water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 11 The graphs of critical pressure 𝑝̃𝑐 against microbubble radius 𝑅0 As is obtained the one-to-one relation between acoustic pressure 𝑝̃ and microbubble 𝑅0 so we can get the size ranging of activated microbubbles under a given acoustic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Referring to the work by Mellow [46, 47], the approximate analytic solution of acoustic field 𝑝̃ for the vibrating horn is available using the Green function method which formulas indicate that the distribution of 𝑝̃ relates to three variables, the horn radius 𝑎 , the vibrating frequency 𝑓 , and the vibrating amplitude 𝐴ℎ , but is only proportional to 𝐴ℎ since the other two are fixed conditions in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is seen that the acoustic fields in Case A, B and C are identical, and become weaker in Case C and D sequentially for the vibrating amplitude reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 12 illustrate the nondimensionalized analytic solution of acoustic field 𝑝̃ 𝜌𝑙𝑐𝑢0 where the maximum value is located at the center of bottom wall and the semi-ellipsoidal iso-surfaces monotonically decrease down to the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It can be inferred from the background experimental pictures which represent the peak bulk of cavity that the region surrounded by the red iso-surface enables to provide sufficient driving force to develop the microbubbles evolving as cavitation nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Thus, in term of the theoretical values of acoustic pressure 𝑝̃ inside the region, the span of qualified microbubbles, shown in Table 3, can be evaluated by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' According to the conclusions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 11, the ranges of 𝑅0 display that the larger nuclei are appeared in the more degassed water (Case A, B and C), likewise situations embodied in weaker acoustic field (Case A and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 32 c/c。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='= 1 16 c/c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 Pressure amplitude / [atm] 8 c/c。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 4 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0625 0 20 40 60 80 100 Microbubbleradius/ [um] 23 Case A Case B Case C Case D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 12 The estimation of cavitation bubble production region Since the value scope of microbubbles 𝑅0 inside the effective region of acoustic field has been acquired, we suggest the multiples about 6~8 of 𝑅0 to define the value range for model parameter 𝑅𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' And the range of maximum radius 𝑅𝑀 can exploit the numerical solution of Rayleigh-Plesset equation with the initial conditions of endpoint values of 𝑝̃ and 𝑅0 that can estimate the parameter 𝑅𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For clarity, all these specified values of five cases are listed in Table 3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Table 3 Estimation of model parameters Case A Case B Case C Case D Pressure amplitude in effective region 𝑝̃ [𝑏𝑎𝑟] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='18~15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='72~15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='27~15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='12~9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='63 Activated microbubbles 𝑅0 [𝜇𝑚] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='36~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='75~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='46~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='61 Model parameters of 𝑅𝑏 [𝜇𝑚] Range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 27~48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='9 Value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 Maximum growth radius 𝑅𝑀 [𝑚𝑚] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='41~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='49~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='56~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='41~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='56 Model parameters of 𝑅𝑚 [𝑚𝑚] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Simulation setup As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 13, we employ a 2D axisymmetric computational domain consistently with the experiment with the dimensions of 40 mm height and 25 mm radius, in which the horn tip of 3 mm diameter is placed vertically from top 30 mm above the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is note that the near-wall grid is densified to ensure the y+ is lower 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0e-01 Nondimensional acoustic pressure picuo 24 than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 13 Computational domain for ultrasonic horn All the walls are used the no-slip velocity boundary condition and zero gradient for pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The top of atmosphere is defined as fixed pressure at 1 atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The horn vibration in a sinusoidal manner at a frequency of 20 kHz, at various amplitudes, depending on the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' To capture the movement a dynamic mesh approach was used that the mesh must constantly be updated by laplacian smoothing and local remeshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It was determined that the mesh, due to very small deformation of the domain, preserves an extremely low value of cell skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Three mesh densities were tested and it is found that it does not influence the outcome of the calculation of cavitation dynamics, but the model parameters have to change slightly for different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Consequently, the following results are calculated on a medium grid with 23550 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Results A series of simulations by the new cavitation model are compared with experiments including the volume evolution of the bubble cloud and the acoustic pressure probed by hydrophone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Considering that the physics of these cavitating flow do not differ significantly between Case A-D, we emphatically analyze the results of Case A whereas others are given more briefly in data charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' A sequence of the spatial structures of cavity beneath the tip of the horn are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 14 graphically displayed as left simulation and right experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is seen that the mushroom-like shape cloud is formed rapidly during the interval of 0 to 40𝜇𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Then, the generated bubbles keep the dynamic balance from 60𝜇𝑠 to 100𝜇𝑠 in which the maximum volume is achieved about 80𝜇𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' After that the cavity contracts at the outer rim and the violent collapse happens at the end of cavitation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' A comparison between the measured and predicted cavity volume is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 15(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is evident that the simulation by NDCM accurately predicts the dynamics of the cavity volume which cycle (5010Hz) is about a quarter of the driving frequency (20kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Noting that the cavity frequency calculated by Schnerr-Sauer model can agree with the experiment, the produced vapor volume is insufficient though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The typical atmosphere horn 40mm axis 30mm walls 25mm 25 period inside pink dash line is magnified shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 15(b), and attached the results obtained by Znidarcic’s model additionally (dash line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We can see that the tendency is almost the same before 30𝜇𝑠, however the shrink of cavity in growth period predicted by Znidarcic model mismatches with the measurements, moreover, the variation rate of cavity volume shifts much faster than the experiment during collapse process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 14 Typical cycle of the oscillation of a large cavity between simulation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='experiment at the driving frequency of 20kHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Some discrepancies should be pointed out that a pinch of bubbles at the tip of cavity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='O μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment20μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment40 μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment60 μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment80μus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment100μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment120 μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment140 μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment180μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment200μs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='Experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='cannot capture perfectly possibly due to the 2D simulation method since the bubble ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='cloud has non-axisymmetric 3D structure shown in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Besides, the convection of corner bubbles which is likely induced by Bjerknes [48] force is unable to take into account here limited by the model capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 15 Comparison between the predicted and measured attached cavity volumes for Case A A comparison of measured and simulated pressure evolutions at a distance of 7 mm from the tip of the horn are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 16(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is indicated that high pressure pulse is emitted at the last stage of cavity collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The peak pressure amplitude seems to be slightly overpredicted which could be caused by the assumption of incompressibility for both phases, and furthermore missing the isolated bubble structures (see the right side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 14) where the cavitation model can only capture the main cavity bulks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Also note that the high frequency components are smoothed out because of the insufficient mesh resolution at the vicinity of probe and low order of temporal scheme under the RANS simulation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Nevertheless, the periodicity of pressure peaks is correctly predicted shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 16(b) of power spectrum density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is evident that the primary and second frequency in PSD are identical with the cavity and the driving horn respectively, which reflect main dynamic characteristics in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=" 16 Comparison between the predicted and measured acoustic pressure (a), PSD 8 Case A-NDCM Schnerr-Sauer Experiment vaporvolume/ [mm' o 4 3 N 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time / [ms]9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Experiment CaseA-NDCM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 - RefbyAnton 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='20 Time / [ms]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 CaseA-NDCM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 Experiment Acoustic pressure/ [bar] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time / [ms]30 5010Hz 25- 20000Hz 20 D 15 PSD 10- 5 0 0 100000 200000 300000 400000 500000 frequency/ [Hz] 27 analysis for the obtained acoustic pressure in Case A (b) As unsuccessful application of Schnerr-Sauer model is well illustrated, we will discuss the rest of Cases B-D which model parameters are orderly defined in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The typical cavity volume evolution for degassed water of Case B and C are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 17(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 18(a) respectively, we can see that the peak volume are declining as the gas content decreased (about 7~8𝑚𝑚3 for Case A, 6~7𝑚𝑚3 for Case B, and 4~5𝑚𝑚3 for Case C), meanwhile accompany with increasing unsteadiness of the cavitation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' This character can also be observed in acoustic pressure variations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 17 (b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 18(b)) which amplitude wiggles violently by degassed extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' There exist complex bubble interactions inferred from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 18(b) (Case C) that the pressure pulse radiates more frequently than the previous two where the calculated pressure reflects these frequency components qualitatively as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, the periodicity of the cavity does not damage which is slightly increased (5010Hz for Case A, 5079Hz for Case B, and 5259Hz for Case C) for more additional high frequency emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Comparing with Cases A, the peak volume is significantly reduced in Case D shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 19(a) which is caused by the weak acoustic field by the low driving power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Differently the cyclic evolution period of cavity (6517Hz) is raised to about one-third of driving frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is seen that the NDCM is capable to predict the cavity dynamics accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Moreover, the proposed model parameters, 𝑅𝑏 and 𝑅𝑚, are physically based where good agreements can be obtained by setting reasonable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) 8 Experiment CaseB-NDCM 6 vaporvolume/ [mm 5 3 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time/ [ms]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 CaseB-NDCM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 Experiment Acoustic pressure / [bar] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time / [ms]35 5079Hz 30- 25 20000Hz 20 - 8 P15- 10- 5- 0- 0 100000 200000 300000 400000 500000 frequency/ [Hz] 28 (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 17 Comparison between the predicted results and measured data for Case B (70% max power, 50% saturated) (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 18 Comparison between the predicted results and measured data for Case C (70% max power, 20% saturated) (a) Experiment 6 CaseC-NDCM 5 vaporvolume/[mm 3- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 Time/ [ms]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 CaseC-NDCM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5- Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- [bar] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 Acousticpressure / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Time/ [ms]25 20000Hz 20 - 15- S P 10- 5259Hz 5- 0- 0 100000 200000 300000 400000 500000 freguency/[Hz]Experiment CaseD-NDCM 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 Time/[ms] 29 (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 19 Comparison between the predicted results and measured data for Case D (30% max power, 100% saturated) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Slender body In this section, we extend the application for NDCM on the hydrodynamic cavitation flow which contains the structures of dispersed bubbly cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The natural cavitation experiments [32] for axisymmetric bodies with blunt and conical heads are employed for elementary investigation, also to compare with the results by Merkle’s model in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The 2D axisymmetric computational domains are adopted shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The diameter of the slender body is 20 mm and the length 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The domain extension has 15d upstream and 20d downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The inlet velocity is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8m/s and fixed outlet pressure based on the cavitation number 𝜎 with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The slender walls are specified as no-slip boundary and the outer ring is set as slip wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The boundary layer grid is generated to ensure y+ less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 20 Computational domain for slender bodies with conical and blunt heads In order to determine the model parameters, it only needs to select the middle mesh with the quantity of 37700 cells and 45400 cells for conical and blunt heads respectively between three resolutions which the simulation results are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For the lack of information to estimate the size of bubble nuclei, the approximate values of 𝑅𝑏 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 CaseD-NDCM Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 Time / [ms]15 6517Hz 12- 20000Hz 9- PSD 9 3- 0 0 100000 200000 300000 400000 500000 frequency/ [Hz]42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5d slipwall outlet 15d inlet 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5d axis axis 10mm 10mm 30 𝑅𝑚 for both heads listed in Table 4 are guessed from several trial calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We suppose that these values are reasonable because the cavitation region has higher negative pressure and wider area as dropping the cavitation number, which enables smaller nuclei to grow (𝑅𝑏) and more residence time leads to bubbles expand larger (𝑅𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Table 4 Model parameters of NDCM model 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Blunt 𝑅𝑏 = 32 𝜇𝑚 𝑅𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='53 𝑚𝑚 𝑅𝑏 = 25 𝜇𝑚 𝑅𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='62 𝑚𝑚 Conical 𝑅𝑏 = 28𝜇𝑚 𝑅𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='43𝑚𝑚 𝑅𝑏 = 18𝜇𝑚 𝑅𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='52𝑚𝑚 The pressure distributions for blunt head with cavitation number of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 21(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' We can see that the results by NDCM achieves good agreements especially for 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 that the cavity length predicted by new model (solid line) is almost identical with the experiment but underestimated in Merkle’s (dash line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The insufficient length of cavity predicted by Merkle’s model is embodied notably in conical head shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 21(b) whereas the NDCM presents a reasonable prediction matching with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' However, some discrepancies still exist limited by 2D geometry that the recovery pressure at the cavity tail is a bit lower than experiment at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 and the cavity length is little overpredicted for blunt at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It is more likely to use the 3D simulation to capture the asymmetric structures of bubble cloud which are shedding periodically that can obtain better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' (a) blunt (b) conical Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 21 Comparison of pressure distribution for blunt and conical head body 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Conclusions In this study, a novel nonlinear dynamic cavitation model (NDCM) is proposed against the bubble cluster structure through strictly mathematical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Firstly, the four thorough assumptions for TFM-type models are explicated that the filtered bubbles are mapped from physical to computational space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Then, we introduced the integral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Sim,NDCM,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Sim,NDCM,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 Ref,Merkle,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Ref, Merkle, =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 Exp,α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0 2 3 4 5 p/X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='g-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 Sim,NDCM,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='8 Ref,Merkle,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 Ref, Merkle, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 Exp,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 Exp,α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='6 0 2 3 4 5 p/X 31 average method to calculate the time derivative term 𝑑 𝑑𝑡 ( 4 3 𝜋𝑛𝑅3) that the second derivative in Rayleigh-Plesset equation can be considered in the characteristic time during growth and collapse, namely 𝜏𝑣 and 𝜏𝑐, solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Consequently, two additional potential functions 𝜓𝑣 and 𝜓𝑐 emerge in model formula which represent the nonlinear effects in cavity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In addition, without any empirical coefficients, there are merely two parameters with definitude physical meaning in which 𝑅𝑏 and 𝑅𝑚 indicate the Blake critical radius and the average maximum growth radius, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' In order to validate the performance of the new model, three simulation cases, from simple to complex, were employed including the collapse of numerical bubble cluster, periodic generation and collapse of real bubble cloud in ultrasonic horn experiment, and hydrodynamic cavitation of slender body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' For the first case, the results showed that the collapse time of NDCM and benchmark agreed well except the speed rate which may cause by different initialization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The layer-by-layer collapse character and pressure shock at last stage were revealed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' On the contrary, the Schnerr-Sauer model with parameter 𝐶𝑐 = 1 overpredicted the collapse time because of insufficient intensity of source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' More importantly, the potential function 𝜓𝑐 𝑆 implied in Schnerr model gives a positive relation with bubble radius 𝑅𝑏 that contradicts with 𝜓𝑐 in NDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Although the collapse time can be remedied by employing large coefficient 𝐶𝑐, the model errors were also augmented that brought great numerical pressure wiggles and incorrect collapse processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Most of spurious pressure can be suppressed by applying NDCM, but the remaining components should be further studied in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The NDCM was then applied to the real bubble cloud generated in acoustic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The main purpose was to confirm the physical relevance of 𝑅𝑏 and 𝑅𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Four experimental conditions were adopted that the theoretical value ranges of model parameters were determined based on the rectified diffusion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' It was found that the variation of those well-matched simulation results was in accord with the laws of specified model parameters, and more sensitive to 𝑅𝑏 which should be given larger values in more degassed water or weaker acoustic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' A detailed comparison of Case A against the results from Znidarcic showed that the second derivative term in Rayleigh-Plesset equation considered in integral form rather than differential can provide better predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Finally, the new model was extended to hydrodynamic cavitation of convective dominated flow that the slender bodies with two heads of conical and blunt were simulated under the cavitation number of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The good agreements of cavity length and pressure distribution further indicated that the NDCM is applicable for the cavitation cavity with dispersed bubble structures.' metadata={'source': 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+page_content=' Futral, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Schmidt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Adams, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Koumoutsakos, 11 PFLOP/s simulations of cloud cavitation collapse, in Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 2013, Association for Computing Machinery: Denver, Colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Article 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Crum, Acoustic cavitation series: Part five rectified diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Ultrasonics, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 22(5): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 215-223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 35 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Mellow, On the sound field of a resilient disk in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Journal of the Acoustical Society of America, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 123(4): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1880-1891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Mellow, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Kärkkäinen, On the sound field of an oscillating disk in a finite open and closed circular baffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Journal of The Acoustical Society of America, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 118: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 1311-1325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' Blake, Bjerknes Forces in Stationary Sound Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' The Journal of the Acoustical Society of America, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 21(5): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} +page_content=' 551-551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQfOgPu/content/2301.03017v1.pdf'} diff --git a/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/2301.02321v1.pdf.txt b/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/2301.02321v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a9886cc0a66168d7b861ae33e29abe555cc72a2 --- /dev/null +++ b/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/2301.02321v1.pdf.txt @@ -0,0 +1,1419 @@ + + +1 +Interlayer Exciton–Phonon Bound State in Bi2Se3/monolayer WS2 van der Waals Heterostructures + + +Zachariah Hennighausen1,*, Jisoo Moon1, Kathleen M. McCreary2, Connie H. Li,2 Olaf M.J. van ’t Erve2, +and Berend T. Jonker2,* +1 NRC Postdoc at the Materials Science and Technology Division, Naval Research Laboratory, +Washington, D.C. 20375, USA +2 Materials Science and Technology Division, Naval Research Laboratory, Washington, D.C. 20375, USA + +Abstract + +The ability to assemble layers of two-dimensional (2D) materials to form permutations of van der Waals +heterostructures provides significant opportunities in materials design and synthesis. Interlayer +interactions provide a path to new properties and functionality, and understanding such interactions is +essential to that end. Here we report formation of interlayer exciton-phonon bound states in Bi2Se3/WS2 +heterostructures, where the Bi2Se3 A1(3) surface phonon, a mode particularly susceptible to electron- +phonon coupling, is imprinted onto the excitonic emission of the WS2. The exciton-phonon bound state +(or exciton-phonon quasiparticle) presents itself as evenly separated peaks superposed on the WS2 +excitonic photoluminescence spectrum, whose periodic spacing corresponds to the A1(3) surface phonon +energy. Low-temperature polarized Raman spectroscopy of Bi2Se3 reveals intense surface phonons and +local symmetry breaking that allows the A1(3) surface phonon to manifest in otherwise forbidden +scattering geometries. Our work advances knowledge of the complex interlayer van der Waals +interactions, and facilitates technologies that combine the distinctive transport and optical properties +from separate materials into one device for possible spintronics, valleytronics, and quantum computing +applications. + +* Authors for correspondence, E-mail: hennigha@mit.edu; berry.jonker@nrl.navy.mil; + +Keywords: exciton-phonon bound state, exciton-phonon quasiparticle, exciton-phonon coupling, interlayer quasiparticle, +monolayer WS2, Bi2Se3 + + + + +2 +Introduction + +Van der Waals heterostructures are formed by stacking monolayers of 2D materials in any sequence of +one’s choosing, enabled by the lack of bonds between layer planes.1 Such stacking often results in new +properties, tuned by either material selection or twist angle.2 Understanding the interaction between the +layers in such heterostructures is essential to discovering new properties, engineering functionality, and +advancing their application into technologies. Previous work showed the interlayer interaction between +stacked two dimensional (2D) materials can facilitate a host of new properties, including long-lived +interlayer excitons,3 magnetic phase switching,4 forbidden Raman modes,5 superconductivity,6 orbital +ferromagnetism,7 and emergent ferromagnetism.8 + +In this work, we grew few-layer Bi2Se3 on monolayer WS2, and observe interlayer exciton-phonon coupling +and formation of an exciton-phonon bound state between localized excitons in monolayer WS2 and the +Bi2Se3 A1(3) surface phonon, a mode particularly susceptible to electron-phonon coupling.9,10 The bound +state is manifested as a series of evenly spaced peaks superposed on the WS2 excitonic photoluminescence +(PL) spectrum, whose periodic spacing corresponds to the A1(3) surface phonon energy. Oscillating features +that match a phonon energy and correspond to a luminescence are indicative of electron-phonon or +exciton-phonon bound states.11–15 In addition, polarized Raman spectroscopy of the Bi2Se3 reveals multiple +pronounced surface phonon modes and crystalline symmetry breaking. Notably, the presence of surface +phonons in a forbidden scattering geometry suggests local symmetry breaking at the surface,9 consistent +with a strong WS2-Bi2Se3 interlayer coupling. Previous work found significant interlayer hybridization in +Bi2Se3/WS2 heterostructures, facilitating electron transfer and modifying the bonding,16–18 conditions which +encourage the formation of interlayer quasiparticles.19 Understanding the interlayer interaction is central +to elucidating how their combined properties evolve, enabling the discovery of advanced capabilities for +spintronics,18,20 valleytronics,21 and quantum computing21,22 applications. + +Several publications have reported exciton-phonon or electron-phonon coupling across the interlayer +region, where a free exciton in a monolayer transition metal dichalcogenide (TMD) facilitates the +emergence of otherwise forbidden Raman modes in the adjacent material.5,23–26 Additionally, interlayer +vibronic exciton-phonon states have been reported in a TMD/TMD diode using photocurrent +measurements.27 Our work is distinct in that we report the formation of an interlayer exciton-phonon + + + +3 +bound state.13,14 Exciton-phonon bound states are uncommon quasiparticles that are formed from particles +whose number is not conserved (i.e., phonons).13,14 This means the exciton-phonon bound states are only +stable if their real decay is forbidden by the exciton laws of conservation of energy and momentum, during +phonon disappearance.14 While intralayer exciton-phonon bound states have been reported in low- +dimensional materials,12 to our knowledge, such a state has never been observed to form across an +interlayer region in a vdW heterostructure. + +Both bismuth selenide (Bi2Se3) and monolayer tungsten disulfide (1L WS2) have independently +demonstrated promise for a variety of advanced technologies, including spintronics18,20 and quantum +computing.21,22 Bi2Se3 is a topological insulator (TI) with gapped bulk states and gapless surface conducting +states that are spin-momentum locked, enabling advanced field effect transistors (FETs),28 magneto-electric +devices,29 topological qubits,22 and spin-orbit torque (SOT) devices.20,30 Monolayer WS2 is a direct band gap +semiconductor with independently optically addressable valleys, a strong light-matter interaction, and +high sensitivity to surrounding fields, enabling FETs,31 sensors,32 valleytronics devices,21 ferroelectric +modulation of exciton populations,33 and various photonic devices (e.g., LEDs, modulators, lasers).34 +Bi2Se3/monolayer WS2 heterostructures offer the possibility to combine strong light-matter interaction and +spin-locked current into one device. + +Results + + +Figure 1: Characterization of 4-6QL Bi2Se3 + 1L WS2 2D heterostructures. (a) AFM scan and (b) line profile corresponding to blue +line in 1a. Blue spot marks a location with continuous film (Section S1) and a representative location studied. (c) TEM SAED image + +4-6QLBi,Se3 +22.1nm +Bi,Se +a. +1LWS, +20.0 +4-6QLBizSe3grown +C +WS2 +ontopmonolayerWS2 +SiO2 +15.0 +3.58A +10.0 +2.75 +2.3nm +5.0 +Sio +10μm +1.1 +0-2°twistangle +20000 +4 K +180 A1。(2) +4-6QLBiSe+1LWS +(XY) +296 2LA(M)-2E*2g(M) +4000 +4-6QL Bi, Se, + 1L WS, ( +74 A1g(1) +(XX) +126 E(2) +3512LA(M) +(2) +160 A,(3) +419 A1g(T) +RamanIntensity( +523 +2000 +323 +1000 +d. +100 +200 +300 +RamanShift（cm-1） +500 + +4 +showing Bi2Se3 grows crystalline on WS2 across a 0-2 twist angle range. (d) Linearly polarized Raman response at 4 K with modes +identified. Porto notation is used (see methods). + +Figure 1 presents data characterizing the as-grown 4-6 quintuple layer (QL) Bi2Se3 + monolayer (1L) WS2 +vdW heterostructure, synthesized using chemical vapor deposition (CVD). Figure 1a-b shows an atomic +force microscope (AFM) scan and corresponding line profile, respectively, of a representative sample. For +reference, each QL of Bi2Se3 is ~ 1nm thick, and a monolayer of WS2 is 0.7 nm thick. Based on the AFM data +and diminished PL intensity, we conclude that Bi2Se3 grew as a uniform QL over 1L WS2, with taller islands +(~3.7nm) on top that merge together to form a continuous multilayer film. Section S1 shows additional +AFM data and fluorescence measurements showing that the Bi2Se3 is initially continuous, followed by +growth of multilayer islands characteristic of Bi2Se3 growth. Transmission electron microscope selected +area electron diffraction (TEM-SAED) measurements demonstrate both the WS2 and Bi2Se3 grow +crystalline, and that the Bi2Se3 grows within a narrow range of twist angles (0-2) around 0 aligned with +the WS2. Our TEM results are in agreement with previous work,35 indicating the interlayer interaction is +sufficiently strong to induce epitaxial growth. Figure 1d shows linearly polarized Raman measurements +taken at 4 K where the well-formed modes correspond closely to the respective materials,9,36 suggesting +good crystalline and stoichiometric sample quality. + + +Figure 2: Manifestation of interlayer exciton-phonon bound state. (a) 4-6QL Bi2Se3 + 1L WS2 2D heterostructure PL at 4 K showing +the free exciton and localized state. Evenly spaced “steps” are observed, whose energy spacing (19.3meV) corresponds well to the +160cm-1 A1(3) surface phonon. Previous work identified the A1(3) mode as particularly susceptible to electron-phonon coupling.9,10 +(b) 1L WS2 PL at 4 K, where a comparatively smooth localized state is observed. + + +4 K +BiSe3-WS,PL +Exciton-Phonon +BoundState +Uniformly spaced +energytransitions +FreeExciton +4 K +MonolayerWS,PL +Localizedstate +Comparativelysmooth +line shape with no +pronouncedsteps +FreeExciton + +5 + +Figure 2a shows the PL spectrum of the Bi2Se3/WS2 heterostructure sample used in Figure 1, which exhibits +a red shift compared to the as-grown monolayer WS2 reference sample (Figure 2b). A similar red shift was +reported in graphene/WS2 heterostructures, and is attributed to charge transfer.37 Both spectra are +dominated by a broad asymmetric feature approximately 130 meV below the free exciton with a long tail +at low energy, generally attributed to contributions from multiple defect-bound exciton complexes.38–40 +Power-dependent (Section S2) and temperature-dependent (Section S3) measurements validate our +assignment of exciton species, where the lower energy peak is a localized exciton (or bound exciton), and +the higher energy peak is a neutral free exciton. +In addition, the heterostructure PL exhibits regularly spaced peaks not observed in the reference +monolayer. The peaks on the high energy side are more prominent, while peaks on the low energy side +are likely obscured by the low energy tail of the bound excitons. We use visual inspection to identify these +peak positions to determine their energy separation. Efforts to quantitatively extract these positions by +analyzing the double derivative extrema yielded comparable results (Section S4). Our analysis found the +peaks to be approximately evenly spaced, with an energy spacing of 19.3meV (156cm-1), which corresponds +to the Bi2Se3 A1(3) surface phonon mode (160cm-1). Of note, previous work found the A1(3) mode produces a +Fano shape under resonant conditions, indicative of electron-phonon coupling at the Bi2Se3 surface.9,10 + +Electron-phonon and exciton-phonon bound states (or quasiparticles) have been shown to produce a series +of equally spaced features, often overlayed on a larger PL or optical reflection curve, where the period of +the emerged features approximately matches the phonon’s energy.11–15,41 As such, the presence of such +quasiparticles can be inferred when evenly spaced peaks are observed that correspond to a Raman phonon +mode.13 The number of peaks and consistency of their spacing is material and environment specific, as well +as the strength of the exciton-phonon coupling.12,13 The task is further complicated by the fact that the +evenly spaced features are frequently overlayed on a larger peak, which obscures their precise peak +position.11–15 Exciton-phonon bound states are frequently strongest when the exciton is localized (or bound) +to an impurity center,13,14 consistent with our observations where the quasiparticle corresponds to the +localized state. We note the formation of an exciton-phonon quasiparticle is distinct from exciton-phonon +scattering, in part because the momenta of the exciton and phonon are coupled.14 As such, exciton-phonon +quasiparticle interactions can result in uncommon observations, such as the emission of distinct features +above the exciton energy.12–14 + + + +6 + +The energy spacing was independently observed and measured in several samples (see Section S4 for data +from additional samples). We therefore attribute these multiple peaks as arising from exciton-phonon +coupling to the Bi2Se3 A1(3) surface phonon, a mode with an energy of ~19.8 meV that is particularly +susceptible to electron-phonon coupling,9,10 which together form am exciton-phonon quasiparticle. The PL +spectrum of as-grown monolayer WS2 shown in Fig 2b does not show such additional peaks, +demonstrating that coupling to the Bi2Se3 is responsible for these dramatic features in the PL. + + +Figure 3: Schematic of exciton-phonon quasiparticle recombination pathways. (a) A bound exciton coupled with a phonon to +form a bound exciton-phonon quasiparticle, where the phonon interaction modifies the exciton recombination energy. (b) The +phonon interaction can induce quantized energy levels corresponding to the phonon mode energy, which leads to multiple +recombination pathways that manifest as equally spaced features in experiment. Note, configurational coordinates represent the +relative real-space displacement of the initial (electron) and final (hole) states. Inset is data from Figure 2. + +Figure 3 presents a graphical explanation of the exciton-phonon bound state quasiparticle,14 and a +mechanism for generating evenly spaced peaks in exciton-phonon systems.42 The exciton-phonon bound +state quasiparticle is only understood by combining several concepts from quantum physics. The +quasiparticle is composed of an exciton and a phonon, which are coupled together through virtual +transitions that are related to the Fröhlich interaction.14,42 Virtual transitions are short-lived, unobservable +quantum effects that can facilitate a different measurement (or effect), which is physically detectable. For +example, Raman spectroscopy is detectable, but often requires virtual states and virtual transitions to +manifest, which themselves cannot be observed.43 The Fröhlich interaction describes the coupling of +electrons and phonons through the movement and ionization of a lattice.44 + +We provide two frameworks to qualitatively understand the exciton-phonon mechanism for producing +the evenly spaced peaks. First, when the exciton’s and phonon’s center-of-mass are coupled together, the +exciton recoil term and phonon creation-annihilation operators become coupled, thereby modulating + +1.94 +1.92 +Electron +1.9 +1.88 +1.86 +Phonon +3000 +1500 +Hole + +7 +exciton recombination energies at quantized phonon intervals.14 The second framework is shown in Figure +3b, where it assumes the electron and hole are positioned at different locations within the heterostructure, +but still bound to form an exciton. As the locations vibrate along a phonon mode, they change atomic +coordinates (e.g., configurational coordinates), which alters the overlap of the electron and hole +wavefunctions. We assume the Born-Oppenheimer approximation, where the recombination and photon +emission is instantaneous compared to the much slower atomic movements. As the electron and hole +positions oscillate, the conduction and valence bands change relative coordinates, thereby changing the +recombination pathways to different initial and final states, but only at quantized intervals that correspond +to the phonon energy.13,14 Note, configurational coordinate diagrams are plotted in real-space, which +displays the valence band as a parabola pointed down. + +Previous work found notable interlayer hybridization between Bi2Se3 and monolayer WS2 that greatly +impacts the WS2 excitonic activity,17,18 strains atoms at the interface,16 and induces the formation of a pure +electronic moiré lattice at the interface.16 Such interlayer hybridization facilitates the exchange of electrons +between the materials and modifies interlayer bonding, thereby setting conditions that encourage the +formation of interlayer quasiparticles.19 We note, the adjacent Bi2Se3 dramatically reduces the PL intensity +of the WS2, due in part to the emergence of non-radiative recombination pathways and charge transfer +from the monolayer WS2 into Bi2Se3.17,18 We also observe a strong reduction (>2x) in PL intensity. But in +addition, our work clearly reveals the presence of these exciton-phonon bound state interactions. + +Section S3 show Bi2Se3-WS2 temperature-dependent measurements, which reveal the localized exciton state +(LS) exhibits an unusual negative thermal quenching, where the LS intensity increases and peaks at ~75 K, +before decreasing.45 Previous theory work proposed that as the temperature increases, electrons are +thermally excited from nearby bound states into a primary state that allows for radiative recombination. +The LS and free exciton peak positions shift lower in energy as the temperature increases, in agreement +with the Varshni equation, demonstrating temperature dependent recombination energy.46 Together, the +PL results demonstrate complex behavior, consistent with notable interlayer hybridization between Bi2Se3 +and monolayer WS2.16,18 + +We believe it is unlikely that a WS2 phonon is forming the bound state. First, we observe no indications of +an exciton-phonon bound state in 60+ pristine monolayer WS2 samples probed, and we could not find any + + + +8 +literature reporting such an effect, suggesting Bi2Se3 grown on top is required to facilitate the quasiparticle. +Second, the WS2 phonon modes detected using Raman spectroscopy are likely too high energy. For +example, the nearest WS2 Raman mode, LA(M) at ~180cm-1 (or 22.3meV), is 15.5% above the measured +energy spacing of 19.3meV. Third, WS2 LA(M) is an acoustic mode, which are less likely to form bound +states with excitons compared to optical modes.13 In contrast, the Bi2Se3 A1(3) surface phonon is only 2.6% +above the measured energy spacing, it is an optical mode, and it is particularly susceptible to electron- +phonon coupling.9,10 + +It is unlikely that potential defects in WS2 resulting from the Bi2Se3 growth could alone – without the +presence of the Bi2Se3 on top – produce the evenly spaced peaks. Limited growth temperature (210°C) and +low growth time (27 min.) constrain potential selenium or bismuth doping. Further, selenization (i.e., +forming WS2xSe2(1-x) alloy) could not be detected in either PL or Raman spectra, suggesting potential +selenium doping has a minimal effect on the phonon or exciton modes.47 + +Exciton-phonon coupling and scattering plays a prominent role in monolayer TMDs, where it influences a +variety of properties, including valley polarization,48,49 spectral broadening,49,50 and mobility.51 Despite this, +phonon scattering alone is unlikely to produce periodic features (or vibronic transitions), and a more +complex mechanism is required. For example, the Franck-Condon Principle (FCP) enables multiple +quantized radiative recombination pathways.52 However, it is primarily applied to small molecules, whose +excited states shift the atomic coordinates.53 Conversely, the exciton-phonon quasiparticle is a theoretical +framework designed for crystals,13,14 where phonon transitions are central, leading us to conclude an +interlayer exciton-phonon quasiparticle is the most appropriate assignment. + +It should be noted that the FCP can be applied to crystalline solids when excitons self-localize and a strong +electron-phonon coupling is present.54,55 Evidence is emerging that conditions in TMD-based van der Waals +heterostructures may be conducive to self-localized excitons,56,57 raising the prospect that the free-excitons +in WS2 are self-trapping when forming the exciton-phonon bound state. Further research is required to +determine the model that best describes the system. + + + + +9 + +Figure 4: Prominent surface phonons and local symmetry breaking. (a)-(b) Linear and (c)-(d) circular polarized Raman response +from CVD-grown Bi2Se3-WS2 heterostructure and MBE-grown Bi2Se3-Al2O3 (reference sample). Black arrow identifies the A1(3) +surface phonon, which is associated with the interlayer exciton-phonon bound state. Phonon modes forbidden in a specific +scattering geometry (i.e., XX, XY, RR, or RL) are labeled with a boxed-outline.9 The presence of A1(2) and A1(3) surface phonons in a +forbidden scattering geometry (i.e., RL), suggests the local symmetry breaking at the surface,9 consistent with a strong WS2 +interlayer coupling. Porto notation is used (see methods). + + +To elucidate the interlayer exciton-phonon coupling, Figure 4 shows linear and circular polarization +Raman measurements at 4 K for CVD-grown Bi2Se3-WS2, and a Bi2Se3 reference sample, grown using +molecular beam epitaxy (MBE) on Al2O3. Previous work demonstrated the high quality of the MBE-grown +Bi2Se3 in our setup.58 Polarized Raman spectroscopy is a reliable method to identify and distinguish +different phonon modes with greater confidence because the Raman response can be separated from each +symmetry channel.9,59,60 More specifically, each crystallographic point group contains symmetries that can +be used to calculate the reduced Raman tensors, which describe scattering of phonon modes.60 Applying +polarization tensors to Raman tensors reveals the scattering efficiency of phonon modes exposed to +polarized light. + +Figure 4a-b (Figure 4c-d) show the linearly (circularly) polarized Raman measurements. Section S5 +contains expanded information on the Raman measurements, including each phonon mode’s attributes +and a literature comparison. Crystal symmetry forbids certain Raman scattering in Bi2Se3.60 If the symmetry +is disrupted, these modes can become allowed.9,60 The presence of A1(2) and A1(3) surface phonons in a +forbidden scattering geometry (i.e., RL), suggests the local symmetry breaking at the surface,9 consistent + +4-6QLBi2Se3+1LWS2(XX) +C. +3000- +一 +4-6QL Bi2Se3-WS2 (RR) +5QL Bi2Se3 + AI203 (XX) +180 A(2) +126 E,(2) +5QLBi2Se3+AI203(RR) +180 Arg(2) +4K +178 Ata(2) +4 K +126 E(2) +1) +178 At(2) +Intensity ( +126 E(2) +126 E,(2) +69 Arg(1) +2 +2000- +A +:74A1g +2 +159 A,(3) +1000- +1000 +136 E,(2) +159 A,(3) +500- +137 A,(2) +158 A,(3) +50 +100 +Raman Shift (cm-1) +200 +50 +100 +Raman Shift (cm-1) +200 +b.. +4-6QLBi2Se3+1LWS2(XY） +d. +126 E(2) +4 +4-6QL Bi2Se3-WS2 (RL) +1200 +5QLBi2Se3+A/203 (XY) +1200 +180 A1g(2) +5QLBi2Se3+AI203 (RL) +4 K +(cn' +126 E(2) +3 +69 E(1) +74 E(1) +1000. +138 A,(2) +178 A1g2 +nsity +Raman Intensity ( +126 E,(2) +160 A,(3) +126E(2) +Inter +800- +800. +RamanI +136 E,(2) +137 E,(2) +600- +179 A1g(2) +600. +137E(2) +76 +138 E,(2) +70 E(1) +400 +400 +50 +100 +Raman Shift (cm-1) +200 +50 +100 +RamanShift (cm-1) +200 + +10 +with a strong WS2 interlayer coupling. Additionally, the A1(2) and A1(3) surface phonon modes are markedly +more intense, consistent with an increased phonon population. + +Compared to the 5QL reference sample, as well as bulk Bi2Se3 from literature,9 we observe forbidden modes +at greater intensity in Bi2Se3-WS2 heterostructures, suggesting the crystal symmetry is being disrupted to a +greater degree. For example, the A1g(2) mode is forbidden in the XY configuration. While the reference +sample shows no detectable peak, we observe it prominently in the Bi2Se3-WS2 heterostructure. Of note, +although all the peaks observed correspond well to either Bi2Se3 or monolayer WS2 Raman modes,9,36 we +cannot exclude the possibility that combination61 or moiré Raman modes62 emerge at overlapping +wavenumbers. + +Numerous effects influence local crystalline symmetry, including defects, strain, and substrate effects. +Previous work found evidence of strain at the Bi2Se3-WS2 interface and a purely electronic 2D lattice +between the materials that alters the surrounding electronic environment.16 Such effects impact the +breakdown of translational symmetry at the interface, which subsequently affects surface phonon modes.63 + +Our Raman measurements are consistent with a strong interlayer coupling that encourages strain and +charge redistribution at the interface. Together, notable interlayer hybridization16,18 and a strong interlayer +coupling form an interfacial environment that facilitates interlayer interactions of different particles, +thereby encouraging the formation of exciton-phonon quasiparticles.19 Additionally, our measurements +demonstrate a rich landscape of phonons, including comparatively intense surface phonons, which are +consistent with an increased phonon population. + +Conclusions + +We have demonstrated formation of an interlayer exciton-phonon quasiparticle (or exciton-phonon bound +state) between localized excitons in monolayer WS2 and the Bi2Se3 A1(3) surface phonon, a mode particularly +susceptible to electron-phonon coupling.9,10 We detect evenly spaced features in the PL spectrum with an +energy separation that matches the Bi2Se3 A1(3) surface phonon, overlayed on the WS2 localized exciton +emission peak. Polarized Raman spectroscopy detects forbidden Bi2Se3 surface phonon modes, suggesting +broken crystalline symmetry at the surface. While several publications have reported exciton/electron- + + + +11 +phonon coupling across the interlayer region,5,23–26 our work is distinct in that we report the formation of +an interlayer exciton-phonon bound state, an uncommon quasiparticle composed of phonons, whose +particle number is not conserved.13,14 Bi2Se3/monolayer WS2 heterostructures offer the possibility to +combine strong light-matter interaction and spin-locked current into one material. Understanding the +interlayer coupling is central to elucidating how their combined properties evolve, enabling devices for +spintronics,18,20 valleytronics,21 and quantum computing21,22 applications. + +Methods +Material Growth – Few-layer Bi2Se3: The Bi2Se3 films were grown on 10 × 10 mm2 c-plane (0001) sapphire +(Al2O3) substrates using molecular beam epitaxy (MBE) with base pressure below 5 × 10−10 Torr. The +substrates were initially annealed ex-situ at 1,000 ℃ under the atmospheric pressure, and ozone cleaned +in-situ under 200 Torr of oxygen pressure. It is then annealed at 600 ℃ for 20 min in the ultra-high vacuum +MBE chamber. Individual sources of high-purity (99.999%) Bi and Se were evaporated from standard +effusion cells during the film growth. Se flux was maintained at least ten times higher than Bi’s to minimize +Se vacancies. To obtain an atomically sharp interface between the Bi2Se3 layer and the substrate, we +adopted the two-step growth scheme.64 First, the initial 3 QL Bi2Se3 is grown at 170 ℃. It is slowly annealed +to 300 ℃, and followed by deposition of the remaining 2QL Bi2Se3 layers. 5QL Bi2Se3 was grown. + +Material Growth – Bi2Se3-WS2 2D Heterostructure: Monolayer WS2 is synthesized at ambient pressure in +2-inch diameter quartz tube furnaces on SiO2/Si substrates (275 nm thickness of SiO2). Prior to use, all +SiO2/Si substrates are cleaned in acetone, IPA, and Piranha etch (H2SO4+H2O2) then thoroughly rinsed in +DI water. At the center of the furnace is positioned a quartz boat containing ~1g of WO3 powder. Two +SiO2/Si wafers are positioned face-down, directly above the oxide precursor. A separate quartz boat +containing sulfur powder is placed upstream, outside the furnace-heating zone. The upstream SiO2/Si +wafer contains perylene-3,4,9,10-tetracarboxylic acid tetrapotassium salt (PTAS) seeding molecules, while +the downstream substrate is untreated. The hexagonal PTAS molecules are carried downstream to the +untreated substrate and promote lateral growth of monolayer WS2. Pure argon (65 sccm) is used as the +furnace heats to the target temperature. Upon reaching the target temperature in the range of 825 to 875 +°C, 10 sccm H2 is added to the Ar flow and maintained throughout the 10-minute soak and subsequent +cooling to room temperature. + +4-6QL Bi2Se3 was grown on top of monolayer WS2 using chemical vapor deposition (CVD) in a two-zone +furnace with a 2” quartz tube. High-purity Bi2Se3 flakes are ground using a mortar and pestle into a fine +dust. The powdered Bi2Se3 is placed in a ceramic boat and inserted into the furnace’s quartz tube, and +pushed into the center of the furnace’s first zone. The monolayer WS2, which is on an SiO2 substrate, is +placed downstream of the Bi2Se3 into the center of the furnace’s second zone. The furnace is pumped down +to ~20mTorr. An argon (Ar) carrier gas is flown into the furnace at 80sccm. The Bi2Se3 is heated to 520C, +and the WS2 is heated to 210C. The ramp rate is ~55C/min, and the total growth is 27 min. + +Raman and photoluminescence measurements at low temperature: A Horiba LabRARM HR Evolution +with both linear and circular polarization attachments, and a low-temperature Montana cryostat, was used +for Raman and photoluminescence (PL) spectroscopy measurements. We use Porto Notation (i.e., 𝑧̅()𝑧) + + + +12 +where () is the incident (scattered) polarization, and 𝑧̅(𝑧) is the incoming (outgoing) direction. We define +𝑅 = 𝑋 + 𝑖𝑌 and 𝐿 = 𝑋 − 𝑖𝑌. Previous work showed that laser exposure of monolayer materials at low +temperature can anneal and laser-dope them.65,66 We attempted to mitigate this using very low powers +(~320nW) and short exposure times (~30s) for most measurements. We verified that the laser exposure had +a minimal effect on the material by collecting multiple successive spectra. A long-distance 50x objective +was used with a laser spot diameter of ~1.9 μm at the lowest powers. + +Transmission Electron Microscopy: Bi2Se3-WS2 2D heterostructures were transferred onto a holey +amorphous SiNx TEM grid using the water-assisted-pick-up transfer method.67 Selected area electron +diffraction (SAED) were performed with a JEOL JEM2200FS operating at 200 kV, equipped with a high- +speed Gatan OneView camera. The SAED patterns were internally calibrated to the WS2, and an aperture +of approximately 200nm was used. We suspect that the wet transfer of Bi2Se3-WS2 2D heterostructures +partially disrupts the crystal order, possibly due to a combination of the force and liquids applied during +the transfer process. + +Photoluminescence Spectra computational analysis and fitting: All code was written in Python using the +Spyder integrated development environment (IDE). Spyder belongs to the MIT License and is distributed +through the Anaconda environment. The curve_fit() function with a variety of initial values and boundary +conditions were used to verify fit robustness. The fitting was corroborated using cross-validation, where +we uniformly removed 20% of the data points from a spectrum and repeated fitting. No notable changes +to the fitting were detected, suggesting noise is not skewing the fit. +The energy location of each feature was extracted by taking the double derivative of the best fit, and finding +the minimum values. The minimum values of the double derivative matched well with the peak locations +of Lorentzian functions, reinforcing our method to quantitatively extract the features. + +Acknowledgments +We thank Dr. Darshana Wickramaratne at the Naval Research Laboratory for their insight and fruitful +discussions. + +Supporting Information +Additional characterization of representative regions probed (Section S1); Power-Dependent +Measurements (Section S2); Temperature-Dependent Measurements (Section S3); Expanded Analysis of +the Interlayer Exciton-Phonon Bound State (Section S4); Expanded analysis of low-temperature (4 K) +Bi2Se3 phonon modes (Section S5); + + + + +13 +References + +(1) +Geim, A. K.; Grigorieva, I. V. Van Der Waals Heterostructures. Nature 2013, 499 (7459), +419–425. https://doi.org/10.1038/nature12385. +(2) +Ciarrocchi, A.; Tagarelli, F.; Avsar, A.; Kis, A. Excitonic Devices with van Der Waals +Heterostructures: Valleytronics Meets Twistronics. Nat. Rev. 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Li,2 Olaf M.J. van ’t Erve2, +and Berend T. Jonker2,* +1 NRC Postdoc at the Materials Science and Technology Division, Naval Research Laboratory, +Washington, D.C. 20375, USA +2 Materials Science and Technology Division, Naval Research Laboratory, Washington, D.C. 20375, USA + + + + + + +19 +Section S1. Additional characterization of representative regions probed + + +Figure S1. Bi2Se3 islands grow together forming a nearly continuous 4-6QL film on monolayer WS2. (a) AFM scan around +location of blue spot in (b). (b) optical image where blue spot identifies representative location probed. (c) Fluorescence image +showing that the photoluminescence (PL) is fully quenched at the center, indicating the Bi2Se3 is a nearly 4-6QL continuous film. + +Previous work found that forming heterostructures by growing 1QL Bi2Se3 on a monolayer (1L) transition +metal dichalcogenide (TMD) significantly reduces the TMD PL intensity compared to their as-grown (i.e., +bare) 1L TMDs counterparts,1–4 likely due in part to the formation of an indirect bandgap and static charge +transfer. Further, it was found that as additional Bi2Se3 layers are grown on top, the PL continues to +diminish and quench because the bandgap becomes increasingly indirect with increasing Bi2Se3.4 This +effect is comparable to the PL intensity evolution as a TMD layer count increases (i.e., monolayer vs. bilayer +vs. trilayer). More specifically, increasing layer count from 1L to 2L dramatically reduces the PL intensity, +and increasing from 2L to 3L further diminishes the PL intensity. As layer count increases, the sample +approaches the bulk properties. + +Oun +41.2nm +10μm +40.0 +38.0 +36.0 +34.0 +32.0 +30.0 +28.0 +26.0 +24.0 +22.0 +20.0 + +20 +Section S2. Power-dependent measurements + +Section S2 shows power-law fitting to power-dependent measurements, which is used to identify the +species of exciton each peak originates from (e.g., localized state, free exciton, biexciton). The localized state +(LS) has a coefficient of 0.688, while the free exciton (FE) is 0.975, and the ratio of localized-to-free exciton +intensity decreases with increasing power, enabling us to label the excitons with high confidence. Note, +there are a diverse number of localized and bound excitons, whose classification depends in part on local +chemistry and the spatial extent of the wavefunction. We cannot identify definitively the type of localized +exciton. +At low powers, the ratio of localized to free exciton is greater. However, as the power increases, the ratio +decreases because the number of electrons excited from the valence band to the conduction band increases. +As the electrons recombine, they begin to saturate the localized states, pushing a higher ratio of electrons +in the free exciton states (Figure S2), which are less easily saturated.5,6 + + + +Figure S2. 4-6QL Bi2Se3 + monolayer WS2 2D heterostructure power-dependent measurements. + + + +a. +b. +2 105 +0.025μW +0.028μw +2.5106 +0.257μw +0.393uW +Free exciton has +0.834uw +linearpower +1.61μW +3.32uw +1.5105 +2106 +6.130W +10.1μW +13.8uw +Bi,Se3-WS2 +PL Intensity (a.u) +22.9uw +PL Intensity (a.u) +28.0uw +1.5106 +46.5μW +1105- +93.50A +190u +398pv +1106 +Localized statehassub- +5104- +5105 +linearpowerdependence +0 +0 +Energy (eV)1.6 +1.7 +1.8 +1.9 +2 +2.1 +Energy(eV)1.6 +1.7 +1.8 +1.9 +-2 +2.1 + +21 + +Figure S3. As-grown monolayer WS2 power-dependent measurements. Graphed evolution or curve peaks is shown below in +Figure S4. + + +Figure S4. As-grown monolayer WS2 power-dependent measurements: Peak intensities plotted from Figure S3. The expected +behavior validates the assignment of the localized and free excitons. (a) Shows the localized state exciton (LS) and (b) the free +exciton (LE), which have a fitting coefficient of less than one and approximately equal to one, respectively. The ratio of LS/FE +decreases with laser power. Together, the results provide high confidence of the locations of the localized and free exciton states.5,6 + +The peak intensities were extracted using Spyder integrated development environment (IDE) and the curve_fit() function. The +results were corroborated with manual inspection of the highest intensity pixel recorded by the equipment for each exciton curve. + + + + +1.2105 +0.025uW +1106 +1105 +0.257uW +0.834uw +8104 +1.61uW +6104 +3.32uw +6.13uW +8105 +13.8uW +2104 +28.0uw +Intensity (a.u.) +Energy(eV)1.9 +2 +6 +6105 +4105 +MonolayerWS2 +2105_ +0 +Energy (eV) +1.8 +1.9 +2 +2.1a. 106 +LocalizedState +b. +C. +FreeExciton +25 +LS/FERatio +105 +20 +Localized state rapidly + Intensity (a.u.) +saturates as laser power +105 +increases,pushing more +PL Intensity ( +104 +15 +electrons into FE +of PL +103 +104 +Power law fitting +=1.2e+6*x^(0.6876 +=1.0e+5*x^(0.9745) +PowerApplied(uw) +Power Applied (μuw) +PowerApplied (uw) +1.61 +0.257 +1.61 +6.13 +28.0 +6.13 +28.0 +1.61 +6.13 +28.0 + +22 +Section S3. Temperature-dependent measurements + + +Figure S5. 4-6QL Bi2Se3 + monolayer WS2 2D heterostructure temperature-dependent measurements. (a) 2D density plot of the +PL spectra with temperature, showing the localized and free exciton states. (b)-(c) PL spectra as a function of temperature. +Interestingly, negative thermal quenching is observed for the localized state, an unusual phenomena where the PL intensity +increases with temperature.7 Previous theory work proposed that as the temperature increases, electrons are thermally excited +from nearby bound states into a primary state that allows for radiative recombination.7 The peak position shifts lower as the +temperature increases, in agreement with the Varshni equation, formalism used to describe the PL evolution in semiconductors.8 + +21 +1600 +1600 +FreeExciton +10k +C. +75k +1400 +15k +80k +180k +1400- +1400- +85k +20k +90k +200k +2.0 +1200 +25k +95k +1200- +30k +1200 +100k +2201 +35k +110k +230k +1000 +15 +240k +1.9 +1000- +40k +125k +250k +45k +130k +260k +800 +50k +140k +280k +800- +55k +150k +270k +18 +600 +160k +60k +600- +65k +600 +400 +70k +1.7 +751 +400 +nsity +200 +LocalizedState +200 +1.6 +10 +OE +50 +75 +5100 +150 +200250290 +0- +0- +Energy (eV)1.7 +1.8 +1.9 +2 +2.1 +Energy (eV) +1.7 +1.8 +1.9 +2 +2.1 + +23 +Section S4. Expanded analysis of the interlayer exciton-phonon bound state + + +Figure S6. Bi2Se3-WS2 data from Figure 2 fitted with Lorentzian functions. We obtain a function that follows the curve and +captures the data form. Inflection points were quantitatively identified by taking a double derivative of the best fit. Note, we only +use the fitting methodology as a tool to quantitatively extract the inflection points. We do not extract insight from the Lorentzian +fitting. + + +Figure S7. Representative data with peaks that are approximately equally spaced. + +Table 1. Spacings between Double Derivative Extrema. Multiple Lorentzian functions were fit to the data using Python +Software (see methods). The double derivative was taken of the best fit function and the minimum extrema were identified. The +spacing between the extrema is calculated. +Spacing +Figure S6 +Figure S7a +Figure S7b +1st +20.75 +18.49 +18.87 +2nd +16.79 +18.06 +20.5 +3rd +18.13 +23.85 + +4th +21.54 + + + + + + +Average +19.3meV +20.1meV +19.7meV + + +Methodology for Measuring Spacing of the Interlayer Exciton-Phonon Bound State Features: + +The exciton-phonon bound state (or exciton-phonon quasiparticle) presents itself as evenly spaced peaks +or features, where the spacing is approximately the phonon energy. As such, the presence of an exciton- + +a.3000 +4 K +Bi2Se3-WS2PL +2500 +Best Fit +Multiple +FELorentzian +localizedstate +Intensity (a.u.) +LS Lorentzians +2000. +transitions +1500. +Uniformlyspaced +energytransitions +1000 +PL +Free Exciton +500 +0. +b. +1.8 +1.9 +Energy (eV) +2a. +b +1200 +800 +Photon Counts (a.u.) +1000 +(a.u.) +008 +600 +PhotonCounts( +600 +400 +400 +200 +200 +0 +0 +1.65 +1.70 +175 +1.80 +1.85 +1.90 +1.95 +2.00 +2.05 +1.65 +1.70 +1.75 +180 +185 +1.90 +1.95 +2.00 +2.05 +ev +ev + +24 +phonon or electron-phonon quasiparticle can be inferred when evenly spaced peaks are observed that +correspond to a Raman phonon mode.9–14 The number of peaks and consistency of the even spacing is +material and environment specific, as well as the strength of the exciton-phonon coupling.10,11 Further, the +task is complicated by the fact that the evenly spaced features are frequently overlayed on a larger peak, +which obscures their precise peak position.9–14 + +When analyzing the PL spectra for a possible exciton-phonon bound state, we start by identifying possible +features that could be peaks and then measuring their spacing. To the best of our knowledge, the +community primarily relies on visual inspection when identifying and measuring possible exciton-phonon +peaks. In this work, we applied two methods for peak identification. The primary method is visual +inspection, while the secondary method is a quantitative analysis of best fit double derivative extrema. +Note, we only use the fitting methodology as a tool to quantitatively extract the inflection points. We do +not extract insight from the Lorentzian fitting. + +Only if both methods yielded the same result, did we label the feature as a peak and measure spacings. We +fit the experimental data with either 7 or 8 Lorentzian functions, which was sufficient to obtain a best fit +that corresponded well to the data moving average. We then took the double derivative of the best fit and +identified the minima extrema. We verified each double derivative minima corresponded to a clear visual +feature. We then measured the spacing between each double derivative minima. The double derivative +extrema frequently corresponded to the location of Lorentzian peaks, but did not overlap exactly. + + + + + +25 +Section S5. Expanded analysis of low-temperature (4 K) Bi2Se3 phonon modes + + +Table S2. Summary of low-temperature Bi2Se3 phonon modes for Bi2Se3-WS2 and Bi2Se3-Al2O3. Increased symmetry breaking +was observed in Bi2Se3-WS2 heterostructures compared to Bi2Se3-Al2O3. +Phonon +Mode +Bulk/Surface +Scattering +Geometry +Bi2Se3-WS2 +(cm-1) +Bi2Se3-Al2O3 +(cm-1) +Literature: Bulk +Bi2Se3-Al2O3 (cm-1) +Notable Symmetry +breaking +A1(2) +Surface +RR & XX +137 +137 +13615 & 12916 +Yes (RL Channel) +A1(3) +Surface +RR & XX +160 +159 +15815 & 16016 +Yes (RL Channel) +A1g(1) +Bulk +RR & XX +69 +74 +7515 & 7316,17 & 7218 +No +A1g(2) +Bulk +RR & XX +180 +178 +18015 & 17516,17 & 17418 Yes (XY Channel) +E(1) +Surface +RL +69 +74 +6715 & 6816 +No +E(2) +Surface +RL +126 +137 +12615 & 12516 +Yes (RR Channel) +Eg(2) +Bulk +RL +126 +136 +13715 & 13316 & 13117,18 Yes (RR Channel) + + +The above Raman modes are identified with the assistance of previous work by matching them to the +wavenumber and polarization response, which reveals the symmetry channel.15,19,20 We primarily relied +upon Kung et al. and Gnezdilov et al. for understanding the symmetry channels and polarized Raman +spectroscopy results.15,16 Our data presented HERSE for low-dimensional Bi2Se3 using polarized Raman +spectroscopy and at low-temperatures are among the few in the literature. 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B 2011, 84 (19), 195118. https://doi.org/10.1103/PhysRevB.84.195118. +(17) Irfan, B.; Sahoo, S.; Gaur, A. P. S.; Ahmadi, M.; Guinel, M. J.-F.; Katiyar, R. S.; Chatterjee, R. +Temperature Dependent Raman Scattering Studies of Three Dimensional Topological Insulators +Bi2Se3. J. Appl. Phys. 2014, 115 (17), 173506. https://doi.org/10.1063/1.4871860. +(18) Zhang, X.; Tan, Q.-H.; Wu, J.-B.; Shi, W.; Tan, P.-H. Review on the Raman Spectroscopy of +Different Types of Layered Materials. Nanoscale 2016, 8 (12), 6435–6450. +https://doi.org/10.1039/C5NR07205K. +(19) Boulares, I.; Shi, G.; Kioupakis, E.; Lošťák, P.; Uher, C.; Merlin, R. Surface Phonons in the +Topological Insulators Bi2Se3 and Bi2Te3. Solid State Commun. 2018, 271, 1–5. +https://doi.org/10.1016/j.ssc.2017.12.012. +(20) Chis, V.; Sklyadneva, I. Yu.; Kokh, K. A.; Volodin, V. A.; Tereshchenko, O. E.; Chulkov, E. V. +Vibrations in Binary and Ternary Topological Insulators: First-Principles Calculations and +Raman Spectroscopy Measurements. Phys. Rev. B 2012, 86 (17), 174304. +https://doi.org/10.1103/PhysRevB.86.174304. +(21) Koster, G. F.; Dimmock, J. O.; Wheeler, R. G.; Statz, H. The Properties of the Thirty-Two Point +Groups; MIT Press: Cambridge, MA, USA, 1963. + +s + diff --git a/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/load_file.txt b/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b576bd19c7858b388cfbf9047a84bbdc232ded77 --- /dev/null +++ b/ndE0T4oBgHgl3EQfZgBD/content/tmp_files/load_file.txt @@ -0,0 +1,2478 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf,len=2477 +page_content='1 Interlayer Exciton–Phonon Bound State in Bi2Se3/monolayer WS2 van der Waals Heterostructures Zachariah Hennighausen1,*, Jisoo Moon1, Kathleen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' McCreary2, Connie H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Li,2 Olaf M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' van ’t Erve2, and Berend T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Jonker2,* 1 NRC Postdoc at the Materials Science and Technology Division, Naval Research Laboratory, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 20375, USA 2 Materials Science and Technology Division, Naval Research Laboratory, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 20375, USA Abstract The ability to assemble layers of two-dimensional (2D) materials to form permutations of van der Waals heterostructures provides significant opportunities in materials design and synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Interlayer interactions provide a path to new properties and functionality, and understanding such interactions is essential to that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Here we report formation of interlayer exciton-phonon bound states in Bi2Se3/WS2 heterostructures, where the Bi2Se3 A1(3) surface phonon, a mode particularly susceptible to electron- phonon coupling, is imprinted onto the excitonic emission of the WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The exciton-phonon bound state (or exciton-phonon quasiparticle) presents itself as evenly separated peaks superposed on the WS2 excitonic photoluminescence spectrum, whose periodic spacing corresponds to the A1(3) surface phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Low-temperature polarized Raman spectroscopy of Bi2Se3 reveals intense surface phonons and local symmetry breaking that allows the A1(3) surface phonon to manifest in otherwise forbidden scattering geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Our work advances knowledge of the complex interlayer van der Waals interactions, and facilitates technologies that combine the distinctive transport and optical properties from separate materials into one device for possible spintronics, valleytronics, and quantum computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Authors for correspondence, E mail: hennigha@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' berry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='jonker@nrl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='navy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='mil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Keywords: exciton phonon bound state, exciton phonon quasiparticle, exciton phonon coupling, interlayer quasiparticle, monolayer WS2, Bi2Se3 2 Introduction Van der Waals heterostructures are formed by stacking monolayers of 2D materials in any sequence of one’s choosing, enabled by the lack of bonds between layer planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 Such stacking often results in new properties, tuned by either material selection or twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='2 Understanding the interaction between the layers in such heterostructures is essential to discovering new properties, engineering functionality, and advancing their application into technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work showed the interlayer interaction between stacked two dimensional (2D) materials can facilitate a host of new properties, including long-lived interlayer excitons,3 magnetic phase switching,4 forbidden Raman modes,5 superconductivity,6 orbital ferromagnetism,7 and emergent ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 In this work, we grew few-layer Bi2Se3 on monolayer WS2, and observe interlayer exciton-phonon coupling and formation of an exciton-phonon bound state between localized excitons in monolayer WS2 and the Bi2Se3 A1(3) surface phonon, a mode particularly susceptible to electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,10 The bound state is manifested as a series of evenly spaced peaks superposed on the WS2 excitonic photoluminescence (PL) spectrum, whose periodic spacing corresponds to the A1(3) surface phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Oscillating features that match a phonon energy and correspond to a luminescence are indicative of electron-phonon or exciton-phonon bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='11–15 In addition, polarized Raman spectroscopy of the Bi2Se3 reveals multiple pronounced surface phonon modes and crystalline symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Notably, the presence of surface phonons in a forbidden scattering geometry suggests local symmetry breaking at the surface,9 consistent with a strong WS2-Bi2Se3 interlayer coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work found significant interlayer hybridization in Bi2Se3/WS2 heterostructures, facilitating electron transfer and modifying the bonding,16–18 conditions which encourage the formation of interlayer quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='19 Understanding the interlayer interaction is central to elucidating how their combined properties evolve, enabling the discovery of advanced capabilities for spintronics,18,20 valleytronics,21 and quantum computing21,22 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Several publications have reported exciton-phonon or electron-phonon coupling across the interlayer region, where a free exciton in a monolayer transition metal dichalcogenide (TMD) facilitates the emergence of otherwise forbidden Raman modes in the adjacent material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5,23–26 Additionally, interlayer vibronic exciton-phonon states have been reported in a TMD/TMD diode using photocurrent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='27 Our work is distinct in that we report the formation of an interlayer exciton-phonon 3 bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13,14 Exciton-phonon bound states are uncommon quasiparticles that are formed from particles whose number is not conserved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', phonons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13,14 This means the exciton-phonon bound states are only stable if their real decay is forbidden by the exciton laws of conservation of energy and momentum, during phonon disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='14 While intralayer exciton-phonon bound states have been reported in low- dimensional materials,12 to our knowledge, such a state has never been observed to form across an interlayer region in a vdW heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Both bismuth selenide (Bi2Se3) and monolayer tungsten disulfide (1L WS2) have independently demonstrated promise for a variety of advanced technologies, including spintronics18,20 and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='21,22 Bi2Se3 is a topological insulator (TI) with gapped bulk states and gapless surface conducting states that are spin-momentum locked, enabling advanced field effect transistors (FETs),28 magneto-electric devices,29 topological qubits,22 and spin-orbit torque (SOT) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='20,30 Monolayer WS2 is a direct band gap semiconductor with independently optically addressable valleys, a strong light-matter interaction, and high sensitivity to surrounding fields, enabling FETs,31 sensors,32 valleytronics devices,21 ferroelectric modulation of exciton populations,33 and various photonic devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', LEDs, modulators, lasers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='34 Bi2Se3/monolayer WS2 heterostructures offer the possibility to combine strong light-matter interaction and spin-locked current into one device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Results Figure 1: Characterization of 4-6QL Bi2Se3 + 1L WS2 2D heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) AFM scan and (b) line profile corresponding to blue line in 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Blue spot marks a location with continuous film (Section S1) and a representative location studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (c) TEM SAED image 4-6QLBi,Se3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1nm Bi,Se a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 1LWS, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 4-6QLBizSe3grown C WS2 ontopmonolayerWS2 SiO2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='58A 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3nm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 Sio 10μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 0-2°twistangle 20000 4 K 180 A1。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='(2) 4-6QLBiSe+1LWS (XY) 296 2LA(M)-2E*2g(M) 4000 4-6QL Bi, Se, + 1L WS, ( 74 A1g(1) (XX) 126 E(2) 3512LA(M) (2) 160 A,(3) 419 A1g(T) RamanIntensity( 523 2000 323 1000 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 100 200 300 RamanShift（cm-1） 500 4 showing Bi2Se3 grows crystalline on WS2 across a 0-2\uf0b0 twist angle range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (d) Linearly polarized Raman response at 4 K with modes identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Porto notation is used (see methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 1 presents data characterizing the as-grown 4-6 quintuple layer (QL) Bi2Se3 + monolayer (1L) WS2 vdW heterostructure, synthesized using chemical vapor deposition (CVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 1a-b shows an atomic force microscope (AFM) scan and corresponding line profile, respectively, of a representative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' For reference, each QL of Bi2Se3 is ~ 1nm thick, and a monolayer of WS2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 nm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Based on the AFM data and diminished PL intensity, we conclude that Bi2Se3 grew as a uniform QL over 1L WS2, with taller islands (~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7nm) on top that merge together to form a continuous multilayer film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Section S1 shows additional AFM data and fluorescence measurements showing that the Bi2Se3 is initially continuous, followed by growth of multilayer islands characteristic of Bi2Se3 growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Transmission electron microscope selected area electron diffraction (TEM-SAED) measurements demonstrate both the WS2 and Bi2Se3 grow crystalline, and that the Bi2Se3 grows within a narrow range of twist angles (0-2\uf0b0) around 0\uf0b0 aligned with the WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Our TEM results are in agreement with previous work,35 indicating the interlayer interaction is sufficiently strong to induce epitaxial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 1d shows linearly polarized Raman measurements taken at 4 K where the well-formed modes correspond closely to the respective materials,9,36 suggesting good crystalline and stoichiometric sample quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 2: Manifestation of interlayer exciton-phonon bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) 4-6QL Bi2Se3 + 1L WS2 2D heterostructure PL at 4 K showing the free exciton and localized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Evenly spaced “steps” are observed, whose energy spacing (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3meV) corresponds well to the 160cm-1 A1(3) surface phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work identified the A1(3) mode as particularly susceptible to electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,10 (b) 1L WS2 PL at 4 K, where a comparatively smooth localized state is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 4 K BiSe3 WS,PL Exciton Phonon BoundState Uniformly spaced energytransitions FreeExciton 4 K MonolayerWS,PL Localizedstate Comparativelysmooth line shape with no pronouncedsteps FreeExciton 5 Figure 2a shows the PL spectrum of the Bi2Se3/WS2 heterostructure sample used in Figure 1, which exhibits a red shift compared to the as-grown monolayer WS2 reference sample (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' A similar red shift was reported in graphene/WS2 heterostructures, and is attributed to charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='37 Both spectra are dominated by a broad asymmetric feature approximately 130 meV below the free exciton with a long tail at low energy, generally attributed to contributions from multiple defect-bound exciton complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='38–40 Power-dependent (Section S2) and temperature-dependent (Section S3) measurements validate our assignment of exciton species, where the lower energy peak is a localized exciton (or bound exciton), and the higher energy peak is a neutral free exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' In addition, the heterostructure PL exhibits regularly spaced peaks not observed in the reference monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The peaks on the high energy side are more prominent, while peaks on the low energy side are likely obscured by the low energy tail of the bound excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We use visual inspection to identify these peak positions to determine their energy separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Efforts to quantitatively extract these positions by analyzing the double derivative extrema yielded comparable results (Section S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Our analysis found the peaks to be approximately evenly spaced, with an energy spacing of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3meV (156cm-1), which corresponds to the Bi2Se3 A1(3) surface phonon mode (160cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Of note, previous work found the A1(3) mode produces a Fano shape under resonant conditions, indicative of electron-phonon coupling at the Bi2Se3 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,10 Electron-phonon and exciton-phonon bound states (or quasiparticles) have been shown to produce a series of equally spaced features, often overlayed on a larger PL or optical reflection curve, where the period of the emerged features approximately matches the phonon’s energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='11–15,41 As such, the presence of such quasiparticles can be inferred when evenly spaced peaks are observed that correspond to a Raman phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 The number of peaks and consistency of their spacing is material and environment specific, as well as the strength of the exciton-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='12,13 The task is further complicated by the fact that the evenly spaced features are frequently overlayed on a larger peak, which obscures their precise peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='11–15 Exciton-phonon bound states are frequently strongest when the exciton is localized (or bound) to an impurity center,13,14 consistent with our observations where the quasiparticle corresponds to the localized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We note the formation of an exciton-phonon quasiparticle is distinct from exciton-phonon scattering, in part because the momenta of the exciton and phonon are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='14 As such, exciton-phonon quasiparticle interactions can result in uncommon observations, such as the emission of distinct features above the exciton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='12–14 6 The energy spacing was independently observed and measured in several samples (see Section S4 for data from additional samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We therefore attribute these multiple peaks as arising from exciton-phonon coupling to the Bi2Se3 A1(3) surface phonon, a mode with an energy of ~19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 meV that is particularly susceptible to electron-phonon coupling,9,10 which together form am exciton-phonon quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The PL spectrum of as-grown monolayer WS2 shown in Fig 2b does not show such additional peaks, demonstrating that coupling to the Bi2Se3 is responsible for these dramatic features in the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 3: Schematic of exciton-phonon quasiparticle recombination pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) A bound exciton coupled with a phonon to form a bound exciton-phonon quasiparticle, where the phonon interaction modifies the exciton recombination energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (b) The phonon interaction can induce quantized energy levels corresponding to the phonon mode energy, which leads to multiple recombination pathways that manifest as equally spaced features in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Note, configurational coordinates represent the relative real-space displacement of the initial (electron) and final (hole) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Inset is data from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 3 presents a graphical explanation of the exciton-phonon bound state quasiparticle,14 and a mechanism for generating evenly spaced peaks in exciton-phonon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='42 The exciton-phonon bound state quasiparticle is only understood by combining several concepts from quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The quasiparticle is composed of an exciton and a phonon, which are coupled together through virtual transitions that are related to the Fröhlich interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='14,42 Virtual transitions are short-lived, unobservable quantum effects that can facilitate a different measurement (or effect), which is physically detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' For example, Raman spectroscopy is detectable, but often requires virtual states and virtual transitions to manifest, which themselves cannot be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='43 The Fröhlich interaction describes the coupling of electrons and phonons through the movement and ionization of a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='44 We provide two frameworks to qualitatively understand the exciton-phonon mechanism for producing the evenly spaced peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' First, when the exciton’s and phonon’s center-of-mass are coupled together, the exciton recoil term and phonon creation-annihilation operators become coupled, thereby modulating 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='92 Electron 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='86 Phonon 3000 1500 Hole 7 exciton recombination energies at quantized phonon intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='14 The second framework is shown in Figure 3b, where it assumes the electron and hole are positioned at different locations within the heterostructure, but still bound to form an exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As the locations vibrate along a phonon mode, they change atomic coordinates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', configurational coordinates), which alters the overlap of the electron and hole wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We assume the Born-Oppenheimer approximation, where the recombination and photon emission is instantaneous compared to the much slower atomic movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As the electron and hole positions oscillate, the conduction and valence bands change relative coordinates, thereby changing the recombination pathways to different initial and final states, but only at quantized intervals that correspond to the phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13,14 Note, configurational coordinate diagrams are plotted in real-space, which displays the valence band as a parabola pointed down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work found notable interlayer hybridization between Bi2Se3 and monolayer WS2 that greatly impacts the WS2 excitonic activity,17,18 strains atoms at the interface,16 and induces the formation of a pure electronic moiré lattice at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='16 Such interlayer hybridization facilitates the exchange of electrons between the materials and modifies interlayer bonding, thereby setting conditions that encourage the formation of interlayer quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='19 We note, the adjacent Bi2Se3 dramatically reduces the PL intensity of the WS2, due in part to the emergence of non-radiative recombination pathways and charge transfer from the monolayer WS2 into Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='17,18 We also observe a strong reduction (>2x) in PL intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' But in addition, our work clearly reveals the presence of these exciton-phonon bound state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Section S3 show Bi2Se3-WS2 temperature-dependent measurements, which reveal the localized exciton state (LS) exhibits an unusual negative thermal quenching, where the LS intensity increases and peaks at ~75 K, before decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='45 Previous theory work proposed that as the temperature increases, electrons are thermally excited from nearby bound states into a primary state that allows for radiative recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The LS and free exciton peak positions shift lower in energy as the temperature increases, in agreement with the Varshni equation, demonstrating temperature dependent recombination energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='46 Together, the PL results demonstrate complex behavior, consistent with notable interlayer hybridization between Bi2Se3 and monolayer WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='16,18 We believe it is unlikely that a WS2 phonon is forming the bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' First, we observe no indications of an exciton-phonon bound state in 60+ pristine monolayer WS2 samples probed, and we could not find any 8 literature reporting such an effect, suggesting Bi2Se3 grown on top is required to facilitate the quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Second, the WS2 phonon modes detected using Raman spectroscopy are likely too high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' For example, the nearest WS2 Raman mode, LA(M) at ~180cm-1 (or 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3meV), is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5% above the measured energy spacing of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Third, WS2 LA(M) is an acoustic mode, which are less likely to form bound states with excitons compared to optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 In contrast, the Bi2Se3 A1(3) surface phonon is only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6% above the measured energy spacing, it is an optical mode, and it is particularly susceptible to electron- phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,10 It is unlikely that potential defects in WS2 resulting from the Bi2Se3 growth could alone – without the presence of the Bi2Se3 on top – produce the evenly spaced peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Limited growth temperature (210°C) and low growth time (27 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') constrain potential selenium or bismuth doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Further, selenization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', forming WS2xSe2(1-x) alloy) could not be detected in either PL or Raman spectra, suggesting potential selenium doping has a minimal effect on the phonon or exciton modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='47 Exciton-phonon coupling and scattering plays a prominent role in monolayer TMDs, where it influences a variety of properties, including valley polarization,48,49 spectral broadening,49,50 and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='51 Despite this, phonon scattering alone is unlikely to produce periodic features (or vibronic transitions), and a more complex mechanism is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' For example, the Franck-Condon Principle (FCP) enables multiple quantized radiative recombination pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='52 However, it is primarily applied to small molecules, whose excited states shift the atomic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='53 Conversely, the exciton-phonon quasiparticle is a theoretical framework designed for crystals,13,14 where phonon transitions are central, leading us to conclude an interlayer exciton-phonon quasiparticle is the most appropriate assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' It should be noted that the FCP can be applied to crystalline solids when excitons self-localize and a strong electron-phonon coupling is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='54,55 Evidence is emerging that conditions in TMD-based van der Waals heterostructures may be conducive to self-localized excitons,56,57 raising the prospect that the free-excitons in WS2 are self-trapping when forming the exciton-phonon bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Further research is required to determine the model that best describes the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 9 Figure 4: Prominent surface phonons and local symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a)-(b) Linear and (c)-(d) circular polarized Raman response from CVD-grown Bi2Se3-WS2 heterostructure and MBE-grown Bi2Se3-Al2O3 (reference sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Black arrow identifies the A1(3) surface phonon, which is associated with the interlayer exciton-phonon bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phonon modes forbidden in a specific scattering geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', XX, XY, RR, or RL) are labeled with a boxed-outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 The presence of A1(2) and A1(3) surface phonons in a forbidden scattering geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', RL), suggests the local symmetry breaking at the surface,9 consistent with a strong WS2 interlayer coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Porto notation is used (see methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' To elucidate the interlayer exciton-phonon coupling, Figure 4 shows linear and circular polarization Raman measurements at 4 K for CVD-grown Bi2Se3-WS2, and a Bi2Se3 reference sample, grown using molecular beam epitaxy (MBE) on Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work demonstrated the high quality of the MBE-grown Bi2Se3 in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='58 Polarized Raman spectroscopy is a reliable method to identify and distinguish different phonon modes with greater confidence because the Raman response can be separated from each symmetry channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,59,60 More specifically, each crystallographic point group contains symmetries that can be used to calculate the reduced Raman tensors, which describe scattering of phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='60 Applying polarization tensors to Raman tensors reveals the scattering efficiency of phonon modes exposed to polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure 4a-b (Figure 4c-d) show the linearly (circularly) polarized Raman measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Section S5 contains expanded information on the Raman measurements, including each phonon mode’s attributes and a literature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Crystal symmetry forbids certain Raman scattering in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='60 If the symmetry is disrupted, these modes can become allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,60 The presence of A1(2) and A1(3) surface phonons in a forbidden scattering geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', RL), suggests the local symmetry breaking at the surface,9 consistent 4-6QLBi2Se3+1LWS2(XX) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 3000- 一 4-6QL Bi2Se3-WS2 (RR) 5QL Bi2Se3 + AI203 (XX) 180 A(2) 126 E,(2) 5QLBi2Se3+AI203(RR) 180 Arg(2) 4K 178 Ata(2) 4 K 126 E(2) 1) 178 At(2) Intensity ( 126 E(2) 126 E,(2) 69 Arg(1) 2 2000- A :74A1g 2 159 A,(3) 1000- 1000 136 E,(2) 159 A,(3) 500- 137 A,(2) 158 A,(3) 50 100 Raman Shift (cm-1) 200 50 100 Raman Shift (cm-1) 200 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='. 4-6QLBi2Se3+1LWS2(XY） d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=" 126 E(2) 4 4-6QL Bi2Se3-WS2 (RL) 1200 5QLBi2Se3+A/203 (XY) 1200 180 A1g(2) 5QLBi2Se3+AI203 (RL) 4 K (cn' 126 E(2) 3 69 E(1) 74 E(1) 1000." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 138 A,(2) 178 A1g2 nsity Raman Intensity ( 126 E,(2) 160 A,(3) 126E(2) Inter 800- 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' RamanI 136 E,(2) 137 E,(2) 600- 179 A1g(2) 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 137E(2) 76 138 E,(2) 70 E(1) 400 400 50 100 Raman Shift (cm-1) 200 50 100 RamanShift (cm-1) 200 10 with a strong WS2 interlayer coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Additionally, the A1(2) and A1(3) surface phonon modes are markedly more intense, consistent with an increased phonon population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Compared to the 5QL reference sample, as well as bulk Bi2Se3 from literature,9 we observe forbidden modes at greater intensity in Bi2Se3-WS2 heterostructures, suggesting the crystal symmetry is being disrupted to a greater degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' For example, the A1g(2) mode is forbidden in the XY configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' While the reference sample shows no detectable peak, we observe it prominently in the Bi2Se3-WS2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Of note, although all the peaks observed correspond well to either Bi2Se3 or monolayer WS2 Raman modes,9,36 we cannot exclude the possibility that combination61 or moiré Raman modes62 emerge at overlapping wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Numerous effects influence local crystalline symmetry, including defects, strain, and substrate effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work found evidence of strain at the Bi2Se3-WS2 interface and a purely electronic 2D lattice between the materials that alters the surrounding electronic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='16 Such effects impact the breakdown of translational symmetry at the interface, which subsequently affects surface phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='63 Our Raman measurements are consistent with a strong interlayer coupling that encourages strain and charge redistribution at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Together, notable interlayer hybridization16,18 and a strong interlayer coupling form an interfacial environment that facilitates interlayer interactions of different particles, thereby encouraging the formation of exciton-phonon quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='19 Additionally, our measurements demonstrate a rich landscape of phonons, including comparatively intense surface phonons, which are consistent with an increased phonon population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Conclusions We have demonstrated formation of an interlayer exciton-phonon quasiparticle (or exciton-phonon bound state) between localized excitons in monolayer WS2 and the Bi2Se3 A1(3) surface phonon, a mode particularly susceptible to electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9,10 We detect evenly spaced features in the PL spectrum with an energy separation that matches the Bi2Se3 A1(3) surface phonon, overlayed on the WS2 localized exciton emission peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Polarized Raman spectroscopy detects forbidden Bi2Se3 surface phonon modes, suggesting broken crystalline symmetry at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' While several publications have reported exciton/electron- 11 phonon coupling across the interlayer region,5,23–26 our work is distinct in that we report the formation of an interlayer exciton-phonon bound state, an uncommon quasiparticle composed of phonons, whose particle number is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13,14 Bi2Se3/monolayer WS2 heterostructures offer the possibility to combine strong light-matter interaction and spin-locked current into one material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Understanding the interlayer coupling is central to elucidating how their combined properties evolve, enabling devices for spintronics,18,20 valleytronics,21 and quantum computing21,22 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Methods Material Growth – Few-layer Bi2Se3: The Bi2Se3 films were grown on 10 × 10 mm2 c-plane (0001) sapphire (Al2O3) substrates using molecular beam epitaxy (MBE) with base pressure below 5 × 10−10 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The substrates were initially annealed ex-situ at 1,000 ℃ under the atmospheric pressure, and ozone cleaned in-situ under 200 Torr of oxygen pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' It is then annealed at 600 ℃ for 20 min in the ultra-high vacuum MBE chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Individual sources of high-purity (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='999%) Bi and Se were evaporated from standard effusion cells during the film growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Se flux was maintained at least ten times higher than Bi’s to minimize Se vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' To obtain an atomically sharp interface between the Bi2Se3 layer and the substrate, we adopted the two-step growth scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='64 First, the initial 3 QL Bi2Se3 is grown at 170 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' It is slowly annealed to 300 ℃, and followed by deposition of the remaining 2QL Bi2Se3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 5QL Bi2Se3 was grown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Material Growth – Bi2Se3-WS2 2D Heterostructure: Monolayer WS2 is synthesized at ambient pressure in 2-inch diameter quartz tube furnaces on SiO2/Si substrates (275 nm thickness of SiO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Prior to use, all SiO2/Si substrates are cleaned in acetone, IPA, and Piranha etch (H2SO4+H2O2) then thoroughly rinsed in DI water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' At the center of the furnace is positioned a quartz boat containing ~1g of WO3 powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Two SiO2/Si wafers are positioned face-down, directly above the oxide precursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' A separate quartz boat containing sulfur powder is placed upstream, outside the furnace-heating zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The upstream SiO2/Si wafer contains perylene-3,4,9,10-tetracarboxylic acid tetrapotassium salt (PTAS) seeding molecules, while the downstream substrate is untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The hexagonal PTAS molecules are carried downstream to the untreated substrate and promote lateral growth of monolayer WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Pure argon (65 sccm) is used as the furnace heats to the target temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Upon reaching the target temperature in the range of 825 to 875 °C, 10 sccm H2 is added to the Ar flow and maintained throughout the 10-minute soak and subsequent cooling to room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 4-6QL Bi2Se3 was grown on top of monolayer WS2 using chemical vapor deposition (CVD) in a two-zone furnace with a 2” quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' High-purity Bi2Se3 flakes are ground using a mortar and pestle into a fine dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The powdered Bi2Se3 is placed in a ceramic boat and inserted into the furnace’s quartz tube, and pushed into the center of the furnace’s first zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The monolayer WS2, which is on an SiO2 substrate, is placed downstream of the Bi2Se3 into the center of the furnace’s second zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The furnace is pumped down to ~20mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' An argon (Ar) carrier gas is flown into the furnace at 80sccm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The Bi2Se3 is heated to 520\uf0b0C, and the WS2 is heated to 210\uf0b0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The ramp rate is ~55\uf0b0C/min, and the total growth is 27 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Raman and photoluminescence measurements at low temperature: A Horiba LabRARM HR Evolution with both linear and circular polarization attachments, and a low-temperature Montana cryostat, was used for Raman and photoluminescence (PL) spectroscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We use Porto Notation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', 𝑧̅(\uf06d\uf06e)𝑧) 12 where \uf06d(\uf06e) is the incident (scattered) polarization, and 𝑧̅(𝑧) is the incoming (outgoing) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We define 𝑅 = 𝑋 + 𝑖𝑌 and 𝐿 = 𝑋 − 𝑖𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work showed that laser exposure of monolayer materials at low temperature can anneal and laser-dope them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='65,66 We attempted to mitigate this using very low powers (~320nW) and short exposure times (~30s) for most measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We verified that the laser exposure had a minimal effect on the material by collecting multiple successive spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' A long-distance 50x objective was used with a laser spot diameter of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 μm at the lowest powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Transmission Electron Microscopy: Bi2Se3-WS2 2D heterostructures were transferred onto a holey amorphous SiNx TEM grid using the water-assisted-pick-up transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='67 Selected area electron diffraction (SAED) were performed with a JEOL JEM2200FS operating at 200 kV, equipped with a high- speed Gatan OneView camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The SAED patterns were internally calibrated to the WS2, and an aperture of approximately 200nm was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We suspect that the wet transfer of Bi2Se3-WS2 2D heterostructures partially disrupts the crystal order, possibly due to a combination of the force and liquids applied during the transfer process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Photoluminescence Spectra computational analysis and fitting: All code was written in Python using the Spyder integrated development environment (IDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Spyder belongs to the MIT License and is distributed through the Anaconda environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The curve_fit() function with a variety of initial values and boundary conditions were used to verify fit robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The fitting was corroborated using cross-validation, where we uniformly removed 20% of the data points from a spectrum and repeated fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' No notable changes to the fitting were detected, suggesting noise is not skewing the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The energy location of each feature was extracted by taking the double derivative of the best fit, and finding the minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The minimum values of the double derivative matched well with the peak locations of Lorentzian functions, reinforcing our method to quantitatively extract the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Darshana Wickramaratne at the Naval Research Laboratory for their insight and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Supporting Information Additional characterization of representative regions probed (Section S1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Power-Dependent Measurements (Section S2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Temperature-Dependent Measurements (Section S3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Expanded Analysis of the Interlayer Exciton-Phonon Bound State (Section S4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Expanded analysis of low-temperature (4 K) Bi2Se3 phonon modes (Section S5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 13 References (1) Geim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Grigorieva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' V.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Clark, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Navarro-Moratalla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Klein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' R.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 14 (15) Merlin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Güntherodt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Humphreys, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Cardona, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Suryanarayanan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Holtzberg, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Multiphonon Processes in YbS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' B 1978, 17 (12), 4951–4958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Interlayer Coupling Induced Quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' B 2020, 101 (23), 235138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='org/10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Zuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Tian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Yao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Liao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Dai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': 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WS2/BN Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 2022, 22 (7), 2725–2733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='nanolett.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Lian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Lucking, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Beach, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Tongay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Song, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Terrones, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Shi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Excitonic Complexes and Emerging Interlayer Electron–Phonon Coupling in BN Encapsulated Monolayer Semiconductor Alloy: WS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6Se1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Nano Lett.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Jia, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Liao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': 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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Chuang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phillips, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Oleshko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' McCreary, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Sivaram, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Hellberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Jonker, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Twist Angle-Dependent Atomic Reconstruction and Moiré Patterns in Transition Metal Dichalcogenide Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' ACS Nano 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0c00088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 18 Supporting Information Interlayer Exciton–Phonon Bound State in Bi2Se3/monolayer WS2 van der Waals Heterostructures Zachariah Hennighausen1,*, Jisoo Moon1, Kathleen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' McCreary2, Connie H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Li,2 Olaf M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' van ’t Erve2, and Berend T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Jonker2,* 1 NRC Postdoc at the Materials Science and Technology Division, Naval Research Laboratory, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 20375, USA 2 Materials Science and Technology Division, Naval Research Laboratory, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 20375, USA 19 Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Additional characterization of representative regions probed Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Bi2Se3 islands grow together forming a nearly continuous 4-6QL film on monolayer WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) AFM scan around location of blue spot in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (b) optical image where blue spot identifies representative location probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (c) Fluorescence image showing that the photoluminescence (PL) is fully quenched at the center, indicating the Bi2Se3 is a nearly 4-6QL continuous film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Previous work found that forming heterostructures by growing 1QL Bi2Se3 on a monolayer (1L) transition metal dichalcogenide (TMD) significantly reduces the TMD PL intensity compared to their as-grown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', bare) 1L TMDs counterparts,1–4 likely due in part to the formation of an indirect bandgap and static charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Further, it was found that as additional Bi2Se3 layers are grown on top, the PL continues to diminish and quench because the bandgap becomes increasingly indirect with increasing Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='4 This effect is comparable to the PL intensity evolution as a TMD layer count increases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', monolayer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' bilayer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' trilayer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' More specifically, increasing layer count from 1L to 2L dramatically reduces the PL intensity, and increasing from 2L to 3L further diminishes the PL intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As layer count increases, the sample approaches the bulk properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Oun 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='2nm 10μm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 20 Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Power dependent measurements Section S2 shows power-law fitting to power-dependent measurements, which is used to identify the species of exciton each peak originates from (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=', localized state, free exciton, biexciton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The localized state (LS) has a coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='688, while the free exciton (FE) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='975, and the ratio of localized-to-free exciton intensity decreases with increasing power, enabling us to label the excitons with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Note, there are a diverse number of localized and bound excitons, whose classification depends in part on local chemistry and the spatial extent of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We cannot identify definitively the type of localized exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' At low powers, the ratio of localized to free exciton is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' However, as the power increases, the ratio decreases because the number of electrons excited from the valence band to the conduction band increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As the electrons recombine, they begin to saturate the localized states, pushing a higher ratio of electrons in the free exciton states (Figure S2), which are less easily saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5,6 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 4-6QL Bi2Se3 + monolayer WS2 2D heterostructure power-dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 2 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='025μW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='028μw 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='257μw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='393uW Free exciton has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='834uw linearpower 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='61μW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='32uw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5105 2106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='130W 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1μW 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8uw Bi,Se3 WS2 PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9uw PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0uw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5106 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5μW 1105 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='50A 190u 398pv 1106 Localized statehassub 5104 5105 linearpowerdependence 0 0 Energy (eV)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 Energy(eV)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 21 Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As-grown monolayer WS2 power-dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Graphed evolution or curve peaks is shown below in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As-grown monolayer WS2 power-dependent measurements: Peak intensities plotted from Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The expected behavior validates the assignment of the localized and free excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) Shows the localized state exciton (LS) and (b) the free exciton (LE), which have a fitting coefficient of less than one and approximately equal to one, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The ratio of LS/FE decreases with laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Together, the results provide high confidence of the locations of the localized and free exciton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5,6 The peak intensities were extracted using Spyder integrated development environment (IDE) and the curve_fit() function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The results were corroborated with manual inspection of the highest intensity pixel recorded by the equipment for each exciton curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='2105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='025uW 1106 1105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='257uW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='834uw 8104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='61uW 6104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='32uw 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13uW 8105 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8uW 2104 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0uw Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') Energy(eV)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 6 6105 4105 MonolayerWS2 2105_ 0 Energy (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 106 LocalizedState b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' FreeExciton 25 LS/FERatio 105 20 Localized state rapidly Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') saturates as laser power 105 increases,pushing more PL Intensity ( 104 15 electrons into FE of PL 103 104 Power law fitting =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='2e+6 x^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6876 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0e+5 x^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9745) PowerApplied(uw) Power Applied (μuw) PowerApplied (uw) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='257 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='61 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='61 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 22 Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Temperature dependent measurements Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 4-6QL Bi2Se3 + monolayer WS2 2D heterostructure temperature-dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (a) 2D density plot of the PL spectra with temperature, showing the localized and free exciton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' (b)-(c) PL spectra as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Interestingly, negative thermal quenching is observed for the localized state, an unusual phenomena where the PL intensity increases with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 Previous theory work proposed that as the temperature increases, electrons are thermally excited from nearby bound states into a primary state that allows for radiative recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 The peak position shifts lower as the temperature increases, in agreement with the Varshni equation, formalism used to describe the PL evolution in semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 21 1600 1600 FreeExciton 10k C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 75k 1400 15k 80k 180k 1400 1400 85k 20k 90k 200k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='0 1200 25k 95k 1200 30k 1200 100k 2201 35k 110k 230k 1000 15 240k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 1000 40k 125k 250k 45k 130k 260k 800 50k 140k 280k 800 55k 150k 270k 18 600 160k 60k 600 65k 600 400 70k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 751 400 nsity 200 LocalizedState 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='6 10 OE 50 75 5100 150 200250290 0 0 Energy (eV)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 Energy (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1 23 Section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Expanded analysis of the interlayer exciton-phonon bound state Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Bi2Se3-WS2 data from Figure 2 fitted with Lorentzian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We obtain a function that follows the curve and captures the data form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Inflection points were quantitatively identified by taking a double derivative of the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Note, we only use the fitting methodology as a tool to quantitatively extract the inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We do not extract insight from the Lorentzian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Representative data with peaks that are approximately equally spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Spacings between Double Derivative Extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Multiple Lorentzian functions were fit to the data using Python Software (see methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The double derivative was taken of the best fit function and the minimum extrema were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The spacing between the extrema is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Spacing Figure S6 Figure S7a Figure S7b 1st 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='75 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='49 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='87 2nd 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='79 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='06 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='5 3rd 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='13 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='85 4th 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='54 Average 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3meV 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1meV 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='7meV Methodology for Measuring Spacing of the Interlayer Exciton-Phonon Bound State Features: The exciton-phonon bound state (or exciton-phonon quasiparticle) presents itself as evenly spaced peaks or features, where the spacing is approximately the phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' As such, the presence of an exciton- a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='3000 4 K Bi2Se3 WS2PL 2500 Best Fit Multiple FELorentzian localizedstate Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') LS Lorentzians 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' transitions 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Uniformlyspaced energytransitions 1000 PL Free Exciton 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9 Energy (eV) 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' b 1200 800 Photon Counts (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') 1000 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=') 008 600 PhotonCounts( 600 400 400 200 200 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='70 175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='75 180 185 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='05 ev ev 24 phonon or electron-phonon quasiparticle can be inferred when evenly spaced peaks are observed that correspond to a Raman phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9–14 The number of peaks and consistency of the even spacing is material and environment specific, as well as the strength of the exciton-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='10,11 Further, the task is complicated by the fact that the evenly spaced features are frequently overlayed on a larger peak, which obscures their precise peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='9–14 When analyzing the PL spectra for a possible exciton-phonon bound state, we start by identifying possible features that could be peaks and then measuring their spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' To the best of our knowledge, the community primarily relies on visual inspection when identifying and measuring possible exciton-phonon peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' In this work, we applied two methods for peak identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The primary method is visual inspection, while the secondary method is a quantitative analysis of best fit double derivative extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Note, we only use the fitting methodology as a tool to quantitatively extract the inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We do not extract insight from the Lorentzian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Only if both methods yielded the same result, did we label the feature as a peak and measure spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We fit the experimental data with either 7 or 8 Lorentzian functions, which was sufficient to obtain a best fit that corresponded well to the data moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We then took the double derivative of the best fit and identified the minima extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We verified each double derivative minima corresponded to a clear visual feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' We then measured the spacing between each double derivative minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' The double derivative extrema frequently corresponded to the location of Lorentzian peaks, but did not overlap exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 25 Section S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Expanded analysis of low-temperature (4 K) Bi2Se3 phonon modes Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Summary of low-temperature Bi2Se3 phonon modes for Bi2Se3-WS2 and Bi2Se3-Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Increased symmetry breaking was observed in Bi2Se3-WS2 heterostructures compared to Bi2Se3-Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phonon Mode Bulk/Surface Scattering Geometry Bi2Se3-WS2 (cm-1) Bi2Se3-Al2O3 (cm-1) Literature: Bulk Bi2Se3-Al2O3 (cm-1) Notable Symmetry breaking A1(2) Surface RR & XX 137 137 13615 & 12916 Yes (RL Channel) A1(3) Surface RR & XX 160 159 15815 & 16016 Yes (RL Channel) A1g(1) Bulk RR & XX 69 74 7515 & 7316,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='17 & 7218 No A1g(2) Bulk RR & XX 180 178 18015 & 17516,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='17 & 17418 Yes (XY Channel) E(1) Surface RL 69 74 6715 & 6816 No E(2) Surface RL 126 137 12615 & 12516 Yes (RR Channel) Eg(2) Bulk RL 126 136 13715 & 13316 & 13117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='18 Yes (RR Channel) The above Raman modes are identified with the assistance of previous work by matching them to the wavenumber and polarization response,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' which reveals the symmetry channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='15,19,20 We primarily relied upon Kung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' and Gnezdilov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' for understanding the symmetry channels and polarized Raman spectroscopy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='15,16 Our data presented HERSE for low-dimensional Bi2Se3 using polarized Raman spectroscopy and at low-temperatures are among the few in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' While our samples were between 4-6QL Bi2Se3, the literature values are from bulk Bi2Se3 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Bi2Se3 has the D3d point group symmetry in the rhombohedral crystal structure, suggesting one or more of the following point symmetry groups is being broken: S6, D3, C3v, C3, C2h, C2, Cs, or Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='21 Definitively identifying which symmetry point groups are being disrupted is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' 26 References (1) Hennighausen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Lane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Benabbas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': 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with Tunable Circular Polarization at Room Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Interfaces 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content='1021/acsami.' metadata={'source': 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(12) Toyozawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Hermanson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Exciton-Phonon Bound State: A New Quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Lett.' metadata={'source': 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Güntherodt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Humphreys, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Cardona, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Suryanarayanan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Holtzberg, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Multiphonon Processes in YbS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfZgBD/content/2301.02321v1.pdf'} 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Research, Mumbai 400005, India +Aveek Bid∗ +Department of Physics, Indian Institute of Science, Bangalore-560012, India +Study +of +Berezinskii-Kosterlitz-Thouless +transitions +in +clean, +layered +two- +dimensional superconductors promises to provide insight into a host of novel phe- +nomena like re-entrant vortex-dynamics, underlying unconventional metallic phases, +and topological superconductivity. In this letter, we report the study of charge carrier +dynamics in a novel 2-dimensional superconducting van der Waals heterostructure +comprising of monolayer MoS2 and few-layer NbSe2 (∼ 15 nm). Using low-frequency +conductance fluctuation spectroscopy, we show that the superconducting transition +in the system is percolative. +We present a phenomenological picture of different +phases across the transition correlating with the evaluated noise. The analysis of the +higher-order statistics of fluctuation reveals non-Gaussian components around the +transition indicative of long-range correlation in the system. +I. +INTRODUCTION +With the practical realization of graphene [1], the past decade has seen an extensive ex- +ploration of layered systems. The van der Waals heterostacking of these crystalline layered +materials promises to exhibit parameter-driven exotic phenomena including topologically +non-trivial states [2–5] and strongly correlated phases [6, 7]. An example is atomically thin +superconductors in the ‘true’ 2-dimensional (2D) limit. Contrary to the 3-dimensional super- +∗ aveekbid@iisc.ac.in +arXiv:2301.03473v1 [cond-mat.supr-con] 9 Jan 2023 + +2 +conductors, for a superconductor in the 2D limit, the transition occurs through Berezinskii- +Kosterlitz-Thouless (BKT) mechanism [8, 9]. Below a characteristic critical temperature +TBKT, the vortices and antivortices are bound – the thermal unbinding of these pairs at +T > TBKT gives rise to the transition from the dissipationless to a finite resistive state in +the system. +There are two distinct experimental strategies employed to identify a BKT transition +as one approaches it from above – (i) measurement of the superfluid density nS which is +expected to go to zero discontinuously at the transition [11, 31] and (2) extrapolation of +the temperature dependence of the resistivity measured at T > TBKT to lower T using +the formalism developed by Halperin and Nelson [12]. The first technique of looking for +discontinuity in the superfluid density as a signature of BKT physics does not work well +for superconductors buried inside heterostructures. The second method fails for disordered +superconductors due to two reasons: (i) the presence of impurities tends to broaden the +transition [13–15] and (ii) the inhomogeneities change the value of the vortex-core energy +from that predicted within the 2D XY model [16]. +The study of carrier dynamics through resistance fluctuation spectroscopy has emerged +as a powerful alternative probe to identify BKT transitions [14, 17]. Though this technique +has been employed to probe the BKT physics in thin-film superconductors, the study of +fluctuation statistics is not well explored in layered systems, specifically in van der Waals +heterostructures. With increasing interest in such systems as platforms for realizing low +dimensional superconductivity in the clean limit and topological superconductivity, there is +an urgent need for a detailed study of such systems. +In a previous study, we have reported the observation of two-dimensional Ising supercon- +ductivity in a van der Waals heterostructure comprising of single-layer-MoS2 (SL-MoS2) and +bulk NbSe2 [18]. We established that the reduced dimensionality comes from an effective +thinning of NbSe2 due to the coupling with the MoS2 layer making it a perfect example of +a ‘buried’ van der Waals superconductor. Thus, the conductance fluctuation spectroscopy +technique becomes very relevant to probe the BKT physics in this system. +In this letter, we report a detailed study of the carrier dynamics of this heterostructure +through low-frequency conductance fluctuation spectroscopy around the BKT transition. +Through systematic measurements, we establish that superconductivity has a percolative +nature. We also find proof of correlated dynamics arising from long-range interaction of + +3 +vortices-antivortices near the TBKT establishing universal BKT nature of the superconduct- +ing transition in this system. +II. +DEVICE FABRICATION +To fabricate the device, we mechanically exfoliated single-layer flakes of MoS2 from a bulk +crystal [18]. The thickness of the flake was confirmed from optical contrast and through +Raman spectroscopy. The flake was then transferred onto gold contact probes pre-patterned +on hBN substrates. +Subsequently, a flake of NbSe2 exfoliated inside a glove box (with +oxygen and moisture levels maintained at less than one ppm) was transferred on top of the +MoS2 flake. The thickness of the NbSe2 flake was estimated from its optical contrast to be +∼ 15 nm. Before extraction from inside the glove box, the heterostack of SL-MoS2/NbSe2 was +covered by a hBN flake of thickness ∼ 30 nm to protect it from environmental degradation. +Subsequently, the stack was annealed at 200◦ C to increase the coupling between the layers. +III. +RESULTS AND DISCUSSION +For the initial characterization of superconducting properties of the heterostructures, +electrical transport measurements were done using a DC current source and a nano voltmeter +in a four-probe configuration (Fig. 1(a)). The temperature dependence of resistance R shows +a metallic behavior at high temperatures followed by a transition to a superconducting state +(Fig. 1(c)) at T 0 +C = 6 K. From the R versus T plot and also from non-linear current-voltage +relations, we estimate TBKT to be 6.1±0.1 K.(see Supplementary Information S1 for details, +also Ref. [18]). +We investigated the fluctuation statistics of the system around the TBKT using a 4-probe +resistance fluctuation spectroscopy technique [20, 21, 35] – the details are discussed in Sup- +plementary Information S2. Briefly, the device was current biased, the voltage developed +across it pre-amplified and detected by a dual-phase lock-in-amplifier (LIA). The demod- +ulated output of the LIA was digitized at a sampling rate of 1024 points/s using a 16 bit +analog-to-digital conversion card and transferred in the computer memory for further pro- +cessing. The biasing current was always maintained at a value much smaller than the critical +current of the superconductor. The acquired time series of resistance fluctuations were dec- + +4 +imated and filtered digitally to eliminate aliasing and related digital artifacts. The power +spectral density of the resistance fluctuation SR (f) was then calculated over a frequency +range 4 mHz–4 Hz. +Fig. 2(a) is a plot of the time traces of resistance fluctuations for our device measured +at a few representative temperatures, T. The fluctuations increase in amplitude with T +approaching TBKT from above. The corresponding SR (f) were found to have a frequency +dependence of the form SR (f) ∝ 1/f α (Fig. 2(b)). One can see that the power spectral +density SR (f) increases by several orders of magnitude with decreasing temperature, re- +flecting the observed increased resistance fluctuation in Fig. 2(a). Additionally, the value +of the exponent α for f < 0.5 Hz (the method of evaluation the exponent is discussed in +Supplementary Information S3) increases monotonically from ∼ 1 at higher temperature to +∼ 2.4 near TBKT (Fig. 3(a)). There can be two possible reasons for this increase in α – (i) +transition of the system across different vortex phases, e.g., from an ordered to a disordered +regime [22] or (ii) fluctuation in the domain parameter of different phases across the transi- +tion range [23]. Discriminating between these two scenarios requires further analysis and is +beyond the scope of the current letter. +The relative variance of resistance fluctuations (we refer to this as the noise) was evaluated +by integrating SR (f) over the bandwidth of measurement [20, 21, 35]: +R = ⟨δR2⟩ +⟨R2⟩ = 1 +R2 +ˆ +SR(f)df. +(1) +Fig. 3(b) shows the plots of relative variance and the normalized resistance against T/Tcritical +(T/TBKT) for the heterostructure region. +We observe that R in the normal state is ∼ +10−10. This value is almost five orders of magnitude lower than that reported for a typical +semiconducting TMD [24] attesting to the high quality of our heterostructure. +With decreasing T, R increases by nearly six orders of magnitude over a very narrow +temperature window near TBKT. As we discuss later, this divergence in noise can be well +explained in terms of a percolation network model of superconducting fluctuations [14]. +Moreover according to the percolation model in the transition regime the system should +follow the relation, SR (f) /R2 ∝ R−lrs where lrs is the percolation exponent which takes +up the value ∼ 0.9 in the classical picture [36]. In our case, the exponent lrs comes out +to be 0.89 ± 0.03 (see Supplementary Information S4), establishing the percolative nature +of the system. Notably, when compared with that of the pristine NbSe2, the noise in the + +5 +heterostructure is almost an order higher around the respective transition temperatures (c.f. +Fig. 3(b)). +Before we proceed further, the effect of thermal fluctuations on the measured noise needs +to be considered. dR/dT diverges close to the critical temperature for a superconductor. +Consequently, any minor fluctuation in temperature can give rise to large resistance fluctu- +ations near TBKT. To eliminate this trivial effect as the the origin of the large resistance +fluctuations seen in our device, we evaluated the relative contribution of temperature fluc- +tuations to the noise using the relation [(dR/dT) × (δT/R)]2. Here δT is the temperature +fluctuation in the measurement system, which has been measured to be < 5 mK in our +case. The evaluated value of this relative variance at TBKT is ∼ 10−7 (see Supplementary +Information S5). This value is at least two orders of magnitude smaller than the measured +R near TBKT, thus ruling out any significant contributions of temperature fluctuations in +the measured noise. +In Fig. 4 we present a phenomenological explanation of the effect of percolation dynamics +on the resistance fluctuations in a 2D superconductor in terms of a percolation network model +of superconducting fluctuations [14]. The squares below the plot show the microscopic status +of the system schematically in terms of a superconducting-normal network in different T- +regimes. +Region-I is a purely superconducting phase. On approaching TBKT from below, small +patches of dissipative (metallic) domains begin nucleating in the superconducting back- +ground (region-II). With increasing temperature, fluctuations in the superconducting order +parameter result in the formation of a dynamic network of interconnected superconduct- +ing and normal (dissipative) regions [36]. This effect is especially severe in the case of 2D +superconductors. The enhanced noise in this T-regime has two major components – (i) +resistance fluctuations in dissipative regions; (ii) fluctuations in the number/size of the su- +perconducting clusters [26, 27]. Beyond T = TBKT, the system crosses into region III, where +the proportion of superconducting and non-superconducting domains become almost equal. +At this temperature (which we denote as Tmax), the resistance of the device is nearly half +of the normal state resistance, and the resistance fluctuation is at its maximum. The other +boundary of region III comes at TBCS, which is at ∼ 6.5 K for the system. For T > TBCS, +the fraction of the superconducting phase decreases sharply with increasing T till the entire +system becomes dissipative. Consequent to this decrease in electronic phase segregation of + +6 +the system, the variance of resistance fluctuations decreases as the system approaches the +metallic phase. +We turn now to the nature of the correlations between the fluctuating entities at different +T-ranges. In 2D superconductors undergoing BKT transition, the XY model predicts the +fluctuations to be non-Gaussian around TBKT [28]. These non-Gaussian resistance fluctua- +tions have emerged as a unique signature of BKT physics and have successfully been used to +discriminate between 2D and 3D superconductors [14, 17]. We quantify the non-Gaussianity +of the resistance fluctuations through their ‘Second spectrum,’ which is the four-point corre- +lation function of δR, calculated over a frequency octave (fl, fh). Being extremely sensitive +to the presence of non-Gaussian components (NGC), this parameter is a highly effective +tool to probe correlation in a system [29, 30]. To estimate the second spectrum, repeated +measurements of the PSD, SR (f) is done over a selected frequency range (fl, fh). The power +spectrum of this series over a frequency octave gives the second spectrum [29, 30]: +Sf1 +R (f2) = +ˆ ∞ +0 +� +δR2 (t) δR2 (t + τ) +� +cos (2πf2τ)dτ, +σ(2) = +ˆ fl−fh +0 +Sf1 +R (f2) df2 +��ˆ fh +fl +SR (f) df +�2 +(2) +Here f1 is the center frequency of the chosen octave and f2 is the spectral frequency. σ(2) is +the normalized second spectrum; it equals 3 for Gaussian fluctuations. +Fig. 5 shows the plot of measured σ(2) as a function of T/TBKT for the heterostructure +region. As can be observed, while decreasing the temperature, σ(2) increases from a baseline +value of ∼ 3 at higher T to ∼ 30 near TBKT before decaying again to the Gaussian base +value for T < TBKT. This enhancement of σ(2) in the narrow window of T in Region-III +establishes clearly the appearance of non-Gaussian resistance fluctuations near T = TBKT. +In contrast, for the pristine NbSe2 region σ(2) remains at the baseline value of ∼ 3 (black +solid circle in Fig. 5) throughout the temperature range around the transition indicating a +Gaussian distribution of fluctuations as expected for a 3D superconductor. +Non-Gaussian fluctuations in superconductors can have different origins – (1) long-range +correlation among the vortices as has been observed in previous studies [17], (2) the dom- +inance of percolation kinetics around superconducting transition seen in inhomogeneous +superconductors [14], and (3) dynamic current redistribution which appears as a conse- +quence of substantial transport inhomogeneity and large local resistivity fluctuations that + +7 +translate to the necessary condition of δR/R >> 1 [30]. The third cause can be immediately +ruled out by noting that in our systemδR/R << 1. To discriminate between the remaining +two scenarios, note that a comparison of the T dependence of R and σ(2) (Fig. 5) reveals +that significant fluctuations in the resistance extends beyond the onset of normal state (i.e. +R/RN = 1) at TBCS of ∼ 6.5 K. In the low-temperature limit, it extends to T < TBKT. On +the other hand, the deviation from the Gaussian value in σ(2) stays confined in region III +within TBKT < T < TBCS. This deviation indicates that the increase of σ(2) that marks the +existence of non-Gaussian fluctuations in the fluctuation is a consequence of an independent +process which is unlikely to be the percolation kinetics that dominates the spectrum of re- +sistance fluctuations. This strongly suggests that the first scenario of correlated vortices is +at play in inducing the non-Gaussian fluctuations in the system, as has been reported earlier +for clean, homogeneous superconductors [17]. Further theoretical and experimental studies +are essential to establish unequivocally if this is the case. +IV. +CONCLUSION +In summary, we have studied the carrier dynamics of SL-MoS2/NbSe2 heterostructures +near the superconducting transition by probing the low-frequency conductance fluctuations +of the system. The first spectrum (resistance noise) shows signatures of the percolative +nature of the superconducting transition. We provide a phenomenological explanation of +the different phase-space regions around the transition temperature in terms of a percolative +microstructure picture and correlate the resulting fluctuations with it. Furthermore, we es- +tablish the presence of strong correlations in the system around TBKT arising most probably +from the interacting vortices and thus established that the superconducting transition in +the system is of the universal BKT type. +Acknowledgments: +The authors acknowledge device fabrication facilities in NNFC, +CeNSE, IISc. A.B. acknowledges funding from SERB (No. HRR/2015/000017) and DST +(No. DST/SJF/PSA01/2016-17) + +8 +Si++ +SiO2 +hBN sunken probe +1L - MoS2 +NbSe2 +top BN +v- +Isd +v+ +10 µm +Bottom BN +MoS2 +Top BN +NbSe2 +Overlap +(a) +(b) +1 +0 +2 +3 +4.7 +5.7 +6.7 +7.7 +(c) +Figure 1. (a) A schematic of the device structure. (b) False colour Differential Interference Con- +trast image of the device with different colors defining different layers of the heterostructure – +SL-MoS2(green) and pristine NbSe2 (orange) and overlap region (light-red). (c) Temperature de- +pendence of the four-probe resistance of the heterostructure. + +9 +Figure 2. (a) Time series of resistance fluctuations of the heterostructure region at a few repre- +sentative temperatures. (b) Plots of SR (f) as function of frequency at the same values of T as in +(a). + +10 +Figure 3. (a) Plot of exponent α versus T/TBKT for heterostructure. (b) Plots of the relative +variance of resistance fluctuations R for heterostructure (solid green circles) and for the pristine +3D NbSe2 region (open orange triangles) as function of T/Tcritical. Here T/Tcritical is T/TBKT for +the heterostructure and T/T 0 +c for the pristine NbSe2. On the right-axis are plotted the normalized +resistance R/RN for the heterostructure (red line). + +11 +I +II +III +IV +V +10-6 +10-7 +10-8 +10-9 +0.9 +1.0 +1.1 +1.2 +1.3 +T / TBKT +0 +0.5 +1.0 +R / RN +(a) +1.0 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +R/RN +I +II +III +IV +V +(b) +Figure 4. (a) Plots of the variance of noise +� +δR2� +(left-axis, yellow filled circles) and of the nor- +malized resistance (right-axis, black solid line) as a function of T/TBKT for the heterostructure +region. The color gradient indicates the transition from superconducting (blue) to normal state +(red). (b) Schematics representing the microscopic status of the system in each electronic phase +(for details see text). The color bar gives the value of R/RN – blue represents the zero resistance +i.e. superconducting state and red represents the normal state. + +12 +I +II +III +IV +V +10-6 +10-7 +10-8 +10-9 +0.9 +1.0 +1.1 +1.2 +1.3 +T / Tcritical +0 +5 +10 +15 +20 +25 +30 +35 +Figure 5. +Plots of the variance of resistance fluctuations +� +δR2� +(filled yellow circles, left axis) +and the normalized second spectrum σ(2) for heterostructure (filled blue circles, right axis) and for +pristine 3D NbSe2 (filled black circles, right axis) as a function of T/Tcritical. Here T/Tcritical is +T/TBKT for the heterostructure and T/T 0 +c for the pristine NbSe2. The color gradient has the same +connotation as in Fig. 4. (for details see text). + +13 +SUPPLEMENTARY INFORMATION +S1. +EVALUATION OF BKT TRANSITION TEMPERATURE +Figure S1. (a) Zero magnetic field current-voltage characteristics of the heterostructure with varying +temperature ranging from 4.5 K to 10.0 K. The red dashed line represents the range of current +within which linear fit was done for each curve. (b) Plot of γ as a function of temperature, T to +evaluate TBKT for heterostructure. (c) Plot of (dlnR/dT)−2/3 as a function of temperature, T for +the heterostructure where the intersect of the black dashed line with the x axis gives TBKT . +For the initial characterization of superconducting properties of the heterostructure, elec- +trical transport measurements were done using a DC current source and a nano voltmeter in a +four-probe configuration. As reported in our previous work, the superconductivity in system +is of 2D nature. We thus expect the observed superconducting transition to be of Berezin- +skii–Kosterlitz–Thouless (BKT) type. For a 2D-superconductor there exists a characteristic +temperature, TBKT below which a finite electric current can unbind the vortex-antivortex +pairs system, giving rise to a dissipation which is reflected in the current-voltage character- +istic as a non-linear behavior of the form, V ∝ Iγ(T) [31, 32]. γ is a temperature dependent +exponent that takes the value 3 at T = TBKT and eventually goes to 1 in the normal ohmic +state. Fig. S1(a) shows the zero field DC non-linear current-voltage characteristics. The +value of the exponent γ, evaluated through linear fitting of each curve within the marked +region, are shown in Fig. S1(b). From this plot, we evaluate TBKT to be 6.13 K. We also +evaluated TBCS (defined as the temperature where onset of transition occurs or in other + +14 +words where the IV becomes linear i.e. γ = 1) to be 6.5 K. +The BKT transition temperature can also be obtained from resistance vs temperature +plot as near TBKT the resistance takes the form R = R0exp[−bR/(T − TBKT)1/2] (where bR +gives the vortex-antivortex interaction strength) [31, 33, 34]. To evaluate TBKT we reduced +the formula to a form, (dlnR/dT)−2/3 = (2/bR)2/3 (T − TBKT). As shown in Fig. S1(c), the +intercept of the plot of (dlnR/dT)−2/3 vs T gives TBKT as to be 6.14 K. +S2. +DETAILS OF NOISE MEASUREMENT TECHNIQUE +PA +LIA +DAQ card +Rseries +(>>Rsample) +I+ +I- +V- +V+ +B +A +Y +X +Input +Output +A/D conversion ++ acquisition +Digitized +data +~ +LIA +Sine-out +Vac +Cryostat +DEVICE +iac +Figure S2. Schematic diagram of the noise measurement setup. The sample is probed in four-probe +geometry. PA represents the pre-amplifier and LIA represents the dual phase lock-in amplifier. +We investigated the fluctuation statistics of the system around superconducting transition +through analysis of the zero field temperature dependent resistance fluctuations acquired +using an ac technique that allows us to measure the fluctuations from system and the +background simultaneously [35]. Fig. S2 is schematic of the measurement setup. The sample +was current-biased using the sine wave output of a lock-in amplifier (SR830). A resistor, +Rseries in series with the sample controls the current through it. The value of the excitation +current was always maintained to be lower than the critical current of the superconductor. + +15 +The voltage developed across the sample was detected using the dual-phase lock-in-amplifier +coupled with a preamplifier (SR552). The excitation frequency of the current was kept at +the eye of the noise figure of the preamplifier to minimize the contribution of amplifier noise +in measured the background noise. The time constant were set to be 30 ms with a filter roll +off of 24 dB/octave - this subsequently determines the upper cutoff frequency of the power +spectral density (PSD). The output of the LIA was digitized at a sampling rate of 1024 +points/s using a 16 bit analog-to-digital conversion card and transferred in the computer +memory for further processing. The in-phase channel (X-channel) picks up the excess noise +from the sample as well as the background whereas the quadrature channel (Y-channel) +picks up only the fluctuations from background. +At every temperature, the time series +of the resistance fluctuations was acquired for a duration of 30 minutes (∼ 1.8X106 data +points). These were subsequently decimated with a factor of 128 and digitally filtered to +eliminate aliasing and related digital artifacts. These filtered time series were then used to +calculate the power spectral density (PSD) over the specific frequency range. The PSD of +the sample noise were finally obtained by subtracting the PSD of X-channel fluctuation from +that of the Y-channel. +S3. +EVALUATION OF THE EXPONENT α FROM SR (f) +As mentioned in the main manuscript, the power spectral density, SR (f) has a frequency +dependency SR (f) ∝ 1/f α. To evaluate the exponent α we plotted SR (f) as function of +frequency, f in log-log scale, as shown in Fig. S3. The slope of these plots gives the value of +α. As can be seen here the slope i.e. α is ∼ 1 at 8 K in which the system is in normal state +whereas at 6.2 K which is near to TBKT the value becomes ∼ 2.4. +S4. +CLASSICAL PICTURE OF PERCOLATION +For a system having percolative nature it follows that the spectral density of relative +resistance fluctuation at a certain frequency, SR (f) /R2 grows as power law to the decreasing +resistance towards superconductivity with a form given by [36] +SR (f) +R2 +∝ R−lrs +(S1) + +16 +Figure S3. Plot of SR (f) as a function frequency, f for two different temperatures around the +transition, 8 K (violet) and 6.2 K (brown). The dashed lines show the linear fits to the plots. The +boxes with each plot show value of the slope for the respective fitting. +where lrs is the percolation exponent, which takes the value ∼ 0.9 in the classical picture. +As can be seen in Fig. S4 the percolation exponent for the heterostructure comes out to be +0.89 ± 0.03 which matches quite well with the classical percolation picture. +S5. +CONTRIBUTION OF TEMPERATURE FLUCTUATIONS TO THE +MEASURED NOISE +As mentioned in the main manuscript, we have evaluated the relative contribution of the +temperature fluctuation of the measurement system to the measured noise. The temperature +stability of a system depends mainly on the PID value of the temperature controller used +in the experiment. We have fixed this PID value in such a way that we were able to have a +temperature fluctuation, δT < 5 mK at all temperatures at which noise measurements were +done. +In Fig S5(a), we show a plot of T −Tsetpoint versus time over a period of 30 minutes. This + +17 +lrs = 0.89±0.03 +Figure S4. Plot of SR/R2 at 0.0625 Hz as function of R(olive solid circle). Both the axis are in log +scale. The red line is the linear fit to the data that yields a slope of ∼ 0.89 +is the typical time for a single noise run. Here Tsetpoint is the target temperature value (in +this case, 12 K), and T(t) is the instantaneous value of temperature. From Fig. S5(a), the +maximum fluctuation is about 3 mK indicating that taking 5 mK as δT in our calculation is +a safe choice. Fig. S5(b) shows the plots of the measured relative variance (olive solid circles) +and that estimated from temperature fluctuations. One can see that near TBKT, the value +of relative variance of resistance fluctuations estimated from the temperature fluctuations +is almost two orders of magnitude smaller than the measured relative variance of resistance +fluctuations, R of the heterojunction. This establishes that temperature fluctuations play a +negligibly small role in the measured noise. + +18 +Figure S5. (a) Plot of T − Tsetpoint as a function time measured at 12 K. (b) Plot of the measured +relative variance +� +δR2� +/⟨R⟩2 for heterostructure (olive solid circles) and that estimated from ther- +mal fluctuation (blue solid circles) as function of T/TBKT . The lines are guide to the eye. The +right-axis shows a plot of the resistance for the heterostructure region (solid red line). +S6. +NOISE DATA FROM ANOTHER DEVICE +Fig. S6 shows the relative variance of resistance fluctuations of another device D2 having +identical structure to the device D1 whose data were presented in the main text. As is +evident from the plot, the data from D2 is very similar to that from D1 – it shows percolative +transition near TBKT. Similar to the data for device D1, for D2 also we observe an order of +magnitude higher value of the relative variance for heterostructure region in comparison to +pristine NbSe2 section of the device near T/TBKT. +Fig. S7 shows the variance of the resistance fluctuations (olive solid circle, left axis) along +with the normalized resistance, R/RN (red line, right axis) of the heterostructure region +measured for device D2 as a function of T/TBKT. As can be seen, the increased fluctuation +extends beyond TBCS into the normal state just as for D1 in the main text. +The deviation of the normalized second spectrum, σ(2) from the baseline value of 3 in +Fig. S8 are constricted within the region bounded by TBKT and TBCS suggesting that, like +device D1 in main text, the non-Gaussian nature for device D2 also occurs due to the long +range correlations between the vortex-antivortex pair around the transition. Moreover, as + +19 +Figure S6. +Plots of the relative variance of resistance fluctuations, +� +δR2� +/ +� +R2� +for the heterostruc- +ture (solid olive circles) and for the pristine 3D NbSe2 region (solid orange circles) as function of +T/Tcritical for device D2. Here T/Tcritical is T/TBKT for the heterostructure and T/T 0 +c for the pris- +tine NbSe2. On the right-axis are plotted the normalized resistance R/RN for the heterostructure +(red line). +expected we observed σ(2) to be ∼ 3 for pristine NbSe2 region of device D2 throughout the +temperature range indicating the Gaussian nature of the fluctuations. This similarity in the +evaluated results for the two different devices of similar heterostructure thus proves that the +main observed phenomenons are inherent to the system and not device specific. + +20 +Figure S7. Plots of the variance of noise +� +δR2� +(left-axis, olive solid circles) and of the normalized +Resistance (right-axis, red solid line) as a function of T/TBKT for the heterostructure region of +device D2. + +21 +0.9 +1.0 +1.1 +1.2 +10-9 +10-8 +10-7 +10-6 +10-5 +<δR2> (R2) +T/Tcritical +TBKT +TBCS +0 +5 +10 +15 +20 +σ(2) +Figure S8. Plots of the variance of resistance fluctuations +� +δR2� +(solid olive circles, left axis) and the +normalized second spectrum σ(2) for heterostructure (solid red circles, right axis) and for pristine +3D NbSe2 (solid blue circles, right axis) as a function of T/Tcritical for device D2. Here T/Tcritical +is T/TBKT for the heterostructure and T/T 0 +c for the pristine NbSe2 + +22 +[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. +Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science +306, 666–669 (2004). +[2] L. Fu and C. L. 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Kogan, Electronic noise and fluctuations in solids (Cambridge University Press, 2008). + diff --git a/o9E1T4oBgHgl3EQf2AUg/content/tmp_files/load_file.txt b/o9E1T4oBgHgl3EQf2AUg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9440a5980b47a1906fbf180be62382189d32a18 --- /dev/null +++ b/o9E1T4oBgHgl3EQf2AUg/content/tmp_files/load_file.txt @@ -0,0 +1,612 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf,len=611 +page_content='Correlated carrier dynamics in a superconducting van der Waals heterostructure Prakiran Baidya Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Indian Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Bangalore 560012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' India Vivas Bagwe and Pratap Raychaudhuri Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Mumbai 400005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' India Aveek Bid∗ Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Indian Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Bangalore-560012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' India Study of Berezinskii-Kosterlitz-Thouless transitions in clean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' layered two- dimensional superconductors promises to provide insight into a host of novel phe- nomena like re-entrant vortex-dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' underlying unconventional metallic phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' and topological superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In this letter, we report the study of charge carrier dynamics in a novel 2-dimensional superconducting van der Waals heterostructure comprising of monolayer MoS2 and few-layer NbSe2 (∼ 15 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Using low-frequency conductance fluctuation spectroscopy, we show that the superconducting transition in the system is percolative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We present a phenomenological picture of different phases across the transition correlating with the evaluated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The analysis of the higher-order statistics of fluctuation reveals non-Gaussian components around the transition indicative of long-range correlation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' INTRODUCTION With the practical realization of graphene [1], the past decade has seen an extensive ex- ploration of layered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The van der Waals heterostacking of these crystalline layered materials promises to exhibit parameter-driven exotic phenomena including topologically non-trivial states [2–5] and strongly correlated phases [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' An example is atomically thin superconductors in the ‘true’ 2-dimensional (2D) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Contrary to the 3-dimensional super- ∗ aveekbid@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='03473v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='supr-con] 9 Jan 2023 2 conductors, for a superconductor in the 2D limit, the transition occurs through Berezinskii- Kosterlitz-Thouless (BKT) mechanism [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Below a characteristic critical temperature TBKT, the vortices and antivortices are bound – the thermal unbinding of these pairs at T > TBKT gives rise to the transition from the dissipationless to a finite resistive state in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' There are two distinct experimental strategies employed to identify a BKT transition as one approaches it from above – (i) measurement of the superfluid density nS which is expected to go to zero discontinuously at the transition [11, 31] and (2) extrapolation of the temperature dependence of the resistivity measured at T > TBKT to lower T using the formalism developed by Halperin and Nelson [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The first technique of looking for discontinuity in the superfluid density as a signature of BKT physics does not work well for superconductors buried inside heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The second method fails for disordered superconductors due to two reasons: (i) the presence of impurities tends to broaden the transition [13–15] and (ii) the inhomogeneities change the value of the vortex-core energy from that predicted within the 2D XY model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The study of carrier dynamics through resistance fluctuation spectroscopy has emerged as a powerful alternative probe to identify BKT transitions [14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Though this technique has been employed to probe the BKT physics in thin-film superconductors, the study of fluctuation statistics is not well explored in layered systems, specifically in van der Waals heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' With increasing interest in such systems as platforms for realizing low dimensional superconductivity in the clean limit and topological superconductivity, there is an urgent need for a detailed study of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In a previous study, we have reported the observation of two-dimensional Ising supercon- ductivity in a van der Waals heterostructure comprising of single-layer-MoS2 (SL-MoS2) and bulk NbSe2 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We established that the reduced dimensionality comes from an effective thinning of NbSe2 due to the coupling with the MoS2 layer making it a perfect example of a ‘buried’ van der Waals superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Thus, the conductance fluctuation spectroscopy technique becomes very relevant to probe the BKT physics in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In this letter, we report a detailed study of the carrier dynamics of this heterostructure through low-frequency conductance fluctuation spectroscopy around the BKT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Through systematic measurements, we establish that superconductivity has a percolative nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We also find proof of correlated dynamics arising from long-range interaction of 3 vortices-antivortices near the TBKT establishing universal BKT nature of the superconduct- ing transition in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' DEVICE FABRICATION To fabricate the device, we mechanically exfoliated single-layer flakes of MoS2 from a bulk crystal [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The thickness of the flake was confirmed from optical contrast and through Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The flake was then transferred onto gold contact probes pre-patterned on hBN substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Subsequently, a flake of NbSe2 exfoliated inside a glove box (with oxygen and moisture levels maintained at less than one ppm) was transferred on top of the MoS2 flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The thickness of the NbSe2 flake was estimated from its optical contrast to be ∼ 15 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Before extraction from inside the glove box, the heterostack of SL-MoS2/NbSe2 was covered by a hBN flake of thickness ∼ 30 nm to protect it from environmental degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Subsequently, the stack was annealed at 200◦ C to increase the coupling between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' RESULTS AND DISCUSSION For the initial characterization of superconducting properties of the heterostructures, electrical transport measurements were done using a DC current source and a nano voltmeter in a four-probe configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The temperature dependence of resistance R shows a metallic behavior at high temperatures followed by a transition to a superconducting state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 1(c)) at T 0 C = 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' From the R versus T plot and also from non-linear current-voltage relations, we estimate TBKT to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='(see Supplementary Information S1 for details, also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We investigated the fluctuation statistics of the system around the TBKT using a 4-probe resistance fluctuation spectroscopy technique [20, 21, 35] – the details are discussed in Sup- plementary Information S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Briefly, the device was current biased, the voltage developed across it pre-amplified and detected by a dual-phase lock-in-amplifier (LIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The demod- ulated output of the LIA was digitized at a sampling rate of 1024 points/s using a 16 bit analog-to-digital conversion card and transferred in the computer memory for further pro- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The biasing current was always maintained at a value much smaller than the critical current of the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The acquired time series of resistance fluctuations were dec- 4 imated and filtered digitally to eliminate aliasing and related digital artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The power spectral density of the resistance fluctuation SR (f) was then calculated over a frequency range 4 mHz–4 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 2(a) is a plot of the time traces of resistance fluctuations for our device measured at a few representative temperatures, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The fluctuations increase in amplitude with T approaching TBKT from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The corresponding SR (f) were found to have a frequency dependence of the form SR (f) ∝ 1/f α (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' One can see that the power spectral density SR (f) increases by several orders of magnitude with decreasing temperature, re- flecting the observed increased resistance fluctuation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Additionally, the value of the exponent α for f < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 Hz (the method of evaluation the exponent is discussed in Supplementary Information S3) increases monotonically from ∼ 1 at higher temperature to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='4 near TBKT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' There can be two possible reasons for this increase in α – (i) transition of the system across different vortex phases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=', from an ordered to a disordered regime [22] or (ii) fluctuation in the domain parameter of different phases across the transi- tion range [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Discriminating between these two scenarios requires further analysis and is beyond the scope of the current letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The relative variance of resistance fluctuations (we refer to this as the noise) was evaluated by integrating SR (f) over the bandwidth of measurement [20, 21, 35]: R = ⟨δR2⟩ ⟨R2⟩ = 1 R2 ˆ SR(f)df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 3(b) shows the plots of relative variance and the normalized resistance against T/Tcritical (T/TBKT) for the heterostructure region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We observe that R in the normal state is ∼ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This value is almost five orders of magnitude lower than that reported for a typical semiconducting TMD [24] attesting to the high quality of our heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' With decreasing T, R increases by nearly six orders of magnitude over a very narrow temperature window near TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As we discuss later, this divergence in noise can be well explained in terms of a percolation network model of superconducting fluctuations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Moreover according to the percolation model in the transition regime the system should follow the relation, SR (f) /R2 ∝ R−lrs where lrs is the percolation exponent which takes up the value ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 in the classical picture [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In our case, the exponent lrs comes out to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='03 (see Supplementary Information S4), establishing the percolative nature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Notably, when compared with that of the pristine NbSe2, the noise in the 5 heterostructure is almost an order higher around the respective transition temperatures (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Before we proceed further, the effect of thermal fluctuations on the measured noise needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' dR/dT diverges close to the critical temperature for a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Consequently, any minor fluctuation in temperature can give rise to large resistance fluctu- ations near TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' To eliminate this trivial effect as the the origin of the large resistance fluctuations seen in our device, we evaluated the relative contribution of temperature fluc- tuations to the noise using the relation [(dR/dT) × (δT/R)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here δT is the temperature fluctuation in the measurement system, which has been measured to be < 5 mK in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The evaluated value of this relative variance at TBKT is ∼ 10−7 (see Supplementary Information S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This value is at least two orders of magnitude smaller than the measured R near TBKT, thus ruling out any significant contributions of temperature fluctuations in the measured noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 4 we present a phenomenological explanation of the effect of percolation dynamics on the resistance fluctuations in a 2D superconductor in terms of a percolation network model of superconducting fluctuations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The squares below the plot show the microscopic status of the system schematically in terms of a superconducting-normal network in different T- regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Region-I is a purely superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' On approaching TBKT from below, small patches of dissipative (metallic) domains begin nucleating in the superconducting back- ground (region-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' With increasing temperature, fluctuations in the superconducting order parameter result in the formation of a dynamic network of interconnected superconduct- ing and normal (dissipative) regions [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This effect is especially severe in the case of 2D superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The enhanced noise in this T-regime has two major components – (i) resistance fluctuations in dissipative regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (ii) fluctuations in the number/size of the su- perconducting clusters [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Beyond T = TBKT, the system crosses into region III, where the proportion of superconducting and non-superconducting domains become almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' At this temperature (which we denote as Tmax), the resistance of the device is nearly half of the normal state resistance, and the resistance fluctuation is at its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The other boundary of region III comes at TBCS, which is at ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 K for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' For T > TBCS, the fraction of the superconducting phase decreases sharply with increasing T till the entire system becomes dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Consequent to this decrease in electronic phase segregation of 6 the system, the variance of resistance fluctuations decreases as the system approaches the metallic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We turn now to the nature of the correlations between the fluctuating entities at different T-ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In 2D superconductors undergoing BKT transition, the XY model predicts the fluctuations to be non-Gaussian around TBKT [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' These non-Gaussian resistance fluctua- tions have emerged as a unique signature of BKT physics and have successfully been used to discriminate between 2D and 3D superconductors [14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We quantify the non-Gaussianity of the resistance fluctuations through their ‘Second spectrum,’ which is the four-point corre- lation function of δR, calculated over a frequency octave (fl, fh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Being extremely sensitive to the presence of non-Gaussian components (NGC), this parameter is a highly effective tool to probe correlation in a system [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' To estimate the second spectrum, repeated measurements of the PSD, SR (f) is done over a selected frequency range (fl, fh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The power spectrum of this series over a frequency octave gives the second spectrum [29, 30]: Sf1 R (f2) = ˆ ∞ 0 � δR2 (t) δR2 (t + τ) � cos (2πf2τ)dτ, σ(2) = ˆ fl−fh 0 Sf1 R (f2) df2 ��ˆ fh fl SR (f) df �2 (2) Here f1 is the center frequency of the chosen octave and f2 is the spectral frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' σ(2) is the normalized second spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' it equals 3 for Gaussian fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 5 shows the plot of measured σ(2) as a function of T/TBKT for the heterostructure region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As can be observed, while decreasing the temperature, σ(2) increases from a baseline value of ∼ 3 at higher T to ∼ 30 near TBKT before decaying again to the Gaussian base value for T < TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This enhancement of σ(2) in the narrow window of T in Region-III establishes clearly the appearance of non-Gaussian resistance fluctuations near T = TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In contrast, for the pristine NbSe2 region σ(2) remains at the baseline value of ∼ 3 (black solid circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 5) throughout the temperature range around the transition indicating a Gaussian distribution of fluctuations as expected for a 3D superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Non-Gaussian fluctuations in superconductors can have different origins – (1) long-range correlation among the vortices as has been observed in previous studies [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (2) the dom- inance of percolation kinetics around superconducting transition seen in inhomogeneous superconductors [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' and (3) dynamic current redistribution which appears as a conse- quence of substantial transport inhomogeneity and large local resistivity fluctuations that 7 translate to the necessary condition of δR/R >> 1 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The third cause can be immediately ruled out by noting that in our systemδR/R << 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' To discriminate between the remaining two scenarios, note that a comparison of the T dependence of R and σ(2) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 5) reveals that significant fluctuations in the resistance extends beyond the onset of normal state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' R/RN = 1) at TBCS of ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In the low-temperature limit, it extends to T < TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' On the other hand, the deviation from the Gaussian value in σ(2) stays confined in region III within TBKT < T < TBCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This deviation indicates that the increase of σ(2) that marks the existence of non-Gaussian fluctuations in the fluctuation is a consequence of an independent process which is unlikely to be the percolation kinetics that dominates the spectrum of re- sistance fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This strongly suggests that the first scenario of correlated vortices is at play in inducing the non-Gaussian fluctuations in the system, as has been reported earlier for clean, homogeneous superconductors [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Further theoretical and experimental studies are essential to establish unequivocally if this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' CONCLUSION In summary, we have studied the carrier dynamics of SL-MoS2/NbSe2 heterostructures near the superconducting transition by probing the low-frequency conductance fluctuations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The first spectrum (resistance noise) shows signatures of the percolative nature of the superconducting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We provide a phenomenological explanation of the different phase-space regions around the transition temperature in terms of a percolative microstructure picture and correlate the resulting fluctuations with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Furthermore, we es- tablish the presence of strong correlations in the system around TBKT arising most probably from the interacting vortices and thus established that the superconducting transition in the system is of the universal BKT type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Acknowledgments: The authors acknowledge device fabrication facilities in NNFC, CeNSE, IISc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' acknowledges funding from SERB (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' HRR/2015/000017) and DST (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' DST/SJF/PSA01/2016-17) 8 Si++ SiO2 hBN sunken probe 1L - MoS2 NbSe2 top BN v- Isd v+ 10 µm Bottom BN MoS2 Top BN NbSe2 Overlap (a) (b) 1 0 2 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='7 (c) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) A schematic of the device structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) False colour Differential Interference Con- trast image of the device with different colors defining different layers of the heterostructure – SL-MoS2(green) and pristine NbSe2 (orange) and overlap region (light-red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (c) Temperature de- pendence of the four-probe resistance of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) Time series of resistance fluctuations of the heterostructure region at a few repre- sentative temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) Plots of SR (f) as function of frequency at the same values of T as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) Plot of exponent α versus T/TBKT for heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) Plots of the relative variance of resistance fluctuations R for heterostructure (solid green circles) and for the pristine 3D NbSe2 region (open orange triangles) as function of T/Tcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here T/Tcritical is T/TBKT for the heterostructure and T/T 0 c for the pristine NbSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' On the right-axis are plotted the normalized resistance R/RN for the heterostructure (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 11 I II III IV V 10-6 10-7 10-8 10-9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='3 T / TBKT 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 R / RN (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1 0 R/RN I II III IV V (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) Plots of the variance of noise � δR2� (left-axis, yellow filled circles) and of the nor- malized resistance (right-axis, black solid line) as a function of T/TBKT for the heterostructure region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The color gradient indicates the transition from superconducting (blue) to normal state (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) Schematics representing the microscopic status of the system in each electronic phase (for details see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The color bar gives the value of R/RN – blue represents the zero resistance i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' superconducting state and red represents the normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 12 I II III IV V 10-6 10-7 10-8 10-9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='3 T / Tcritical 0 5 10 15 20 25 30 35 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plots of the variance of resistance fluctuations � δR2� (filled yellow circles, left axis) and the normalized second spectrum σ(2) for heterostructure (filled blue circles, right axis) and for pristine 3D NbSe2 (filled black circles, right axis) as a function of T/Tcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here T/Tcritical is T/TBKT for the heterostructure and T/T 0 c for the pristine NbSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The color gradient has the same connotation as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (for details see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 13 SUPPLEMENTARY INFORMATION S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' EVALUATION OF BKT TRANSITION TEMPERATURE Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) Zero magnetic field current-voltage characteristics of the heterostructure with varying temperature ranging from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 K to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The red dashed line represents the range of current within which linear fit was done for each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) Plot of γ as a function of temperature, T to evaluate TBKT for heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (c) Plot of (dlnR/dT)−2/3 as a function of temperature, T for the heterostructure where the intersect of the black dashed line with the x axis gives TBKT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' For the initial characterization of superconducting properties of the heterostructure, elec- trical transport measurements were done using a DC current source and a nano voltmeter in a four-probe configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As reported in our previous work, the superconductivity in system is of 2D nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We thus expect the observed superconducting transition to be of Berezin- skii–Kosterlitz–Thouless (BKT) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' For a 2D-superconductor there exists a characteristic temperature, TBKT below which a finite electric current can unbind the vortex-antivortex pairs system, giving rise to a dissipation which is reflected in the current-voltage character- istic as a non-linear behavior of the form, V ∝ Iγ(T) [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' γ is a temperature dependent exponent that takes the value 3 at T = TBKT and eventually goes to 1 in the normal ohmic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S1(a) shows the zero field DC non-linear current-voltage characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The value of the exponent γ, evaluated through linear fitting of each curve within the marked region, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' From this plot, we evaluate TBKT to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='13 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We also evaluated TBCS (defined as the temperature where onset of transition occurs or in other 14 words where the IV becomes linear i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' γ = 1) to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The BKT transition temperature can also be obtained from resistance vs temperature plot as near TBKT the resistance takes the form R = R0exp[−bR/(T − TBKT)1/2] (where bR gives the vortex-antivortex interaction strength) [31, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' To evaluate TBKT we reduced the formula to a form, (dlnR/dT)−2/3 = (2/bR)2/3 (T − TBKT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S1(c), the intercept of the plot of (dlnR/dT)−2/3 vs T gives TBKT as to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='14 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' DETAILS OF NOISE MEASUREMENT TECHNIQUE PA LIA DAQ card Rseries (>>Rsample) I+ I- V- V+ B A Y X Input Output A/D conversion + acquisition Digitized data ~ LIA Sine-out Vac Cryostat DEVICE iac Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Schematic diagram of the noise measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The sample is probed in four-probe geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' PA represents the pre-amplifier and LIA represents the dual phase lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We investigated the fluctuation statistics of the system around superconducting transition through analysis of the zero field temperature dependent resistance fluctuations acquired using an ac technique that allows us to measure the fluctuations from system and the background simultaneously [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S2 is schematic of the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The sample was current-biased using the sine wave output of a lock-in amplifier (SR830).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' A resistor, Rseries in series with the sample controls the current through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The value of the excitation current was always maintained to be lower than the critical current of the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 15 The voltage developed across the sample was detected using the dual-phase lock-in-amplifier coupled with a preamplifier (SR552).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The excitation frequency of the current was kept at the eye of the noise figure of the preamplifier to minimize the contribution of amplifier noise in measured the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The time constant were set to be 30 ms with a filter roll off of 24 dB/octave - this subsequently determines the upper cutoff frequency of the power spectral density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The output of the LIA was digitized at a sampling rate of 1024 points/s using a 16 bit analog-to-digital conversion card and transferred in the computer memory for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The in-phase channel (X-channel) picks up the excess noise from the sample as well as the background whereas the quadrature channel (Y-channel) picks up only the fluctuations from background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' At every temperature, the time series of the resistance fluctuations was acquired for a duration of 30 minutes (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='8X106 data points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' These were subsequently decimated with a factor of 128 and digitally filtered to eliminate aliasing and related digital artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' These filtered time series were then used to calculate the power spectral density (PSD) over the specific frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The PSD of the sample noise were finally obtained by subtracting the PSD of X-channel fluctuation from that of the Y-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' EVALUATION OF THE EXPONENT α FROM SR (f) As mentioned in the main manuscript, the power spectral density, SR (f) has a frequency dependency SR (f) ∝ 1/f α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' To evaluate the exponent α we plotted SR (f) as function of frequency, f in log-log scale, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The slope of these plots gives the value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As can be seen here the slope i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' α is ∼ 1 at 8 K in which the system is in normal state whereas at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 K which is near to TBKT the value becomes ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' CLASSICAL PICTURE OF PERCOLATION For a system having percolative nature it follows that the spectral density of relative resistance fluctuation at a certain frequency, SR (f) /R2 grows as power law to the decreasing resistance towards superconductivity with a form given by [36] SR (f) R2 ∝ R−lrs (S1) 16 Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plot of SR (f) as a function frequency, f for two different temperatures around the transition, 8 K (violet) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 K (brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The dashed lines show the linear fits to the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The boxes with each plot show value of the slope for the respective fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' where lrs is the percolation exponent, which takes the value ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 in the classical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S4 the percolation exponent for the heterostructure comes out to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='03 which matches quite well with the classical percolation picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' CONTRIBUTION OF TEMPERATURE FLUCTUATIONS TO THE MEASURED NOISE As mentioned in the main manuscript, we have evaluated the relative contribution of the temperature fluctuation of the measurement system to the measured noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The temperature stability of a system depends mainly on the PID value of the temperature controller used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' We have fixed this PID value in such a way that we were able to have a temperature fluctuation, δT < 5 mK at all temperatures at which noise measurements were done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' In Fig S5(a), we show a plot of T −Tsetpoint versus time over a period of 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This 17 lrs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='03 Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plot of SR/R2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0625 Hz as function of R(olive solid circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Both the axis are in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The red line is the linear fit to the data that yields a slope of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='89 is the typical time for a single noise run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here Tsetpoint is the target temperature value (in this case, 12 K), and T(t) is the instantaneous value of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S5(a), the maximum fluctuation is about 3 mK indicating that taking 5 mK as δT in our calculation is a safe choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S5(b) shows the plots of the measured relative variance (olive solid circles) and that estimated from temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' One can see that near TBKT, the value of relative variance of resistance fluctuations estimated from the temperature fluctuations is almost two orders of magnitude smaller than the measured relative variance of resistance fluctuations, R of the heterojunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This establishes that temperature fluctuations play a negligibly small role in the measured noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 18 Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (a) Plot of T − Tsetpoint as a function time measured at 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' (b) Plot of the measured relative variance � δR2� /⟨R⟩2 for heterostructure (olive solid circles) and that estimated from ther- mal fluctuation (blue solid circles) as function of T/TBKT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The lines are guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The right-axis shows a plot of the resistance for the heterostructure region (solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' NOISE DATA FROM ANOTHER DEVICE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S6 shows the relative variance of resistance fluctuations of another device D2 having identical structure to the device D1 whose data were presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As is evident from the plot, the data from D2 is very similar to that from D1 – it shows percolative transition near TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Similar to the data for device D1, for D2 also we observe an order of magnitude higher value of the relative variance for heterostructure region in comparison to pristine NbSe2 section of the device near T/TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S7 shows the variance of the resistance fluctuations (olive solid circle, left axis) along with the normalized resistance, R/RN (red line, right axis) of the heterostructure region measured for device D2 as a function of T/TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' As can be seen, the increased fluctuation extends beyond TBCS into the normal state just as for D1 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' The deviation of the normalized second spectrum, σ(2) from the baseline value of 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S8 are constricted within the region bounded by TBKT and TBCS suggesting that, like device D1 in main text, the non-Gaussian nature for device D2 also occurs due to the long range correlations between the vortex-antivortex pair around the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Moreover, as 19 Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plots of the relative variance of resistance fluctuations, � δR2� / � R2� for the heterostruc- ture (solid olive circles) and for the pristine 3D NbSe2 region (solid orange circles) as function of T/Tcritical for device D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here T/Tcritical is T/TBKT for the heterostructure and T/T 0 c for the pris- tine NbSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' On the right-axis are plotted the normalized resistance R/RN for the heterostructure (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' expected we observed σ(2) to be ∼ 3 for pristine NbSe2 region of device D2 throughout the temperature range indicating the Gaussian nature of the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' This similarity in the evaluated results for the two different devices of similar heterostructure thus proves that the main observed phenomenons are inherent to the system and not device specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 20 Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plots of the variance of noise � δR2� (left-axis, olive solid circles) and of the normalized Resistance (right-axis, red solid line) as a function of T/TBKT for the heterostructure region of device D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content='2 10-9 10-8 10-7 10-6 10-5 <δR2> (R2) T/Tcritical TBKT TBCS 0 5 10 15 20 σ(2) Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Plots of the variance of resistance fluctuations � δR2� (solid olive circles, left axis) and the normalized second spectrum σ(2) for heterostructure (solid red circles, right axis) and for pristine 3D NbSe2 (solid blue circles, right axis) as a function of T/Tcritical for device D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Here T/Tcritical is T/TBKT for the heterostructure and T/T 0 c for the pristine NbSe2 22 [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Novoselov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Geim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Morozov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Dubonos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQf2AUg/content/2301.03473v1.pdf'} +page_content=' Grigorieva, and A.' 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sha256:07fb0364660809854d4b253c623b9f2aeedf1ce54582558333f99915523186d8 +size 144929 diff --git a/yNE4T4oBgHgl3EQfxw02/content/tmp_files/2301.05260v1.pdf.txt b/yNE4T4oBgHgl3EQfxw02/content/tmp_files/2301.05260v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e25b0ba9ccee74782d28334591d60a701fe1ef52 --- /dev/null +++ b/yNE4T4oBgHgl3EQfxw02/content/tmp_files/2301.05260v1.pdf.txt @@ -0,0 +1,1916 @@ + + +On the switching mechanism and optimisation of ion +irradiation enabled 2D MoS2 memristors +Samuel Aldana,*a Jakub Jadwiszczak a and Hongzhou Zhang a +Memristors are prominent passive circuit elements with promising futures for energy-efficient in-memory +processing and revolutionary neuromorphic computation. State-of-the-art memristors based on two-dimensional +(2D) materials exhibit enhanced tunability, scalability and electrical reliability. However, the fundamental of the +switching is yet to be clarified before they can meet industrial standards in terms of endurance, variability, +resistance ratio, and scalability. This new physical simulator based on the kinetic Monte Carlo (kMC) algorithm +reproduces the defect migration process in 2D materials and sheds light on the operation of 2D memristors. The +present work employs the simulator to study a two-dimensional 2H-MoS2 planar resistive switching (RS) device +with an asymmetric defect concentration introduced by ion irradiation. The simulations unveil the non-filamentary +RS process and propose practical routes to optimize the device's performance. For instance, the resistance ratio +can be increased by 53% by controlling the concentration and distribution of defects, while the variability can be +reduced by 55% by increasing 5-fold the device size from 10 to 50 nm. Our simulator also explains the trade-offs +between the resistance ratio and variability, resistance ratio and scalability, and variability and scalability. Overall, +the simulator may enable an understanding and optimization of devices to expedite cutting-edge applications. +1. Introduction +The +development +of +Information +and +Communication +Technology (ICT), such as 5G circuits and the internet of things, +demands breakthroughs in non-volatile memory1 since the +state-of-the-art flash technology is hitting its physical limits in +terms of power consumption and device scalability. 2 Among the +emerging memory technologies, memristors have attracted +much research interest and become the herald of next- +generation computational architecture, revolutionizing ICT. 3-6 +The potential of memristors is rooted in their superior +performance (e.g., ultrafast switching, low power consumption, +data retention and endurance), intrinsically high scalability, +stackability, compatibility with Complementary Metal Oxide +Semiconductor (CMOS) technology and flexibility for wearable +applications. 1, 2, 7 However, device variability, including cell-to- +cell variability and cycle-to-cycle variability, hinders the +industrial deployment of memristor technology. The cell-to-cell +variability is the inhomogeneity between devices via the same +fabrication process and cycle-to-cycle variability is related to the +operation of individual devices, 1, 7 which emerges from the +stochastic processes during the resistive switching (RS). For +example, in filamentary RS devices the location and morphology +of conducting filaments may vary between cycles and cells. 8, 9 +Device variability is inevitable in RS devices which relies on +material defects10-13, while the forming process exacerbates the +problem by varying defect concentration and distribution14. +Although the cycle-to-cycle variability can be insignificant in +some applications exemplified by neuromorphic imaging +recognition, 1 it imposes major limitations on memristor +applications, rendering a range of high-end applications +impractical (e.g., multilevel information processing and long- +term storage7). For example, device variability causes signal +degradation in crossbar arrays15 and hampers projections of +device lifetime, demanding excessive budget in testing. 16 +Verification and iterative approaches may mitigate the +resistance +variability +in +multilevel +information17 +and +radiofrequency applications respectively. +18 However, a +consensus on a figure of merit for the variability issue is yet to +be achieved and a lack of comparable statistics on device +variability remains a main obstacle to the technology. 7 +Recently, memristive behaviour has been observed in a range +of two-dimensional (2D) materials. 19-25 This may expedite the +industry deployment of the memristor technology since 2D +memristors exbibit superior tunability, scalability and electrical +reliability. 4, 19, 26-34 These characteristics arise from the ultrathin +nature and unique mechanical, electric and optoelectronic +properties of 2D materials. 26, 27 The 2D memristor landscape +shows diverse device architectures and switching mechanisms. +For example, the switching of 2D vertical memristors20, 35-38 +depends on the formation and rupture of conductive filaments. +2D planar memristors may rely on phase transition, 39, 40 charge +trapping/de-trapping, 41, 42 electron tunnelling modulated by +polarization, 43 electrochemical processes44 and Schottky barrier +modulation. 29 Defect migration plays a crucial role in these +processes. 32, 45-47 Compared with their 3D counterparts, the +device variability of 2D memristors has rarely been explored, +while it is of utmost importance to gain in-depth knowledge of +the switching process and mitigate the variability issues in 2D +memristors. +Physical simulators are indispensable for understanding the +resistive switching process and they can greatly facilitate +investigations on device viability. 7, 48, 49 Ab initio calculations can +accurately relate the resistive switching to the defect creation, +50 the electronic structure and transport properties in +nanostructures. 51 However, they are limited to relatively small +volumes (a few nm as maximum) and short times (shorter than +ns) and can hardly reproduce RS processes at the device level. +48 Continuous models are apt to describe the average +behaviours of individual devices or even circuits. However, they +overlook the microscopic characteristics of the system, such as +particle migration52-55, and are hence not suitable to investigate +the stochasticity that emerges from the evolution of +a. Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), +Advanced Materials and Bioengineering Research (AMBER) Research Centers, +School of Physics, Trinity College Dublin, Dublin, D02 PN40, Ireland. E-mail: +aldanads@tcd.ie +Electronic Supplementary Information (ESI) available. + + +microscopic configuration of the system during the switching +process. The kinetic Monte Carlo (kMC) algorithm is an +established technique to study the microscopic evolution and +its impacts on device performance. 11, 12, 56-59 For example, the +kMC algorithm can reproduce particle diffusion and the +formation and rupture of percolation paths involving several RS +cycles of memristors. 11, 56-58 Although the kMC algorithm and +continuous models have been employed to explore defect +accumulation and electrical conduction in 2D materials, 29, 38 the +simulation of 2D memristors at the device level is scarce and the +variability of 2D memristors has been rarely explored. In this +work, we build a new physical kMC simulator for the defect +migration process in 2D materials and shed light on the +operation of 2D memristors. We investigate the effects of initial +vacancy distribution, the scaling limits and the factors that +regulate device endurance and variability. We use experimental +data from the MoS2 memristor enabled by site-specific defects +to collate and verify the model. 19 The simulator helps +understand the physics of resistive switching in 2D materials, +offering practical guidance to optimize device performance. +2. Experimental results and simulation +approach +Figure 1a shows a device schematic. The device exhibits a planar +metal-semiconductor-metal structure and the semiconductor +channel is a 2D 2H-MoS2 with an asymmetric defect +concentration introduced by focused helium ion irradiation (see +more details about the device fabrication in Supplementary +Information, section 1). We note that the irradiation-induced +defects enable the resistive switching, while devices of pristine +MoS2 do not switch. 19 The defect region, referred to as the +fissure, bisects the channel with an asymmetric concentration +along the horizontal direction, i.e., across the electrodes. For +example, the right tail of the peak (tens of nanometres) in Figure +1a is much wider than the left (< 1 nm). +In our simulation, we have assessed three initial defect +distributions with different asymmetries (see Supplementary +Information, section 2), i.e., a skewed Gaussian distribution +(Figure S1a), a triangle function (Figure S1b) and a step function +(Figure S1c). The skewed Gaussian distribution emulates the +asymmetric distribution of defects observed experimentally19, +and the triangle and step functions of different asymmetries are +to explore possible routes for device optimization. The source +electrode as convention is grounded and placed at the left side +of the asymmetric peak. The polarity of applied voltage +regulates the direction of the fields with respect to the +asymmetric defect distribution, i.e., a positive voltage (𝑉𝑑 > 0 ) +indicates the electric field points from the abrupt edge of the +fissure to the long tail side. +Figure 1b shows 14 experimental quasi-static pinched +hysteresis loops by selecting every 15th loop from 1200 +consecutive cycles measured at a triangular voltage ramp +between ± 35 V with a rate of 6 V s−1. The device shows +bipolar operation, and the SET (RESET) process occurs at a +positive (negative) voltage. The current level increases with the +cycling. Figure 1c shows the resistance ratio over cycle, which +shows that cycling the device leads to an exponential decay +(shown as a dashed line) in resistance ratio (21% after 1000 +cycles). These degradation processes will be further discussed +alongside the simulation work. +To investigate the device switching process, we implemented +the kinetic Monte Carlo (kMC) algorithm60 using MATLAB in a +simulation domain of 50 × 50 nm. The vacancy distribution +within the fissure evolves with the external electric field applied +via the electrodes, 19 while no defects escape the simulation +domain during the switching. Since the fissure region dominates +the device resistance, the size of the simulation domain is +sufficient to include the main physical processes involved in the +resistive switching. The defects are doubly charged sulfur +vacancies because the helium-ion irradiation preferentially +removes sulfur atoms from the MoS2 lattice. 33 The generation +of new defects under the applied electric field is negligible due +to the high activation energy (5.85 eV 61 for vacancy and > 5 eV +for antisite defect62). The sulfur vacancy migrates via the +exchange of the vacancy position with one of the adjacent sulfur +atoms, 29, 63 as shown in the Figure 1a. In the quasi-static +switching, the vacancy migration events occur at a much longer +time scale (~ s) than the lattice vibration (10−13 s), so the +system is in thermodynamic equilibrium for any vacancy +distributions. 60 Furthermore, the field-driven migration renders +the reverse migration negligible, validating the kMC approach. + +Figure 1: Device. a) Schematic device structure with the defect distribution in the MoS2 +channel. The 50 × 50 nm simulation domain represents the defect profile of a skewed +Gaussian distribution, while the channel is on the micrometre scale. The arrows in the +lattice structure indicate the possible routes of vacancy migration, i.e., a S atom (yellow) +exchanges its position with a nearest-neighbour vacancy site (grey). b) 14 representative +experimental I-V hysteresis loops sampled from continuous 1200 cycles. c) The evolution +of the resistance ratio (calculated at -10 V) over cycles (time in the top x-axis). The dashed +line is an exponential fitting, and the shaded indicates the 95% prediction interval. + +a +b +101 +40 +S vacancy +Mo +4 +100 +20 +2 +Cycle 1 +10 +103 +Cycle 5 +1 +Cycle 10 +10-4 +Cycle 15 +20 +40 +30 +20 +-10 +0 +10 +20 +30 +x[nm] +Voltage [V] +c +8.0 +Time (s] +0 +5000 +10000 +15000 +20000 +Vd +25000 +Fissure +7.5 +V,=0 +Y. +Drain +Source +MoS2 +SiO2 +5.5 +5.0 +0 +200 +400 +600 +800 +1000 +1200 +Cycles + +In our simulation, we combine the electric response with the +thermal effect since the lattice temperature modulates the +transition rate, which is given by Maxwell-Boltzmann statistics +and Transition State Theory (TST), i.e. 𝛤 = 𝜈 · exp (−𝐸𝐴/𝐾𝐵𝑇), +where 𝜈 = 7 × 1013 s−1 is the vibration constant of the +particle, 𝐾𝐵 the Boltzmann constant, 𝑇 the temperature and 𝐸𝐴 +the activation energy of the migration. The activation energy is +modulated by the local electric field as follows: 𝐸𝐴 = 𝐸𝐴 +0 − 𝒃 · +𝑭(𝑥, 𝑦), 12 where 𝑭(𝑥, 𝑦) is the electric field, 𝒃 the polarization +factor and 𝐸𝐴 +0 = 2.297 eV the activation energy for migration in +the zero-field condition. 4 The kMC algorithm weighs all the +possible migrations and chooses the evolvement path. It should +be noted that the larger the transition rate, the smaller the time +step 𝑡 = − ln(𝑟𝑎𝑛𝑑) / ∑ 𝛤, where 𝑟𝑎𝑛𝑑 stands for a random +number between 0 and 1 and ∑ 𝛤 is the summation of the +transition rates for all possible migrations. For each defect +distribution +during +the +switching, +the +local +vacancy +concentration 𝜌𝑑 (𝒓⃗⃗ ) is averaged over 6 × 6 grid points (3.2 +nm2), which determines the local resistance 𝑅(𝒓⃗⃗ ) via the +empirical relationship 𝑅(𝒓⃗⃗ ) ∝ 𝜌𝑑 +𝑛 (𝒓⃗⃗ ), 33 (𝑛 is a parameter +extracted from the experimental results, see Supplementary +Information, section 4). The electric field screening is assumed +to be a linear function of the local resistance since the dielectric +constant in MoS2 strongly depends on the distribution and +number of sulfur vacancies. 64 For a given applied voltage, the +local electric field decreases with the increase in the defect + +Figure 2: Switching mechanism. The initial defect profile of the device exhibits a skewed Gaussian distribution with a peak density of 5.64 Vs nm−2 and a width +of 8 nm. The voltage ramp starts from positive polarity with a rate of 0.71 V s−1 between 35 V and −35 V. a) a typical simulated I-V pinched hysteresis loop. +Numerals on the loop mark four representative states of the switching process. The SET process takes place from I to II, and for the RESET from III to IV. The colour +maps shown in the right panel correspond to the microscopic configuration of defects in the LRS (top) and HRS (bottom). b), c) and d) are the defect density, local +resistance and electric field profiles along the x-axis during the SET, respectively. e), f) and g) are the correspondent profiles during the RESET processes. + +a +State Il:LRS +40 +III +100 +RESET +2 +口 +[μA] +SET +IV +10 +2030 +40 +50 +x[nm] +State I: HRS +40 +[nm] +2 +10-2. +-30 +-20 +-10 +0 +10 +20 +30 +10 +2030 +40 +50 +Voltage [V] +x [nm] +b +c +d +7 +300 +8.6 V start +8.6 V start +8.6 V start +6 +17.1 V +17.1 V +17.1 V +28.6 V +28.6 V +250 +28.6 V +m-11 +34.3 V +Resistance [MQ] +34.3 V +34.3 V +28.6Vback +6 +28.6Vback +200 +28.6Vback +Defect density I +4 +8.6Vback +8.6Vback +8.6Vback +4 +3 +1 +2 +100 +2 +1 +50- +0 +0 +0 +20 +25 +30 +35 +40 +20 +25 +30 +35 +40 +20 +25 +30 +35 +40 +x [nm] +x [nm] +x [nm] +e +f +6 +7 +0 +-8.6Vstart +-8.6 V start +8 +6 +-17.1 V +-17.1 v +50十 +-28.6 V +-28.6 V +IV +-34.3 V +-34.3 V +-100 +t density [Vs +-28.6Vback +-28.6back +[MV r +4 +-8.6 Vback +Resistance +-8.6 Vback +field +IV +-150 +II +3 +-8.6V start +Electric +Defect +IV +-200 +-17.1 V +2 +2 +-28.6 V +-250 +-34.3 V +III +1 +-28.6vback +0 +II +IV +-8.6back +-300- +0 +20 +25 +30 +35 +40 +20 +25 +30 +35 +40 +20 +25 +30 +35 +40 +x[nm] +x[nm] +x [nm] +density, indicating the defect migration is a self-limiting process. +Further details about the simulation can be found in the +Supplementary Information, section 3. +3. Simulation results and discussions +Figure 2a shows a typical simulated pinched hysteresis loop +from a defect distribution of skewed Gaussian profile under a +voltage ramp rate of 0.71 V s−1 between 35 V and -35 V. Prior +to the switching, the fissure region is 8 nm wide with a peak +density 𝜌 = 5.64 Vs nm−2 (see the probability distributions in +Supplementary Information, section 2). The I-V sweep follows +the directions indicated by the arrows. The loop shows a SET +process with a positive voltage and a RESET process with a +negative voltage, suggesting the same bipolar switching +behaviour +observed +experimentally. +The +simulation +corroborates that the operation of the device does not need a +forming process, facilitating circuit simplicity. 65-67 The switching +is progressive, in contrast to an abrupt resistance change, +suggesting the absence of filamentary conduction. This explains +the observed low power consumption. 7 The resistance ratio +(𝑟 = 𝑅𝑜𝑓𝑓/𝑅𝑜𝑛) is 1.44 calculated at -4 V during the RESET +process and the maximum current level is 3 μA. It is interesting +to note the simulated loop reproduces the asymmetry found +experimentally between the SET and RESET processes (see +Figure 1b). +The switching is due to the reconfiguration of the defect +distribution within the fissure by the electric field. The two +colour-maps attached to Figure 2a show the microscopic +distributions of defects in the High Resistance State (HRS) and +the Low Resistance State (LRS), corresponding to the state +labelled by I) and II) on the loop respectively (more microscopic +details about the states I-IV can be found in Supplementary +Information, section 5). The defects accumulate within a small +region in the HRS, leading to a much higher density than the LRS. +Figure 2b details the defect evolution during the SET process. +The defect profiles are extracted at a series of sequential +voltages from state I to II (see Figure 2a) and the lightness of the +curves reduces with increasing time. The initial vacancy +concentration exhibits a skewed Gaussian distribution (the +most intense red curve), mimicking the defect profile by the ion +irradiation. The external field of positive polarity gradually +drives the defects away from the fissure region, lowering the +peak and leading to a plateau of the distribution across a 10-nm +wide region. The local resistance varies with the defect profile +(see Figure 2c). The spatial-varying resistance regulates the +electric potential distribution in the channel when an external +voltage is applied (see Figure 2d). The larger the resistance of a +region, the larger the electric field, and the more significant the +vacancy drifting. Therefore, the self-adaptive electric field +reduces the vacancy concentration and hence the resistance. +From State I to II, the peak resistance reduces by an order of +magnitude, leading to an overall reduction in the channel +resistance and hence the SET process. +Figure 2e reveals the evolution of the defect profile during the +RESET process from state III to IV (see Figure 2a). The defects +move towards and accumulate at the abrupt edge of the fissure +(𝑥 = 22 nm in Figure 2e), which increases the local resistance +(see Figure 2f) and hence the local electric field (see Figure 2g). +On the long tail of the peak (i.e., 𝑥 > 22 nm), the field pushes +defects from a wide region (22 nm < 𝑥 < 30 nm) towards the +peak, while on the left side of the peak the field drops drastically +within a 1 nm region. This asymmetric distribution of the field +limits the escape of the defects from the fissure into the left side +of the channel and causes the defect accumulation at the peak, +recovering the initial defect configuration. The drift of defects +to the left side of the fissure may become important if the field +is sufficiently high where the RESET process will fail, leading to +device failure (see Figure S4). The simulator can predict the +operational range of the voltage for the device. For example, a +device with a peak density of 5.64 Vs nm−2 and a width of +12 nm can be stressed with -30 V for 17 s or -40 V for 2 s before +the device fails (see Figure S4a). +The simulator allows us to investigate the device performance. +Figure 3a shows the temporal evolution of the device resistance +to a triangular voltage ramp (2.1 V s−1) between 25.2 and +−25.2 V. The initial defect profile has a skewed Gaussian +distribution with a peak density of 5.64 Vs nm−2 and a width of +9 nm. The voltage ramp starts from a positive polarity +𝑉𝑑. The +device switches over 45 cycles (2161 s). Both the 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 +drop 57% after the first ten cycles and reaches a steady state +where 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 drop slower (17% in 35 cycles). The +resistance drop occurs due to a progressive reduction in the +peak density of the defect profile, which dominates the device +resistance. This is evident in Figure 3b and 3c, which show the +evolution of the density profile with consecutive RESET and SET +processes. Here, we can see that cycling gradually relocates the +defects into the originally low-density (right tails) regions +(below 3.5 Vs nm−2) at the expense of the peak density. This +reduces the defects population of the peak region from 31% in +cycle 1 to 22% in cycle 45, as can be seen in Figure S5a. The +defect accumulation in the tail region (> 25 nm onward in the x- +axis) stems from the low electric field in the region, which is 16% +Figure 3: Device fatigue. The initial defect profile of the device exhibits a skewed +Gaussian distribution with a peak density of 5.64 Vs nm−2 and a width of 9 nm. The +voltage ramps from a positive polarity with a rate of 2.1 V s−1 between 25.2 V and −25.2 +V for 45 cycles (2161 s). a) shows the resistance evolution over time with two exponential +fittings for the LRS (in red) and the HRS (in blue) of the cycles. The shaded regions +correspond to the 95% prediction interval of the fitting. b) and c) correspond to the +density profile along the x-axis (averaged over the y-axis) for successive RESETs and SETs +respectively. d) resistance ratio projection based on the simulation. + +a +Time (s) +b +500 +1000 +1500 +2000 +6 +350 +Consecutive RESETs +Cycle 1 +Resistance +Cycle 15 +300 +Rorr fitting +Cycle 30 +[OW] +Row fitting +Cycle 45 +Resi +150 +100 +0+ +0 +10 +20 +30 +40 +15 +20 +25 +30 +35 +40 +Cycles +d +x [nm] +U +6 +1.8 +Time (s) +Consecutive SETs +Cycle 1 +102 +103 +104 +Cycle 15 +1.7 +Cycle 30 +RorF/Row +1.6 +Fitted line +Cycle 45 +01.3 +1.2 +1.1 +15 +20 +25 +30 +35 +40 +101 +102 +x [nm] +Cycles + +of the electric field found in the peak (see Figure S5b). The low +field modulates the migration barrier by 1% (see Figure S5c), +leading to negligible field-driven migration in the tail region. The +further the defects migrate away from the peak into the tail +during the SET process, the harder for them to move back to the +peak during the RESET process. Consequently, the defect +distribution flattens across the fissure during the cycling leading +to a gradual reduction in the device resistance. This flattening +of the defect profile progressively reduces the difference +between the OFF and ON states, reducing the ON/OFF ratio and +eventually leading to device failure (see also the resistive +switching loops in Figure S6). As observed experimentally, the +power consumption increases (see the rise from 20 μW to more +than 50 μW in 45 cycles in Figure S7) with the resistance +reduction. Nevertheless, the simulation qualitatively explains +the fatigue behaviour (see Figure 1c) and allows quantitative +prediction. For devices with long cycling tolerance, we can fit +the behaviours of 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 (red and blue dashed lines in +Figure 3a respectively) and project the evolvement of ON/OFF +ratio for longer times (Figure 3d). +The most significant advantage of the simulator is its capability +of exploring device variability since the kMC algorithm is apt to +investigate stochastic processes. Figure 4a shows the cycle-to- +cycle variability of the resistance ratio in one 45-cycle +simulation, starting with a skewed Gaussian distribution of +vacancies, negative polarity, with a peak density 𝜌1 = +5.64 Vs nm−2 and a width of 9 nm. The value of resistance ratio +distributes uniformly in the range of 1.2 to 1.4 with the mean of +1.29. We define the cycle-to-cycle variability of two consecutive +cycles as ∆r = |𝑟𝑖 − r𝑖+1|. The standard deviation of the cycle- +to-cycle variability is 0.03. In Figure 4b, we investigate the cell- +to-cell variability by running 26 independent simulations for a +given macroscopic distribution (see section 3 in Supplementary +Information for more details about running different +simulations). Each simulation starts with a unique initial +microscopic vacancy configuration in the lattice, while the +macroscopic defect distribution remains the same. This +scenario mimics the device fabrication process, where the +sputtering process and the creation of vacancies are stochastic +at the nanometre scale, 19, 33 introducing cell-to-cell variability. +We note the device fabrication is limited by many other +parameters (e.g., ion beam stability, focusing, sample +cleanliness, etc.) and the cell-to-cell variability here represents +the upper limit of the ideal situation. For each such simulation, +we average the resistance over 15 cycles and use the standard +deviation of the cycle-to-cycle variability as the error. Here, we +can observe a uniform distribution of resistance ratios around +the mean of 1.31 with a standard deviation of 0.02. + + +We investigate the potential of device scalability by evaluating +the effects of the device dimensions (i.e., height and width) on +the device resistance and the resistance ratio. For the +simulations shown in Figure 5, the initial defect profile has a +skewed Gaussian distribution with a peak density of +5.64 Vs nm−2. The device is under a voltage ramp rate of 2.1 +V s−1 between 25.2 and -25.2 V, which starts from the positive +polarity. Figure 5a shows the temporal evolution of the +resistance. It is evident that the reduction in the vertical +dimension, i.e., device height, increases the overall device +resistance. This corroborates the non-filamentary conduction +and the resistance increase is due to the reduction of +conducting channels along the vertical direction. The limit of +scaling down the vertical dimension is shown in Figure 5b. Both +the resistance ratio and the device variability deteriorate as the +device height reduces. A trade-off needs to be identified +between the device conductance and the resistance ratio +(variability), indicating a limit on the scalability of the y- +dimension. The scaling of the horizontal dimension, i.e., along +the x-axis, is regulated by the fissure width and the range of +defect drift. As shown in Figure 5c, the device resistance +decreases with the fissure width, and the resistance ratio and +the device variability also degrade with the scaling of the fissure +width (Figure 5d). The simulation reveals the scaling limit of the +devices, and it can be further extended to achieve practical +device design and optimization for a set of predefined +Figure 4: Device variability. The initial defect profile of the device exhibits a skewed +Gaussian distribution with a peak density of 5.64 Vs nm−2 and a width of 9 nm. The +voltage ramps from a negative polarity with a rate of 2.1 V s−1 between 25.2 V and - +25.2 V. a) shows simulated cycle-to-cycle variability of the resistance ratio with an +average of 1.29 (the dashed line). b) corresponds to the resistance ratio variability of +26 independent 15-cycle simulations initiated with the same parameters but varying +microscopic defect configurations. For each simulation, the resistance ratio is the +average over the 15 cycles and the error bar is the standard deviation of the cycle-to- +cycle variability. +Figure 5: Device scalability. The initial defect profile of the device exhibits a skewed +Gaussian distribution with a peak density of 5.64 Vs nm−2. The voltage ramps from a +negative polarity with a rate of 2.1 V s−1 between 25.2 V and -25.2 V for 15 cycles. The +resistance ratio is averaged over the 15 cycles and the error is the standard deviation of +the cycle-to-cycle variability. a) and b) show the effect of the device height (y-axis) with +a 12 nm wide fissure on the resistance ratio. c) and d) reveals the effect of the fissure +width (x-axis) with a fixed device height of 50 nm. + +a +b +1.40 +1.40 +1.35 +ratio +1.30 +1.20 +1.25 +1.15 +0 +10 +20 +30 +40 +1.20 +0 +5 +10 +15 +20 +25 +Cycles +Simulationsa +b +1.450 +2500 +50 nm +40 nm +1.425 +WU OE +1.400 +20 nm +10 nm +1.375 +1500 +1.350 +istan +1.300 +500 +1.275 +1.250 +0 +100 +200 +300400 +500 +600 +700 +10 +20 +30 +40 +50 +Time [s] +p +Height [nm] +c +500 +1.45 +12 nm +11 nm +1.40 +400 +9 nm +1.35 +[UW] +7 nm +01.30 +5 nm +1.25 +2 1.15 +1.10 +100 +1.05 +0 +100 +200 +300 +400 +500 +009 +700 +1.00 +4 +6 +8 +10 +12 +Time [s] +Width [nm] +performance metrics (e.g., resistance level, resistance ratio and +variability level, etc.). + + +The asymmetric nature of the initial defect distribution is crucial +to the resistive switching. This indicates that device +optimization may be possible by tuning the initial defect +distributions. In Figure 6 we show the resistance ratio for an +initial defect distribution with a triangle shape (Figure 6a) and a +step function shape (Figure 6b) with varying peak densities and +fissure region widths. The skewed Gaussian distribution case is +in Figure S8. In the triangle case (Figure 6a), when the peak +density is higher than 3.95 Vs nm−2, the resistance ratio +appears to exhibit a maximum when the fissure width is varied +from 4 nm to 32 nm. The value of the maximum ratio decreases +and occurs at a larger fissure width when the peak density +decreases. Devices with the step function distribution (Figure +6b) exhibits a similar maximum ratio. In contrast to the triangle +case, the maximum appears in all the peak densities simulated +and on the right side of the peak the resistance ratio falls more +rapidly with increase in the fissure width compared with the +triangle distribution. However, the triangle distribution enables +a higher resistance ratio at a narrower fissure than both the +Gaussian and step function cases, so it may offer a better option +for device scaling and further performance optimization. We +also note that in all the distributions simulated, the resistance +ratio and variability can be enhanced by increasing the initial +peak density. This is because a larger defect population enables +more significant differences between the HRS and LRS. +Finally, we note that although the voltage is high in this MoS2- +based device, the current is low, so the device energy +consumption is reasonable. Besides, as the voltage drops mainly +in the fissure region, we cannot address the scaling of the +switching voltage by reducing the length between the drain and +the source. In this sense, the minimum electric field needed to +move the vacancies determine the voltage scale. Hence, it +might be possible to lower the operating voltage and further +reduce the power consumption by increasing the density peaks +(see in Figure S4 how the electric field has a stronger influence +in higher densities), selecting materials with suitable activation +energies63 or by defect engineering (defects migrate easier +through grain boundaries4). +Conclusions +We have developed a kMC simulator for 2D planar memristors +based on defect migration using the case of a MoS2-based +device enabled by a Helium Ion Beam Microscope. The +simulator reproduces the asymmetric resistive switching cycle +with a performance close to observed experimentally. Besides, +this approach is helpful for insights into the switching +mechanism, in addition to study the device variability, the +device endurance and the resistance ratio. We studied the +device degradation with cycling, which produces defect +relocations into low-density regions, causing a drop in the +resistance ratio and reducing the switching window. Some +trade-offs are proposed to tailor the device features by +controlling the device size, the number of defects introduced in +the channel and their distribution. For example, the device can +be miniaturized at the expense of reducing the resistance ratio, +increasing the variability and the resistance, or higher peak +densities can be used to increase the resistance ratio and +variability. We note that to reduce the switching voltage of a +device based on defect migration, we must enhance the +migration of defects by employing defect engineering, using +higher peak densities or other Transition Metal Dichalcogenides +with lower activation energy for defect migration. Besides, the +conduction mechanism during the ON state is a distributed one, +which explains the drop of resistance when the device size in +the 𝑦-axis is increased. We have also assessed different +distributions of defect density in the channel to find some +routes for device optimization. Comparing the skewed Gaussian +distribution, the step function and the triangle distribution, we +can conclude the triangle shape enables larger resistance ratios +for narrower fissure regions. Besides, the fissure width of these +distributions also affects the resistance ratio. +Author Contributions + +J. J. fabricated the devices and collected the experimental data. +S. A. developed the simulator and conducted the simulation. S. +A. and H. Z. analysed the data and wrote the manuscript. H.Z. +conceived the study and supervised the project. All authors +have given approval to the final version of the manuscript. +Conflicts of interest +There are no conflicts to declare. +Acknowledgements +The authors gratefully acknowledge financial support by the +Science Foundation Ireland under 20/FFP-P/8727. +References +Figure 6: Device optimization. The dependence of the resistance ratio on the initial peak +density (5.64 Vs nm−2, 5.08 Vs nm−2, 4.52 Vs nm−2 and 3.95 Vs nm−2) and device +width for two density profiles: (a) a triangle and (b) step distributions. The devices are +stressed under consecutive voltage ramps with a rate of 2.1 V s−1 between 25.2 to -25.2 +V. 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[2-3] The fissure region extension is usually over 10 nm +and shows an asymmetric distribution. [4] +This fissure region is the current-limiting resistor and is responsible for the resistive +switching behavior. In this sense, the resistance of the device is independent of the channel +length. [4] The defects concentration is what determine the resistance in the channel [3] and +so the proposed mechanism to explain the resistive switching process is the field-driven +migration of defects. [4] + + + + +3 +SECTION 2: Probability distributions used to initialize the defect distribution in the +channel +Figure S1 show three probability distribution used in the device x-axis (i.e., across the +electrodes, see the axes in Figure 1a) employed for the initialization of defect distribution in the +channel. In the Figure, ℎ stands for the probability of finding a defect in the grid points and width +(equivalent to two standard deviations 𝜎 in the skewed Gaussian case and the base of the triangle +and the step function in the other two cases) stands for the extension of the fissure region. Figure +S1a corresponds to the skewed Gaussian distribution case and we use the Pearson distribution [5] +with 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = 1.2 and 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 = 4.4 to calculate it. Figure S1b stands for a right triangle +and Figure S1c for a step function. + +Figure S1: Probability distributions where ℎ stands for the probability of finding a defect in the +grid points. a) The skewed Gaussian function where width correspond to two standard deviations +𝜎 and ℎ𝑚𝑎𝑥 the maximum probability point. b) Triangle function with base width and height ℎ𝑚𝑎𝑥. +c) Step function with base width and height ℎ𝑚𝑎𝑥. Note that h is constant in y-axis for each position +in x-axis. + + + + +a) +b) +0.8 +0.8 +0.8 +Mean +hmax +max +0.6 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0 +0 +0 +Width +Width +Width +-0.2 +-0.2. +-0.2 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +X axis (nm) +X axis (nm) +X axis (nm) +4 +SECTION 3: Simulator details +Regarding the hexagonal crystal structure in Figure 1a, the lattice constants are: 𝑎 = 𝑏 = +0.639 nm and 𝛾 = 60°, typical values for 2H-MoS2. [6] We discretize the simulation domain using +a step size of 0.277 nm and 0.32 nm for 𝑥 and 𝑦 axis respectively for a grid of 50 × 50 nm2. This +discretization is the basis for the defect migration processes and to solve the heat and Poisson +equation using the finite difference method. The Poisson equation has the following form: +∇2𝜑(𝑥, 𝑦) = (𝜕2𝜑(𝑥, 𝑦) +𝜕𝑥2 ++ 𝜕2𝜑(𝑥, 𝑦) +𝜕𝑦2 +) = − 𝜌𝑐(𝑥, 𝑦) +𝜀 + +(S1) +Where 𝜑(𝑥, 𝑦) is the electric potential for a given charge distribution 𝜌𝑐(𝑥, 𝑦) in a MoS2 +monolayer with relative permittivity 𝜀 = 3.7. [7] On the other hand, the steady-state heat equation +has the following form: +𝛾∇2𝑇(𝑥, 𝑦) = 𝛾 (𝜕2𝑇(𝑥, 𝑦) +𝜕𝑥2 ++ 𝜕2𝑇(𝑥, 𝑦) +𝜕𝑦2 +) = 𝑓(𝑥, 𝑦) +(S2) +Here, 𝛾 = +𝐾 +𝐶𝑝𝜌 is the thermal diffusivity, 𝐾 = 34.5 W K−1 m−1 is the thermal conductivity, [8] 𝐶𝑝 = +373 J kg−1 K−1 is the specific heat capacity, [9] 𝜌 = 5060 kg m−3 is the mass density for MoS2 +and 𝑓(𝑥, 𝑦) is the power density dissipated by means of Joule heating. We implemented the +Dirichlet boundary conditions at 𝑥 = 0 and 𝑥𝑚𝑎𝑥, and Neumann and adiabatic boundary +conditions for the electric field and temperature at 𝑦 = 0 and 𝑦𝑚𝑎𝑥. +Concerning the activation energy for vacancy migration, the electric field may promote the +movement in one direction while hindering the opposite as follow: 𝐸𝐴 = 𝐸𝐴 +0 − 𝑏 · 𝑭(𝑥, 𝑦). [10] +Here, the electric field 𝑭(𝑥, 𝑦) is defined for every grid point (𝑥, 𝑦), 𝐸𝐴 +0 = 2.297 eV is the + + +5 +activation energy without an electric field [11] and 𝒃 = 𝒑𝟎 · [(2 + 𝜀)/3] the polarization factor. We +have used the MoS2 molecular dipole moment 𝑝0 = 15 e nm. [12] +The time step 𝑡 = − +ln(𝑟𝑎𝑛𝑑) +∑ 𝛤 + for the simulation is inversely proportional to the transition rate, +𝑟𝑎𝑛𝑑 stands for a random number between 0 and 1 and ∑ 𝛤 is the summation of all the transition +rates of migrations, which each one depends on the local electric field and temperature. [13-15] It +should be noted that the smaller the activation energy, the smaller the time step. Besides, the +probability of a transition event that takes place in 𝑡 time is 𝑃 = 1 − exp (−𝛤 · t). +Regarding the generation of random numbers, there is a technical issue when running several +simulations with the same parameters. If it is used the same seed to generate pseudorandom +numbers, the sequence of random numbers for each MATLAB session will be the same. In this +case, you can program a queue with many simulations (one simulation after the other), which the +risk of filling the java heap memory and receiving a java heap memory error. A different way to +deal with this issue is to use the command rng('shuffle', 'twister') to reseed the random number +generator based on the current time. This command will enable variations between runs since it +will have different pseudorandom numbers, whether you are using the same MATLAB session or +not. + + + + +6 +SECTION 4: Dependence of resistance with local density + + + +Figure S2: Shows the resistance as a function of the local defect density used during the +simulation. +The relationship between the defect concentration and the MoS2 monolayer resistance has been +studied previously. [3] Following this dependence, we propose a model 𝑅(𝑟⃗) ∝ 𝜌𝑑 +𝑛 (𝑟⃗), where the +resistance is in the range found for the MoS2 monolayer (see Figure S2). However, this has been +chosen as an example, as different materials may have different resistance values and the simulator +can be adapted. + + + +1010 +109 +(U) +108 +Resistance ( +107 +106 +105 +104 +103 +102 +0 +6 +Density (Vs/nm²) +7 +SECTION 5: Colour maps of defect density, resistance and electric field distributions in +the channel +The simulation allows us to disclose the switching mechanism via enabling detailed knowledge +of the vacancy migration process. Figure S3.I, II, III and IV (2D color maps for defect density, +resistance and electric field in this order) correspond with the states labeled by I), II), III) and IV) +on the loop in Figure 2a. The SET process takes place from I to II, and for the RESET from III to +IV. Regarding the density in Figure S3.I, we can observe a peak density (5.64 Vs nm−2 density +region in intense yellow) at the abrupt edge of the fissure region 𝑥 = 22 nm. That peak density is +the main contribution to the channel resistance reaching 1700 MΩ (see the intense yellow region +in the resistance figure in Figure S3.I). Figure S3.II shows the device ON state when most of the +defect migrations have taken place, lowering the peak density to a plateau of the distribution across +a 10-nm wide region with a mean density below 3.5 Vs nm−2. The resistance drops from 1700 MΩ +to 460 MΩ in S3.II and lead to an 80% decrease in the electric field in the same region due to the +screening effect (see electric field figures). It is worth noting that between Figure S3.II and S3.III +almost no defect migrates, as the electric field is not high enough, so the state keeps unchanged. +However, between S3.III and S3.IV, migrations regarding the RESET process take place. The +defects move towards and accumulate at the abrupt edge of the fissure region at 𝑥 = 22 nm, which +raises the resistance and drives the device back to the high resistance state (HRS). Compared with +state I, the peak density in state IV (the yellow region in density figures) is significantly lower, +leading to a 75% lower resistance in the new HRS. This leads to the same asymmetry in the RS +loop found experimentally. [4] Besides, this lower resistance in state IV results also in a lower +electric field. + + +8 + +Figure S3: Color maps of defect density (left figures), resistance (figures in the middle) and +electric field (right figures) distributions for the states labelled by I), II), III) and IV) in Figure 2a. +The SET process takes place from I to II, and for the RESET from III to IV. + + + +D +250 +1500 +10 +0 +10 +200 +Electric field (MV/m) +4 +Resistance (MQ) +20 +1000 +150 +3 +2 +100 +500 +40 +1 +40 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +X axis (nm) +Xaxis (nm) +Xaxis (nm) +II) +250 +1500 +10 +10 +10 +200 +Electric field (MV/m) +4 +Resistance (MQ) +20 +1000 +150 +2 +30 +100 +500 +40 +40 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +Xaxis (nm) +Xaxis (nm) +Xaxis (nm) +II) +1500 +10 +10 +10 +Yaxis (nm) +Resistance (MQ) +Electric field (MV/m) +Yaxis (nm) +20 +1000 +20 +-100 +30 +30 +-150 +500 +40 +1 +40 +40 +200 +0 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +10 +20 +30 +40 +50 +X axis (nm) +X axis (nm) +IV) +Xaxis(nm) +1500 +5 +10 +Electric field (MV/m) +10 +-50 +10 +Electric field (MV/m) +(wu) +4 +Yaxis (nm) +2 +20 +20 +1000 +-100 +Yaxis( +3 +2 +30 +30 +30 +-150 +500 +40 +40 +40 +-200 +0 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +X axis (nm) +X axis (nm) +Xaxis (nm) +9 +SECTION 6: Stability of initial distribution of defects under high RESET electric field + +Figure S4: Device resistance stability for the initial state under different constant applied voltages +(-20 V, -25 V, -30 V, -35V and -40 V) for 𝜌1 = 5.64 𝑉𝑠 𝑛𝑚−2 (a) and 𝜌3 = 4.52 𝑉𝑠 𝑛𝑚−2 (b). The +simulation starts with a skewed Gaussian distribution of defects with fissure region width of 12 nm. +The evolution of the defect density profile along the x-axis for the case of 𝜌1 = 5.64 𝑉𝑠 𝑛𝑚−2 +shown in a) for -20 V c) and -40 V d). + + + +b) +55 +20 +-20 V +id +110 +50 +-25 V +Resistance (M2) +(MQ) +-30 V +100 +45 +-35 V +Resistance ( +-40 V +90 +-20 V +40 +80 +-25 V +p1 +-30 V +35 +70 +-35 V +-40 V +60 +30 +5 +10 +15 +20 +5 +10 +15 +20 +Time (s) +Time (s) +c) +d +6 +6 +t=1s +t=1s +t = 24 s +5 +t = 24 s +5 +-40 V +-20 V +4 +3 +3 +2 +2 +0 +15 +20 +25 +30 +35 +15 +20 +25 +30 +35 +X axis (nm) +X axis (nm) +10 +SECTION 7: The net diffusion of defects with cycles + + +Figure S5: The data was extracted from the same simulation used in Figure 3: The initial defect +profile of the device exhibits a skewed Gaussian distribution of defects with a peak density of +5.64 Vs nm−2, a width of 9 nm and a voltage ramps for a positive polarity of 2.1 V s-1 in the 25.2 +V to -25.2 V range for 45 cycles. a) represents the percentage of density sections with a density +over 3.5 Vs nm−2 among all non-zero density sections in the simulation domain for different +cycles. b) evolution of electric field distribution along cycles c) the dependence between the +activation energy for defect migration and the electric field for different local defect densities. + + + +b) +(e +c) +(eV) +32 +p1 +Defect migration ( +-20 +lectric field (MV/m) +Cvcle +2 +-40 +28 +-60 +Cycle 45 +1.6 +5.64 Vs/nm2 +3.5 Vs/nm² +-80 +24 +Consecutive RESETs +1.2 +3.2 Vs/nm² +energy: +-100 +3 Vs/nm² +-120 +0.8 +20 +Act. +10 +20 +30 +40 +18 +20 +22 +24 +26 +28 +30 +106 +107 +108 +Cycles +X axis (nm) +Electric field (V/m) +11 +SECTION 8: Degradation process and device power consumption + + +Figure S6: 45 resistive switching I-V curves of the simulation data used in Figure 3. The +simulation starts from a skewed Gaussian distribution of defects with polarity +𝑉𝑑 and a width of +9 nm for 𝜌1 = 5.64 Vs nm−2. The ramp voltage rate was 2.1 V s-1 in the 25.2 V to -25.2 V range. + +Figure S7: Energy consumed to SET (black squares) and RESET (red dots) the device at each +cycle for the case 𝜌1 = 5.64 𝑉𝑠 𝑛𝑚−2, a width of 9 nm and −𝑉𝑑 showed in Figure 3. The ramp +voltage rate was 2.1 V s-1 in the 25.2 V to -25.2 V range for 45 cycles. The energy consumption is + +Current (μA) +10-1 +10-2 +-20 +-10 +0 +10 +20 +Voltage (V)Energy consumption (μWs) +55 +4535 +SET +RESET +20 +0 +10 +20 +30 +40 +Cycles +12 +calculated as 𝐼 × 𝑉 × 𝑡 along the sweep that leads the device to the ON state (SET) and OFF state +(RESET). + + + + + +13 +SECTION 9: Polarity effect in Gaussian distribution of defects + + +Figure S8. a) and b) simulations start from a skewed Gaussian distribution of defects and consist +of 15 RS cycles under a ramp voltage rate of 2.1 𝑉 𝑠−1 between the 25.2 V to -25.2 V range using +the following peak densities: 𝜌1 = 5.64 𝑉𝑠 𝑛𝑚−2, 𝜌2 = 5.08 𝑉𝑠 𝑛𝑚−2, 𝜌3 = 4.52 𝑉𝑠 𝑛𝑚−2 and +𝜌4 = 3.95 𝑉𝑠 𝑛𝑚−2. a) shows the resistance ratio over the fissure region width for different +densities and polarity −𝑉𝑑. b) shows the resistance ratio over the fissure region width for different +densities and polarity +𝑉𝑑. The line between points is for ease of reading. We average the +resistance ratio over the 15 cycles and use the standard deviation of the cycle-to-cycle variability +as the error. + +a) +1.5 +P4 +-Vd +id +Roff/Ron ratio +1.4 +P2 +1.3 +1.2 +1.1 +1.0 +5 +6 +7 +8 +12 +Width (nm) +b) +1.5 +p4 ++Vd +1.4 +td +Roff/Ron ratio +1.3 +P2 +p +1.2 +1.1 +1.0 +0.9 +5 +6 +7 +8 +9 +10 +11 +12 +Width (nm) +14 +In Figure S8, we have assessed the effect of the asymmetry distribution of defects of the skewed +Gaussian distribution using both polarities: −𝑉𝑑 in Figure S8a and +𝑉𝑑 in Figure S8b. For these +15-cycles simulations, each point is the average of the cycles, while the error bar is the standard +deviation of cycle-to-cycle variability. The ramp voltage rate used is 2.1 V s−1 for the 25.2 V to - +25.2 V range. For every 𝜌 value (𝜌1 = 5.64 Vs nm−2, 𝜌2 = 5.08 Vs nm−2, 𝜌3 = 4.52 Vs nm−2 +and 𝜌4 = 3.95 Vs nm−2), we cover different fissure region width between 5 and 12 nm. It must be +noted the temperature gains importance as the density (and so the resistance) is reduced. But in +general, the influence of the temperature is limited when 𝜌 > 4.52 Vs nm−2 and when the fissure +region grows (see the maximum temperatures for three densities in Figure S9). Both polarities +(Figure S8a and S8b) show a higher resistance ratio for higher defect density values (𝜌) and larger +fissure region widths. The reason is that a higher population of defects (high-density regions over +3.5 Vs nm−2) enables a larger resistance ratio since it lets larger differences between the ON and +OFF states. This dependence is the reason for the similar trends in both polarities. + + +Figure S9: Three 15-cycles simulations starting from a skewed Gaussian distribution of defects +with different densities (𝜌1 = 5.64 Vs nm−2, 𝜌2 = 5.08 Vs nm−2 and 𝜌3 = 4.52 Vs nm−2), a + +340 +p, and w = 9 nm +Temperature max (k) +P2 and w = 9 nm +330 +P3 and w = 9 nm +320 +310 +300 +0 +100 200 300 400 500 600 700 +Time (s) +15 +width of 9 nm and polarity +Vd. The ramp voltage rate was 2.1 V s-1 in the 25.2 V to -25.2 V +range. The plot shows the maximum temperature at every time step for 721 s. + + + + + + +16 +References +[1] +M. O'Brien, N. McEvoy, T. Hallam, H. Y. Kim, N. C. Berner, D. Hanlon, K. H. Lee, J. +N. Coleman, G. S. Duesberg, Sci Rep 2014, 4, 7, 7374. +[2] +S. Kretschmer, M. Maslov, S. Ghaderzadeh, M. Ghorbani-Asl, G. Hlawacek, A. V. +Krasheninnikov, ACS Appl. Mater. Interfaces 2018, 10, 30827. +[3] +D. S. Fox, Y. B. Zhou, P. Maguire, A. O'Neill, C. O'Coileain, R. Gatensby, A. M. +Glushenkov, T. Tao, G. S. Duesberg, I. V. Shvets, M. Abid, M. Abid, H. C. Wu, Y. Chen, J. N. +Coleman, J. F. Donegan, H. Z. Zhang, Nano Lett. 2015, 15, 5307. +[4] +J. Jadwiszczak, D. Keane, P. Maguire, C. P. Cullen, Y. B. Zhou, H. D. Song, C. +Downing, D. Fox, N. McEvoy, R. Zhu, J. Xu, G. S. Duesberg, Z. M. Liao, J. J. Boland, H. Z. +Zhang, ACS Nano 2019, 13, 14262. +[5] +P. Brady, MATLAB Central File Exchange 2022. +[6] +M. Mortazavi, C. Wang, J. K. Deng, V. B. Shenoy, N. V. Medhekar, J. Power Sources +2014, 268, 279. +[7] +D. Davelou, G. Kopidakis, G. Kioseoglou, I. N. Remediakis, Solid State Commun. 2014, +192, 42. +[8] +R. S. Yan, J. R. Simpson, S. Bertolazzi, J. Brivio, M. Watson, X. F. Wu, A. Kis, T. F. +Luo, A. R. H. Walker, H. G. Xing, ACS Nano 2014, 8, 986. +[9] +R. J. Dolleman, D. Lloyd, M. Lee, J. S. Bunch, H. S. J. van der Zant, P. G. Steeneken, +Phys. Rev. Mater. 2018, 2, 8, 114008. +[10] +A. Padovani, L. Larcher, O. Pirrotta, L. Vandelli, G. Bersuker, IEEE Trans. Electron +Devices 2015, 62, 1998. +[11] +L. Wang, W. G. Liao, E. E. H. Wong, Z. G. Yu, S. F. Li, Y. E. F. Lim, X. W. Feng, E. E. +C. Tan, X. Huang, L. Chen, L. Liu, J. S. Chen, X. Gong, C. X. Zhu, X. K. Liu, Y. W. Zhang, D. +Z. Chi, K. W. Ang, Adv. Funct. Mater. 2019, 29, 10, 1901106. +[12] +J. Wierzbowski, J. Klein, M. Kaniber, K. Müller, and J. J. Finley. +[13] +S. Aldana, P. Garcia-Fernandez, R. Romero-Zaliz, M. B. Gonzalez, F. Jimenez-Molinos, +F. Gomez-Campos, F. Campabadal, J. B. Roldan, J. Phys. D-Appl. Phys. 2020, 53, 11, 225106. +[14] +J. Guy, G. Molas, P. Blaise, M. Bernard, A. Roule, G. Le Carval, V. Delaye, A. Toffoli, +G. Ghibaudo, F. Clermidy, B. De Salvo, L. Perniola, IEEE Trans. Electron Devices 2015, 62, +3482. +[15] +A. F. Voter, presented at Conference of the NATO-Advanced-Study-Institute on Radiation +Effects in Solids, Erice, ITALY, Jul 17-29, 2004. + + diff --git a/yNE4T4oBgHgl3EQfxw02/content/tmp_files/load_file.txt b/yNE4T4oBgHgl3EQfxw02/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60d3c0fd25d2f9ab30533eac206173bc8649fa61 --- /dev/null +++ b/yNE4T4oBgHgl3EQfxw02/content/tmp_files/load_file.txt @@ -0,0 +1,2053 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf,len=2052 +page_content='On the switching mechanism and optimisation of ion irradiation enabled 2D MoS2 memristors Samuel Aldana,*a Jakub Jadwiszczak a and Hongzhou Zhang a Memristors are prominent passive circuit elements with promising futures for energy-efficient in-memory processing and revolutionary neuromorphic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' State-of-the-art memristors based on two-dimensional (2D) materials exhibit enhanced tunability, scalability and electrical reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' However, the fundamental of the switching is yet to be clarified before they can meet industrial standards in terms of endurance, variability, resistance ratio, and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This new physical simulator based on the kinetic Monte Carlo (kMC) algorithm reproduces the defect migration process in 2D materials and sheds light on the operation of 2D memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The present work employs the simulator to study a two-dimensional 2H-MoS2 planar resistive switching (RS) device with an asymmetric defect concentration introduced by ion irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=" The simulations unveil the non-filamentary RS process and propose practical routes to optimize the device's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For instance, the resistance ratio can be increased by 53% by controlling the concentration and distribution of defects, while the variability can be reduced by 55% by increasing 5-fold the device size from 10 to 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Our simulator also explains the trade-offs between the resistance ratio and variability, resistance ratio and scalability, and variability and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Overall, the simulator may enable an understanding and optimization of devices to expedite cutting-edge applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Introduction The development of Information and Communication Technology (ICT), such as 5G circuits and the internet of things, demands breakthroughs in non-volatile memory1 since the state-of-the-art flash technology is hitting its physical limits in terms of power consumption and device scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 2 Among the emerging memory technologies, memristors have attracted much research interest and become the herald of next- generation computational architecture, revolutionizing ICT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 3-6 The potential of memristors is rooted in their superior performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', ultrafast switching, low power consumption, data retention and endurance), intrinsically high scalability, stackability, compatibility with Complementary Metal Oxide Semiconductor (CMOS) technology and flexibility for wearable applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 1, 2, 7 However, device variability, including cell-to- cell variability and cycle-to-cycle variability, hinders the industrial deployment of memristor technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The cell-to-cell variability is the inhomogeneity between devices via the same fabrication process and cycle-to-cycle variability is related to the operation of individual devices, 1, 7 which emerges from the stochastic processes during the resistive switching (RS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, in filamentary RS devices the location and morphology of conducting filaments may vary between cycles and cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 8, 9 Device variability is inevitable in RS devices which relies on material defects10-13, while the forming process exacerbates the problem by varying defect concentration and distribution14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Although the cycle-to-cycle variability can be insignificant in some applications exemplified by neuromorphic imaging recognition, 1 it imposes major limitations on memristor applications, rendering a range of high-end applications impractical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', multilevel information processing and long- term storage7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, device variability causes signal degradation in crossbar arrays15 and hampers projections of device lifetime, demanding excessive budget in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 16 Verification and iterative approaches may mitigate the resistance variability in multilevel information17 and radiofrequency applications respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 18 However, a consensus on a figure of merit for the variability issue is yet to be achieved and a lack of comparable statistics on device variability remains a main obstacle to the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 7 Recently, memristive behaviour has been observed in a range of two-dimensional (2D) materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 19-25 This may expedite the industry deployment of the memristor technology since 2D memristors exbibit superior tunability, scalability and electrical reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 4, 19, 26-34 These characteristics arise from the ultrathin nature and unique mechanical, electric and optoelectronic properties of 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 26, 27 The 2D memristor landscape shows diverse device architectures and switching mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, the switching of 2D vertical memristors20, 35-38 depends on the formation and rupture of conductive filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 2D planar memristors may rely on phase transition, 39, 40 charge trapping/de-trapping, 41, 42 electron tunnelling modulated by polarization, 43 electrochemical processes44 and Schottky barrier modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 29 Defect migration plays a crucial role in these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 32, 45-47 Compared with their 3D counterparts, the device variability of 2D memristors has rarely been explored, while it is of utmost importance to gain in-depth knowledge of the switching process and mitigate the variability issues in 2D memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Physical simulators are indispensable for understanding the resistive switching process and they can greatly facilitate investigations on device viability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 7, 48, 49 Ab initio calculations can accurately relate the resistive switching to the defect creation, 50 the electronic structure and transport properties in nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 51 However, they are limited to relatively small volumes (a few nm as maximum) and short times (shorter than ns) and can hardly reproduce RS processes at the device level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 48 Continuous models are apt to describe the average behaviours of individual devices or even circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' However, they overlook the microscopic characteristics of the system, such as particle migration52-55, and are hence not suitable to investigate the stochasticity that emerges from the evolution of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Advanced Materials and Bioengineering Research (AMBER) Research Centers, School of Physics, Trinity College Dublin, Dublin, D02 PN40, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' E-mail: aldanads@tcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='ie Electronic Supplementary Information (ESI) available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' microscopic configuration of the system during the switching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The kinetic Monte Carlo (kMC) algorithm is an established technique to study the microscopic evolution and its impacts on device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 11, 12, 56-59 For example, the kMC algorithm can reproduce particle diffusion and the formation and rupture of percolation paths involving several RS cycles of memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 11, 56-58 Although the kMC algorithm and continuous models have been employed to explore defect accumulation and electrical conduction in 2D materials, 29, 38 the simulation of 2D memristors at the device level is scarce and the variability of 2D memristors has been rarely explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In this work, we build a new physical kMC simulator for the defect migration process in 2D materials and shed light on the operation of 2D memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We investigate the effects of initial vacancy distribution, the scaling limits and the factors that regulate device endurance and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We use experimental data from the MoS2 memristor enabled by site-specific defects to collate and verify the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 19 The simulator helps understand the physics of resistive switching in 2D materials, offering practical guidance to optimize device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Experimental results and simulation approach Figure 1a shows a device schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The device exhibits a planar metal-semiconductor-metal structure and the semiconductor channel is a 2D 2H-MoS2 with an asymmetric defect concentration introduced by focused helium ion irradiation (see more details about the device fabrication in Supplementary Information, section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We note that the irradiation-induced defects enable the resistive switching, while devices of pristine MoS2 do not switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 19 The defect region, referred to as the fissure, bisects the channel with an asymmetric concentration along the horizontal direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', across the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, the right tail of the peak (tens of nanometres) in Figure 1a is much wider than the left (< 1 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In our simulation, we have assessed three initial defect distributions with different asymmetries (see Supplementary Information, section 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', a skewed Gaussian distribution (Figure S1a), a triangle function (Figure S1b) and a step function (Figure S1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The skewed Gaussian distribution emulates the asymmetric distribution of defects observed experimentally19, and the triangle and step functions of different asymmetries are to explore possible routes for device optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The source electrode as convention is grounded and placed at the left side of the asymmetric peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The polarity of applied voltage regulates the direction of the fields with respect to the asymmetric defect distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', a positive voltage (𝑉𝑑 > 0 ) indicates the electric field points from the abrupt edge of the fissure to the long tail side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 1b shows 14 experimental quasi-static pinched hysteresis loops by selecting every 15th loop from 1200 consecutive cycles measured at a triangular voltage ramp between ± 35 V with a rate of 6 V s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The device shows bipolar operation, and the SET (RESET) process occurs at a positive (negative) voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The current level increases with the cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 1c shows the resistance ratio over cycle, which shows that cycling the device leads to an exponential decay (shown as a dashed line) in resistance ratio (21% after 1000 cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' These degradation processes will be further discussed alongside the simulation work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' To investigate the device switching process, we implemented the kinetic Monte Carlo (kMC) algorithm60 using MATLAB in a simulation domain of 50 × 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The vacancy distribution within the fissure evolves with the external electric field applied via the electrodes, 19 while no defects escape the simulation domain during the switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Since the fissure region dominates the device resistance, the size of the simulation domain is sufficient to include the main physical processes involved in the resistive switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defects are doubly charged sulfur vacancies because the helium-ion irradiation preferentially removes sulfur atoms from the MoS2 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 33 The generation of new defects under the applied electric field is negligible due to the high activation energy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='85 eV 61 for vacancy and > 5 eV for antisite defect62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The sulfur vacancy migrates via the exchange of the vacancy position with one of the adjacent sulfur atoms, 29, 63 as shown in the Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In the quasi-static switching, the vacancy migration events occur at a much longer time scale (~ s) than the lattice vibration (10−13 s), so the system is in thermodynamic equilibrium for any vacancy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 60 Furthermore, the field-driven migration renders the reverse migration negligible, validating the kMC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 1: Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) Schematic device structure with the defect distribution in the MoS2 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The 50 × 50 nm simulation domain represents the defect profile of a skewed Gaussian distribution, while the channel is on the micrometre scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The arrows in the lattice structure indicate the possible routes of vacancy migration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', a S atom (yellow) exchanges its position with a nearest-neighbour vacancy site (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) 14 representative experimental I-V hysteresis loops sampled from continuous 1200 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' c) The evolution of the resistance ratio (calculated at -10 V) over cycles (time in the top x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The dashed line is an exponential fitting, and the shaded indicates the 95% prediction interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a b 101 40 S vacancy Mo 4 100 20 2 Cycle 1 10 103 Cycle 5 1 Cycle 10 10-4 Cycle 15 20 40 30 20 10 0 10 20 30 x[nm] Voltage [V] c 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 Time (s] 0 5000 10000 15000 20000 Vd 25000 Fissure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 V,=0 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Drain Source MoS2 SiO2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 0 200 400 600 800 1000 1200 Cycles In our simulation, we combine the electric response with the thermal effect since the lattice temperature modulates the transition rate, which is given by Maxwell-Boltzmann statistics and Transition State Theory (TST), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 𝛤 = 𝜈 · exp (−𝐸𝐴/𝐾𝐵𝑇), where 𝜈 = 7 × 1013 s−1 is the vibration constant of the particle, 𝐾𝐵 the Boltzmann constant, 𝑇 the temperature and 𝐸𝐴 the activation energy of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The activation energy is modulated by the local electric field as follows: 𝐸𝐴 = 𝐸𝐴 0 − 𝒃 · 𝑭(𝑥, 𝑦), 12 where 𝑭(𝑥, 𝑦) is the electric field, 𝒃 the polarization factor and 𝐸𝐴 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='297 eV the activation energy for migration in the zero-field condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 4 The kMC algorithm weighs all the possible migrations and chooses the evolvement path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' It should be noted that the larger the transition rate, the smaller the time step 𝑡 = − ln(𝑟𝑎𝑛𝑑) / ∑ 𝛤, where 𝑟𝑎𝑛𝑑 stands for a random number between 0 and 1 and ∑ 𝛤 is the summation of the transition rates for all possible migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For each defect distribution during the switching, the local vacancy concentration 𝜌𝑑 (𝒓⃗⃗ ) is averaged over 6 × 6 grid points (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 nm2), which determines the local resistance 𝑅(𝒓⃗⃗ ) via the empirical relationship 𝑅(𝒓⃗⃗ ) ∝ 𝜌𝑑 𝑛 (𝒓⃗⃗ ), 33 (𝑛 is a parameter extracted from the experimental results, see Supplementary Information, section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The electric field screening is assumed to be a linear function of the local resistance since the dielectric constant in MoS2 strongly depends on the distribution and number of sulfur vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 64 For a given applied voltage, the local electric field decreases with the increase in the defect Figure 2: Switching mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial defect profile of the device exhibits a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 8 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The voltage ramp starts from positive polarity with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='71 V s−1 between 35 V and −35 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) a typical simulated I-V pinched hysteresis loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Numerals on the loop mark four representative states of the switching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The SET process takes place from I to II, and for the RESET from III to IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The colour maps shown in the right panel correspond to the microscopic configuration of defects in the LRS (top) and HRS (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b), c) and d) are the defect density, local resistance and electric field profiles along the x-axis during the SET, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' e), f) and g) are the correspondent profiles during the RESET processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a State Il:LRS 40 III 100 RESET 2 口 [μA] SET IV 10 2030 40 50 x[nm] State I: HRS 40 [nm] 2 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 30 20 10 0 10 20 30 10 2030 40 50 Voltage [V] x [nm] b c d 7 300 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V start 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V start 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V start 6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V 250 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V m-11 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V Resistance [MQ] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 200 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback Defect density I 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 4 3 1 2 100 2 1 50- 0 0 0 20 25 30 35 40 20 25 30 35 40 20 25 30 35 40 x [nm] x [nm] x [nm] e f 6 7 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vstart 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V start 8 6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 v 50十 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V IV 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V 100 t density [Vs 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6Vback 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6back [MV r 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 Vback Resistance 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 Vback field IV 150 II 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6V start Electric Defect IV 200 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V 2 2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 V 250 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 V III 1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6vback 0 II IV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6back 300- 0 20 25 30 35 40 20 25 30 35 40 20 25 30 35 40 x[nm] x[nm] x [nm] density, indicating the defect migration is a self-limiting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Further details about the simulation can be found in the Supplementary Information, section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Simulation results and discussions Figure 2a shows a typical simulated pinched hysteresis loop from a defect distribution of skewed Gaussian profile under a voltage ramp rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='71 V s−1 between 35 V and -35 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Prior to the switching, the fissure region is 8 nm wide with a peak density 𝜌 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 (see the probability distributions in Supplementary Information, section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The I-V sweep follows the directions indicated by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The loop shows a SET process with a positive voltage and a RESET process with a negative voltage, suggesting the same bipolar switching behaviour observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulation corroborates that the operation of the device does not need a forming process, facilitating circuit simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 65-67 The switching is progressive, in contrast to an abrupt resistance change, suggesting the absence of filamentary conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This explains the observed low power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 7 The resistance ratio (𝑟 = 𝑅𝑜𝑓𝑓/𝑅𝑜𝑛) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='44 calculated at -4 V during the RESET process and the maximum current level is 3 μA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' It is interesting to note the simulated loop reproduces the asymmetry found experimentally between the SET and RESET processes (see Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The switching is due to the reconfiguration of the defect distribution within the fissure by the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The two colour-maps attached to Figure 2a show the microscopic distributions of defects in the High Resistance State (HRS) and the Low Resistance State (LRS), corresponding to the state labelled by I) and II) on the loop respectively (more microscopic details about the states I-IV can be found in Supplementary Information, section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defects accumulate within a small region in the HRS, leading to a much higher density than the LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 2b details the defect evolution during the SET process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defect profiles are extracted at a series of sequential voltages from state I to II (see Figure 2a) and the lightness of the curves reduces with increasing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial vacancy concentration exhibits a skewed Gaussian distribution (the most intense red curve), mimicking the defect profile by the ion irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The external field of positive polarity gradually drives the defects away from the fissure region, lowering the peak and leading to a plateau of the distribution across a 10-nm wide region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The local resistance varies with the defect profile (see Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The spatial-varying resistance regulates the electric potential distribution in the channel when an external voltage is applied (see Figure 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The larger the resistance of a region, the larger the electric field, and the more significant the vacancy drifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Therefore, the self-adaptive electric field reduces the vacancy concentration and hence the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' From State I to II, the peak resistance reduces by an order of magnitude, leading to an overall reduction in the channel resistance and hence the SET process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 2e reveals the evolution of the defect profile during the RESET process from state III to IV (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defects move towards and accumulate at the abrupt edge of the fissure (𝑥 = 22 nm in Figure 2e), which increases the local resistance (see Figure 2f) and hence the local electric field (see Figure 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' On the long tail of the peak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', 𝑥 > 22 nm), the field pushes defects from a wide region (22 nm < 𝑥 < 30 nm) towards the peak, while on the left side of the peak the field drops drastically within a 1 nm region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This asymmetric distribution of the field limits the escape of the defects from the fissure into the left side of the channel and causes the defect accumulation at the peak, recovering the initial defect configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The drift of defects to the left side of the fissure may become important if the field is sufficiently high where the RESET process will fail, leading to device failure (see Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulator can predict the operational range of the voltage for the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, a device with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 12 nm can be stressed with -30 V for 17 s or -40 V for 2 s before the device fails (see Figure S4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulator allows us to investigate the device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 3a shows the temporal evolution of the device resistance to a triangular voltage ramp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1) between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 and −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial defect profile has a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The voltage ramp starts from a positive polarity +𝑉𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The device switches over 45 cycles (2161 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Both the 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 drop 57% after the first ten cycles and reaches a steady state where 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 drop slower (17% in 35 cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The resistance drop occurs due to a progressive reduction in the peak density of the defect profile, which dominates the device resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This is evident in Figure 3b and 3c, which show the evolution of the density profile with consecutive RESET and SET processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Here, we can see that cycling gradually relocates the defects into the originally low-density (right tails) regions (below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 Vs nm−2) at the expense of the peak density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This reduces the defects population of the peak region from 31% in cycle 1 to 22% in cycle 45, as can be seen in Figure S5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defect accumulation in the tail region (> 25 nm onward in the x- axis) stems from the low electric field in the region, which is 16% Figure 3: Device fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial defect profile of the device exhibits a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The voltage ramps from a positive polarity with a rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V and −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V for 45 cycles (2161 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) shows the resistance evolution over time with two exponential fittings for the LRS (in red) and the HRS (in blue) of the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The shaded regions correspond to the 95% prediction interval of the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) and c) correspond to the density profile along the x-axis (averaged over the y-axis) for successive RESETs and SETs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' d) resistance ratio projection based on the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a Time (s) b 500 1000 1500 2000 6 350 Consecutive RESETs Cycle 1 Resistance Cycle 15 300 Rorr fitting Cycle 30 [OW] Row fitting Cycle 45 Resi 150 100 0+ 0 10 20 30 40 15 20 25 30 35 40 Cycles d x [nm] U 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 Time (s) Consecutive SETs Cycle 1 102 103 104 Cycle 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='7 Cycle 30 RorF/Row 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 Fitted line Cycle 45 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 15 20 25 30 35 40 101 102 x [nm] Cycles of the electric field found in the peak (see Figure S5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The low field modulates the migration barrier by 1% (see Figure S5c), leading to negligible field-driven migration in the tail region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The further the defects migrate away from the peak into the tail during the SET process, the harder for them to move back to the peak during the RESET process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Consequently, the defect distribution flattens across the fissure during the cycling leading to a gradual reduction in the device resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This flattening of the defect profile progressively reduces the difference between the OFF and ON states, reducing the ON/OFF ratio and eventually leading to device failure (see also the resistive switching loops in Figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' As observed experimentally, the power consumption increases (see the rise from 20 μW to more than 50 μW in 45 cycles in Figure S7) with the resistance reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Nevertheless, the simulation qualitatively explains the fatigue behaviour (see Figure 1c) and allows quantitative prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For devices with long cycling tolerance, we can fit the behaviours of 𝑅𝑜𝑛 and 𝑅𝑜𝑓𝑓 (red and blue dashed lines in Figure 3a respectively) and project the evolvement of ON/OFF ratio for longer times (Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The most significant advantage of the simulator is its capability of exploring device variability since the kMC algorithm is apt to investigate stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 4a shows the cycle-to- cycle variability of the resistance ratio in one 45-cycle simulation, starting with a skewed Gaussian distribution of vacancies, negative polarity, with a peak density 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The value of resistance ratio distributes uniformly in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 with the mean of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We define the cycle-to-cycle variability of two consecutive cycles as ∆r = |𝑟𝑖 − r𝑖+1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The standard deviation of the cycle- to-cycle variability is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In Figure 4b, we investigate the cell- to-cell variability by running 26 independent simulations for a given macroscopic distribution (see section 3 in Supplementary Information for more details about running different simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Each simulation starts with a unique initial microscopic vacancy configuration in the lattice, while the macroscopic defect distribution remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This scenario mimics the device fabrication process, where the sputtering process and the creation of vacancies are stochastic at the nanometre scale, 19, 33 introducing cell-to-cell variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We note the device fabrication is limited by many other parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', ion beam stability, focusing, sample cleanliness, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=') and the cell-to-cell variability here represents the upper limit of the ideal situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For each such simulation, we average the resistance over 15 cycles and use the standard deviation of the cycle-to-cycle variability as the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Here, we can observe a uniform distribution of resistance ratios around the mean of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='31 with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We investigate the potential of device scalability by evaluating the effects of the device dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', height and width) on the device resistance and the resistance ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For the simulations shown in Figure 5, the initial defect profile has a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The device is under a voltage ramp rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 and -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V, which starts from the positive polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 5a shows the temporal evolution of the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' It is evident that the reduction in the vertical dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', device height, increases the overall device resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This corroborates the non-filamentary conduction and the resistance increase is due to the reduction of conducting channels along the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The limit of scaling down the vertical dimension is shown in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Both the resistance ratio and the device variability deteriorate as the device height reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' A trade-off needs to be identified between the device conductance and the resistance ratio (variability), indicating a limit on the scalability of the y- dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The scaling of the horizontal dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', along the x-axis, is regulated by the fissure width and the range of defect drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' As shown in Figure 5c, the device resistance decreases with the fissure width, and the resistance ratio and the device variability also degrade with the scaling of the fissure width (Figure 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulation reveals the scaling limit of the devices, and it can be further extended to achieve practical device design and optimization for a set of predefined Figure 4: Device variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial defect profile of the device exhibits a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 and a width of 9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The voltage ramps from a negative polarity with a rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V and - 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) shows simulated cycle-to-cycle variability of the resistance ratio with an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='29 (the dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) corresponds to the resistance ratio variability of 26 independent 15-cycle simulations initiated with the same parameters but varying microscopic defect configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For each simulation, the resistance ratio is the average over the 15 cycles and the error bar is the standard deviation of the cycle-to- cycle variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure 5: Device scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The initial defect profile of the device exhibits a skewed Gaussian distribution with a peak density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The voltage ramps from a negative polarity with a rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V and -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V for 15 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The resistance ratio is averaged over the 15 cycles and the error is the standard deviation of the cycle-to-cycle variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) and b) show the effect of the device height (y-axis) with a 12 nm wide fissure on the resistance ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' c) and d) reveals the effect of the fissure width (x-axis) with a fixed device height of 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 0 10 20 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 0 5 10 15 20 25 Cycles Simulationsa b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='450 2500 50 nm 40 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='425 WU OE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='400 20 nm 10 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='375 1500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='350 istan 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='300 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='250 0 100 200 300400 500 600 700 10 20 30 40 50 Time [s] p Height [nm] c 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='45 12 nm 11 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 400 9 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 [UW] 7 nm 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 5 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='05 0 100 200 300 400 500 009 700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='00 4 6 8 10 12 Time [s] Width [nm] performance metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', resistance level, resistance ratio and variability level, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The asymmetric nature of the initial defect distribution is crucial to the resistive switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This indicates that device optimization may be possible by tuning the initial defect distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In Figure 6 we show the resistance ratio for an initial defect distribution with a triangle shape (Figure 6a) and a step function shape (Figure 6b) with varying peak densities and fissure region widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The skewed Gaussian distribution case is in Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In the triangle case (Figure 6a), when the peak density is higher than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 Vs nm−2, the resistance ratio appears to exhibit a maximum when the fissure width is varied from 4 nm to 32 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The value of the maximum ratio decreases and occurs at a larger fissure width when the peak density decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Devices with the step function distribution (Figure 6b) exhibits a similar maximum ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In contrast to the triangle case, the maximum appears in all the peak densities simulated and on the right side of the peak the resistance ratio falls more rapidly with increase in the fissure width compared with the triangle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' However, the triangle distribution enables a higher resistance ratio at a narrower fissure than both the Gaussian and step function cases, so it may offer a better option for device scaling and further performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We also note that in all the distributions simulated, the resistance ratio and variability can be enhanced by increasing the initial peak density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This is because a larger defect population enables more significant differences between the HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Finally, we note that although the voltage is high in this MoS2- based device, the current is low, so the device energy consumption is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Besides, as the voltage drops mainly in the fissure region, we cannot address the scaling of the switching voltage by reducing the length between the drain and the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In this sense, the minimum electric field needed to move the vacancies determine the voltage scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Hence, it might be possible to lower the operating voltage and further reduce the power consumption by increasing the density peaks (see in Figure S4 how the electric field has a stronger influence in higher densities), selecting materials with suitable activation energies63 or by defect engineering (defects migrate easier through grain boundaries4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Conclusions We have developed a kMC simulator for 2D planar memristors based on defect migration using the case of a MoS2-based device enabled by a Helium Ion Beam Microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulator reproduces the asymmetric resistive switching cycle with a performance close to observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Besides, this approach is helpful for insights into the switching mechanism, in addition to study the device variability, the device endurance and the resistance ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We studied the device degradation with cycling, which produces defect relocations into low-density regions, causing a drop in the resistance ratio and reducing the switching window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Some trade-offs are proposed to tailor the device features by controlling the device size, the number of defects introduced in the channel and their distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For example, the device can be miniaturized at the expense of reducing the resistance ratio, increasing the variability and the resistance, or higher peak densities can be used to increase the resistance ratio and variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We note that to reduce the switching voltage of a device based on defect migration, we must enhance the migration of defects by employing defect engineering, using higher peak densities or other Transition Metal Dichalcogenides with lower activation energy for defect migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Besides, the conduction mechanism during the ON state is a distributed one, which explains the drop of resistance when the device size in the 𝑦-axis is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We have also assessed different distributions of defect density in the channel to find some routes for device optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Comparing the skewed Gaussian distribution, the step function and the triangle distribution, we can conclude the triangle shape enables larger resistance ratios for narrower fissure regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Besides, the fissure width of these distributions also affects the resistance ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Author Contributions J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' fabricated the devices and collected the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' developed the simulator and conducted the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' analysed the data and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' conceived the study and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' All authors have given approval to the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Acknowledgements The authors gratefully acknowledge financial support by the Science Foundation Ireland under 20/FFP-P/8727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' References Figure 6: Device optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The dependence of the resistance ratio on the initial peak density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 Vs nm−2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs nm−2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 Vs nm−2) and device width for two density profiles: (a) a triangle and (b) step distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The devices are stressed under consecutive voltage ramps with a rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 between 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The resistance ratio is averaged over 15 cycles and the error is the standard deviation of the cycle-to-cycle variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' e b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 Triangle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 Vs·nm=2 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 Vs - nm2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs- nm2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs - nm=2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 Vs · nm2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 V, · nm2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 V, ·nm=2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs - nm*2 JRoN /RoN 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 5 10 15 Width [m] 20 25 30 Width [m] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Kapur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Saurabh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Grover, IETE Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', 2020, 37, 377-390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Martin-Martinez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Gonzalez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Jimenez-Molinos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Campabadal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Nafria and J.' metadata={'source': 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Gonzalez-Cordero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Cartujo- Cassinello, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Roldan and Ieee, Barcelona, SPAIN, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Gonzalez-Cordero, M.' metadata={'source': 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L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Dellmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Sebastian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Jonnalagadda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Santini, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Koelmans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Rossel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Eleftheriou and Ieee, Bucharest, ROMANIA, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Kreupl, Carbon Memory Assessment, white paper for the ITRS meeting on emerging research devices (ERD), August 25–26, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 1 Supplementary Information Supplementary Information: On the switching mechanism and optimisation of ion irradiation enabled 2D MoS2 memristors Samuel Aldana*, Jakub Jadwiszczak, Hongzhou Zhang 2 SECTION 1: Device description The large crystal triangles of monolayer MoS2 used for the device are grown using a close- proximity CVD method and sulfurization of MoO3 on marked 285 nm SiO2/Si chips inside a CVD furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [1] A unidirectional single-pixel-wide line of He with a scan dose of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 pC μm−1 enables the resistive switching behavior in MoS2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This scan bisects the middle of the channel with a damaged region parallel to the source and drain electrodes (see the device schematic and the irradiation strategy in Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The exposition of MoS2 to this 30 keV ion beam energy introduces sulfur vacancies in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [2-3] The fissure region extension is usually over 10 nm and shows an asymmetric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [4] This fissure region is the current-limiting resistor and is responsible for the resistive switching behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In this sense, the resistance of the device is independent of the channel length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [4] The defects concentration is what determine the resistance in the channel [3] and so the proposed mechanism to explain the resistive switching process is the field-driven migration of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [4] 3 SECTION 2: Probability distributions used to initialize the defect distribution in the channel Figure S1 show three probability distribution used in the device x-axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=', across the electrodes, see the axes in Figure 1a) employed for the initialization of defect distribution in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In the Figure, ℎ stands for the probability of finding a defect in the grid points and width (equivalent to two standard deviations 𝜎 in the skewed Gaussian case and the base of the triangle and the step function in the other two cases) stands for the extension of the fissure region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S1a corresponds to the skewed Gaussian distribution case and we use the Pearson distribution [5] with 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 and 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 to calculate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S1b stands for a right triangle and Figure S1c for a step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S1: Probability distributions where ℎ stands for the probability of finding a defect in the grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) The skewed Gaussian function where width correspond to two standard deviations 𝜎 and ℎ𝑚𝑎𝑥 the maximum probability point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) Triangle function with base width and height ℎ𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' c) Step function with base width and height ℎ𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Note that h is constant in y-axis for each position in x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 Mean hmax max 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 0 0 0 Width Width Width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 X axis (nm) X axis (nm) X axis (nm) 4 SECTION 3: Simulator details Regarding the hexagonal crystal structure in Figure 1a, the lattice constants are: 𝑎 = 𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='639 nm and 𝛾 = 60°, typical values for 2H-MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [6] We discretize the simulation domain using a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='277 nm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='32 nm for 𝑥 and 𝑦 axis respectively for a grid of 50 × 50 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This discretization is the basis for the defect migration processes and to solve the heat and Poisson equation using the finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The Poisson equation has the following form: ∇2𝜑(𝑥, 𝑦) = (𝜕2𝜑(𝑥, 𝑦) 𝜕𝑥2 + 𝜕2𝜑(𝑥, 𝑦) 𝜕𝑦2 ) = − 𝜌𝑐(𝑥, 𝑦) 𝜀 (S1) Where 𝜑(𝑥, 𝑦) is the electric potential for a given charge distribution 𝜌𝑐(𝑥, 𝑦) in a MoS2 monolayer with relative permittivity 𝜀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [7] On the other hand, the steady-state heat equation has the following form: 𝛾∇2𝑇(𝑥, 𝑦) = 𝛾 (𝜕2𝑇(𝑥, 𝑦) 𝜕𝑥2 + 𝜕2𝑇(𝑥, 𝑦) 𝜕𝑦2 ) = 𝑓(𝑥, 𝑦) (S2) Here, 𝛾 = 𝐾 𝐶𝑝𝜌 is the thermal diffusivity, 𝐾 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 W K−1 m−1 is the thermal conductivity, [8] 𝐶𝑝 = 373 J kg−1 K−1 is the specific heat capacity, [9] 𝜌 = 5060 kg m−3 is the mass density for MoS2 and 𝑓(𝑥, 𝑦) is the power density dissipated by means of Joule heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We implemented the Dirichlet boundary conditions at 𝑥 = 0 and 𝑥𝑚𝑎𝑥, and Neumann and adiabatic boundary conditions for the electric field and temperature at 𝑦 = 0 and 𝑦𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Concerning the activation energy for vacancy migration, the electric field may promote the movement in one direction while hindering the opposite as follow: 𝐸𝐴 = 𝐸𝐴 0 − 𝑏 · 𝑭(𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [10] Here, the electric field 𝑭(𝑥, 𝑦) is defined for every grid point (𝑥, 𝑦), 𝐸𝐴 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='297 eV is the 5 activation energy without an electric field [11] and 𝒃 = 𝒑𝟎 · [(2 + 𝜀)/3] the polarization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We have used the MoS2 molecular dipole moment 𝑝0 = 15 e nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [12] The time step 𝑡 = − ln(𝑟𝑎𝑛𝑑) ∑ 𝛤 for the simulation is inversely proportional to the transition rate, 𝑟𝑎𝑛𝑑 stands for a random number between 0 and 1 and ∑ 𝛤 is the summation of all the transition rates of migrations, which each one depends on the local electric field and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [13-15] It should be noted that the smaller the activation energy, the smaller the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Besides, the probability of a transition event that takes place in 𝑡 time is 𝑃 = 1 − exp (−𝛤 · t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Regarding the generation of random numbers, there is a technical issue when running several simulations with the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' If it is used the same seed to generate pseudorandom numbers, the sequence of random numbers for each MATLAB session will be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' In this case, you can program a queue with many simulations (one simulation after the other), which the risk of filling the java heap memory and receiving a java heap memory error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=" A different way to deal with this issue is to use the command rng('shuffle', 'twister') to reseed the random number generator based on the current time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This command will enable variations between runs since it will have different pseudorandom numbers, whether you are using the same MATLAB session or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 6 SECTION 4: Dependence of resistance with local density Figure S2: Shows the resistance as a function of the local defect density used during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The relationship between the defect concentration and the MoS2 monolayer resistance has been studied previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [3] Following this dependence, we propose a model 𝑅(𝑟⃗) ∝ 𝜌𝑑 𝑛 (𝑟⃗), where the resistance is in the range found for the MoS2 monolayer (see Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' However, this has been chosen as an example, as different materials may have different resistance values and the simulator can be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 1010 109 (U) 108 Resistance ( 107 106 105 104 103 102 0 6 Density (Vs/nm²) 7 SECTION 5: Colour maps of defect density, resistance and electric field distributions in the channel The simulation allows us to disclose the switching mechanism via enabling detailed knowledge of the vacancy migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='I, II, III and IV (2D color maps for defect density, resistance and electric field in this order) correspond with the states labeled by I), II), III) and IV) on the loop in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The SET process takes place from I to II, and for the RESET from III to IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Regarding the density in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='I, we can observe a peak density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2 density region in intense yellow) at the abrupt edge of the fissure region 𝑥 = 22 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' That peak density is the main contribution to the channel resistance reaching 1700 MΩ (see the intense yellow region in the resistance figure in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='II shows the device ON state when most of the defect migrations have taken place, lowering the peak density to a plateau of the distribution across a 10-nm wide region with a mean density below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 Vs nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The resistance drops from 1700 MΩ to 460 MΩ in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='II and lead to an 80% decrease in the electric field in the same region due to the screening effect (see electric field figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' It is worth noting that between Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='II and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='III almost no defect migrates, as the electric field is not high enough, so the state keeps unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' However, between S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='III and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='IV, migrations regarding the RESET process take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The defects move towards and accumulate at the abrupt edge of the fissure region at 𝑥 = 22 nm, which raises the resistance and drives the device back to the high resistance state (HRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Compared with state I, the peak density in state IV (the yellow region in density figures) is significantly lower, leading to a 75% lower resistance in the new HRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This leads to the same asymmetry in the RS loop found experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [4] Besides, this lower resistance in state IV results also in a lower electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 8 Figure S3: Color maps of defect density (left figures), resistance (figures in the middle) and electric field (right figures) distributions for the states labelled by I), II), III) and IV) in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The SET process takes place from I to II, and for the RESET from III to IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Electric field (MV/m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Resistance (MQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='X axis (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='X axis (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Xaxis (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='SECTION 6: Stability of initial distribution of defects under high RESET electric field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Figure S4: Device resistance stability for the initial state under different constant applied voltages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='(-20 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' -25 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' -30 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' -35V and -40 V) for 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 𝑉𝑠 𝑛𝑚−2 (a) and 𝜌3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 𝑉𝑠 𝑛𝑚−2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulation starts with a skewed Gaussian distribution of defects with fissure region width of 12 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The evolution of the defect density profile along the x-axis for the case of 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 𝑉𝑠 𝑛𝑚−2 shown in a) for -20 V c) and -40 V d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Resistance (M2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='(MQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Resistance ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='t=1s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='t=1s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='t = 24 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='t = 24 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='40 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='X axis (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='X axis (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='SECTION 7: The net diffusion of defects with cycles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='Figure S5: The data was extracted from the same simulation used in Figure 3: The initial defect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='profile of the device exhibits a skewed Gaussian distribution of defects with a peak density of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2, a width of 9 nm and a voltage ramps for a positive polarity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s-1 in the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range for 45 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) represents the percentage of density sections with a density over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 Vs nm−2 among all non-zero density sections in the simulation domain for different cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) evolution of electric field distribution along cycles c) the dependence between the activation energy for defect migration and the electric field for different local defect densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) (e c) (eV) 32 p1 Defect migration ( 20 lectric field (MV/m) Cvcle 2 40 28 60 Cycle 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs/nm2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 Vs/nm² 80 24 Consecutive RESETs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 Vs/nm² energy: 100 3 Vs/nm² 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='8 20 Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 10 20 30 40 18 20 22 24 26 28 30 106 107 108 Cycles X axis (nm) Electric field (V/m) 11 SECTION 8: Degradation process and device power consumption Figure S6: 45 resistive switching I-V curves of the simulation data used in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The simulation starts from a skewed Gaussian distribution of defects with polarity +𝑉𝑑 and a width of 9 nm for 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The ramp voltage rate was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s-1 in the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S7: Energy consumed to SET (black squares) and RESET (red dots) the device at each cycle for the case 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 𝑉𝑠 𝑛𝑚−2, a width of 9 nm and −𝑉𝑑 showed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The ramp voltage rate was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s-1 in the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range for 45 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The energy consumption is Current (μA) 10-1 10-2 20 10 0 10 20 Voltage (V)Energy consumption (μWs) 55 4535 SET RESET 20 0 10 20 30 40 Cycles 12 calculated as 𝐼 × 𝑉 × 𝑡 along the sweep that leads the device to the ON state (SET) and OFF state (RESET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 13 SECTION 9: Polarity effect in Gaussian distribution of defects Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) and b) simulations start from a skewed Gaussian distribution of defects and consist of 15 RS cycles under a ramp voltage rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 𝑉 𝑠−1 between the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range using the following peak densities: 𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 𝑉𝑠 𝑛𝑚−2, 𝜌2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 𝑉𝑠 𝑛𝑚−2, 𝜌3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 𝑉𝑠 𝑛𝑚−2 and 𝜌4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 𝑉𝑠 𝑛𝑚−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) shows the resistance ratio over the fissure region width for different densities and polarity −𝑉𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' b) shows the resistance ratio over the fissure region width for different densities and polarity +𝑉𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The line between points is for ease of reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' We average the resistance ratio over the 15 cycles and use the standard deviation of the cycle-to-cycle variability as the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 P4 Vd id Roff/Ron ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 P2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 5 6 7 8 12 Width (nm) b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 p4 +Vd 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='4 td Roff/Ron ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='3 P2 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='9 5 6 7 8 9 10 11 12 Width (nm) 14 In Figure S8, we have assessed the effect of the asymmetry distribution of defects of the skewed Gaussian distribution using both polarities: −𝑉𝑑 in Figure S8a and +𝑉𝑑 in Figure S8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For these 15-cycles simulations, each point is the average of the cycles, while the error bar is the standard deviation of cycle-to-cycle variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The ramp voltage rate used is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s−1 for the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to - 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' For every 𝜌 value (𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2, 𝜌2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 Vs nm−2, 𝜌3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs nm−2 and 𝜌4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='95 Vs nm−2), we cover different fissure region width between 5 and 12 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' It must be noted the temperature gains importance as the density (and so the resistance) is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' But in general, the influence of the temperature is limited when 𝜌 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs nm−2 and when the fissure region grows (see the maximum temperatures for three densities in Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Both polarities (Figure S8a and S8b) show a higher resistance ratio for higher defect density values (𝜌) and larger fissure region widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The reason is that a higher population of defects (high-density regions over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='5 Vs nm−2) enables a larger resistance ratio since it lets larger differences between the ON and OFF states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' This dependence is the reason for the similar trends in both polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Figure S9: Three 15-cycles simulations starting from a skewed Gaussian distribution of defects with different densities (𝜌1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='64 Vs nm−2, 𝜌2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='08 Vs nm−2 and 𝜌3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='52 Vs nm−2), a 340 p, and w = 9 nm Temperature max (k) P2 and w = 9 nm 330 P3 and w = 9 nm 320 310 300 0 100 200 300 400 500 600 700 Time (s) 15 width of 9 nm and polarity +Vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The ramp voltage rate was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='1 V s-1 in the 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V to -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content='2 V range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' The plot shows the maximum temperature at every time step for 721 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' 16 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=" O'Brien, N." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' McEvoy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Hallam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Berner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Hanlon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Coleman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Duesberg, Sci Rep 2014, 4, 7, 7374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Kretschmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Maslov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Ghaderzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Ghorbani-Asl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Hlawacek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE4T4oBgHgl3EQfxw02/content/2301.05260v1.pdf'} +page_content=' Krasheninnikov, ACS Appl.' metadata={'source': 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a/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/2301.05488v1.pdf.txt b/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/2301.05488v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d2625427820d33f4e159e4ff53ff6b914959cb4 --- /dev/null +++ b/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/2301.05488v1.pdf.txt @@ -0,0 +1,783 @@ +arXiv:2301.05488v1 [math-ph] 13 Jan 2023 +From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative +Construction for Infinite Dimensions +Frederik vom Ende +Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, +14195 Berlin, Germany +(*Electronic mail: frederik.vom.ende@fu-berlin.de) +(Dated: 16 January 2023) +We present an alternative (constructive) proof of the statement that for every completely +positive, trace-preserving map Φ there exists an ancilla Hilbert space K +in a pure +state |ψ⟩⟨ψ| as well as a unitary operator U on system plus ancilla such that Φ equals +trK (U((·) ⊗ |ψ⟩⟨ψ|)U∗). The main tool of our proof is Sz.-Nagy’s dilation theorem ap- +plied to isometries defined on a subspace. In our construction, the ancilla consists of a +system of dimension the Kraus rank of Φ together with a qubit which, however, only acts +as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an +ancilla of dimension “Kraus rank plus one”. We conclude by providing an example which +illustrates how the constructions differ from each other. +PACS numbers: 02.30.Tb, 02.40.Pc, 03.67.-a +Keywords: quantum channel; unitary dilation; Kraus operators; open quantum system; +1 + +I. +INTRODUCTION +It is well known that every completely positive trace-preserving map can be represented by +first coupling the original system to an ancilla, followed by applying a unitary operation to the +enlarged system, and finally discarding the ancilla, cf., e.g., Thm. 6.18 in1. This representation – +sometimes called Stinespring dilation or Stinespring form of quantum maps – is a natural approach +in the analysis of important properties of completely positive maps. Due to its direct physical +interpretation it can even be applied to certain experimental setups2,3. +The Stinespring form is crucial to many areas of quantum physics. To name just a few: in +quantum thermodynamics this is how the allowed operations in the corresponding resource the- +ory approach are defined4,5, in quantum control it is used to emulate arbitrary quantum maps via +restricting Markovian evolutions of larger systems6,7, and in continuous-variable quantum infor- +mation theory8 the channels which have a Stinespring form (where the unitary on the full system is +Gaussian) have been shown to approximately coincide with the set of linear bosonic channels9. In- +deed, the Stinespring form of bosonic Gaussian channels has been studied a fair bit in the past10–12 +as the main information of these infinite-dimensional systems boils down to the underlying classi- +cal phase space (i.e. something finite-dimensional). This approach has found use in studying, e.g, +the Holevo quantity13 or the black hole quantum information loss problem14,15. +Motivated by the widespread importance of this representation we will take a step back and +have a look at how one proves its existence in the first place. Given a linear map Φ on Cn×n, +n ∈ N, if +Φ ≡ trK +� +U((·)⊗ω)U∗� +(1) +for some complex Hilbert space K , some state ω on K , and some unitary U on Cn ⊗K , then +Eq. (1) is called a Stinespring form of Φ. It is an established fact that a linear map has a Stinespring +form if and only if it is completely positive and trace-preserving, and this characterization even +extends to infinite dimensions16,17. To illustrate the common construction – first in the simpler +case of finite dimensions – one starts with a set of Kraus operators {Ki}ℓ +i=1 of Φ and collects +them “in the first column” of a larger matrix U0. Then one fills up the rest of U0 such that it +becomes unitary: because the Kraus operators satisfy ∑ℓ +i=1K∗ +i Ki = +1 the “columns” of U0 form +an orthonormal system which can then be extended to an orthonormal basis. This results in “the” +desired unitary U, Thm. 6.18 & Eq. (6.23) in1, cf. also Appendix A in18. Note that one can also +obtain Eq. (1) directly from Stinespring’s original dilation theorem for C∗-algebras19 (as done in, +2 + +e.g., Thm. 4.18 ff. in20) which, however, is less explicit in the sense that it translates an abstract +object (the isometry V) into another one (the unitary U). Hence our aim is to start from the Kraus +operators as they are more relevant from an application point-of-view21. +When trying to generalize the idea stated previously to infinite dimensions one immediately +runs into two problems: +• For general spaces, not every isometry can be extended to a unitary. The prime example is +the left-shift on ℓ2(N) truncated to a subspace: +σl,0 : {x ∈ ℓ2(N) : x1 = 0} → ℓ2(N) +(0,x1,x2,x3,...) �→ (x1,x2,x3,...) +This is an isometry which cannot be extended to a unitary (or even to an isometry) on +ℓ2(N); the reason for this is that σl,0 it is already surjective so there is “no room left” in +the co-domain ℓ2(N). Of course, this is a purely infinite-dimensional effect because in +finite dimensions every isometric endomorphism is automatically unitary by the rank-nullity +theorem. +• Even assuming domain and range of the isometry U0 have the same co-dimension, if this +number is infinite, then the above strategy of generating an orthonormal basis relies on the +axiom of choice. This feels counter-intuitive because something as fundamental as describ- +ing physical processes via environment interactions should work in any “reasonable” math- +ematical framework. For more on the extension of isometric operators defined only on a +subspace, as well as the connection to extending symmetric operators refer to p. 387 ff. in22. +In their original proof of Eq. (1) for infinite-dimensional H Hellwig & Kraus16 (resp.17,23,24) +resolved this problem by treating H ⊗K as “dimK -many copies” of H and then simply adding +another H to it. As it turns out this is all the extra space one needs in the co-domain to guarantee +that an isometry can be extended to something unitary. For the reader’s convenience we summarize +their argument in Appendix A. +In contrast, we want to stay closer to the proof idea from finite dimensions (i.e. defining an +isometry on an explicit subspace of H ⊗K via Kraus operators) in this article. For this we need +a result on extensions of isometries on a linear subspace, which is what the next section will be +about. Then in Section III we will present the alternative construction of the Stinespring form of +arbitrary quantum maps; in the spirit of George Pólya “Two proofs are better than one. “It is safe +3 + +riding at two anchors.”” (p. 62 in25). Finally we compare the three methods for constructing a +Stinespring form described in this paper by means of a simple example in Section IV. +II. +PRELIMINARIES: EXTENSIONS OF ISOMETRIES +Sz.-Nagy’s dilation theorem is undoubtedly among the most famous results in operator theory +as a whole, and in the branch of operator dilations in particular. It states that for every contraction +T on a Hilbert space H (i.e. ∥T∥op ≤ 1 with ∥·∥op the usual operator norm) the operator + + +T +√ +1−TT ∗ +√ +1−T ∗T +−T ∗ + +, +(2) +on H ×H —where Eq. (2) is clearly a dilation of T—is unitary (Appendix, Sec. 4 in26, cf. also27 +and Ch. VI in28). +Now assume that we are dealing with an isometry V0 defined on a (closed) subspace M ⊆ H . +One way to extend V0 to an operator on all of H is via V ′ := V0PM where PM be the orthogonal +projection29 onto M . Note that V ′ is a contraction (∥V ′∥op ≤ ∥V0∥op∥PM ∥op = ∥PM ∥op ≤ 1) +which means that we can apply Sz.-Nagy’s dilation theorem to it. For this is it crucial to observe +that (V ′)∗ coincides with the adjoint V ∗ +0 : H → M of V0 (i.e. ⟨V0x,y⟩ = ⟨x,V ∗ +0 y⟩ for all x ∈ M , +y ∈ H ) because +⟨x,(V ′)∗y⟩ = ⟨V ′x,y⟩ = ⟨V0(PM x),y⟩ = ⟨PM x,V ∗ +0 y⟩ += ⟨PM x,V ∗ +0 y⟩+⟨( 1−PM )x,V ∗ +0 y⟩ = ⟨x,V ∗ +0 y⟩ +for all x,y ∈ H . In the second-to-last step we used that V ∗ +0 y ∈ M but ( 1 − PM )x ∈ M ⊥ so the +inner product of the two vanishes. With this as well as the fact that V0 is an isometry (i.e. V ∗ +0 V0 = +1) +Eq. (2) becomes + + +V ′ +� +1−V ′(V ′)∗ +� +1−(V ′)∗V ′ +−(V ′)∗ + + = + + +V0PM +� +1−V0(PMV ∗ +0 ) +� +1−V ∗ +0 V0PM +−V ∗ +0 + + += + + V0PM +� +1−V0V ∗ +0 +√ +1−PM +−V ∗ +0 + +. +Finally, observe that +1−V0V ∗ +0 , +1−PM are orthogonal projections so in particular they are positive +semi-definite (note ⟨x,Px⟩ = ⟨x,P2x⟩ = ⟨x,P∗Px⟩ = ⟨Px,Px⟩ = ∥Px∥2 ≥ 0 for all P∗ = P = P2) +4 + +and thus they are their own square root. This yields the unitary dilation + + V0PM +1−V0V ∗ +0 +1−PM +−V ∗ +0 + +. +(3) +on H ×H of V0. However, the direct product of vector spaces is not as meaningful in a “quantum +physics setting” because multipartite quantum systems are described via the tensor product of the +respective Hilbert spaces. This is why we reformulate our preceding calculations as follows: +Lemma 1. Let M ⊆ H be a closed subspace of a complex Hilbert space H , and letV0 : M → H +be an isometry. There exists U : H ⊗ C2 → H ⊗ C2 unitary which extends V0 in the sense that +U(x⊗e1) = V0x⊗e1 for all x ∈ M . +Proof. In Eq. (3) we already saw that +U0 : H ×H → H ×H + +x +y + + �→ + +V0PM x+( 1−V0V ∗ +0 )y +( 1−PM )x−V ∗ +0 y + + +is a unitary dilation of V0. All that is left to do is to “translate” U0 into a unitary on H ⊗ C2. +For this note that the space H ⊗C2 is isometrically isomorphic to H ×H by means of the map +J : H ×H → H ⊗C2, (x,y) �→ x⊗e1 +y⊗e2 (i.e. J is a unitary transformation, Remark 2.6.8 +in30). Thus we define U : H ⊗C2 → H ⊗C2, x �→ (J ◦U0◦J−1)(x) as visualized in the following +commutative diagram: +H ⊗C2 +H ⊗C2 +H ×H +H ×H +U +J−1 +U0 +J +Obviously U is unitary as it is a composition of unitaries, and U satisfies the desired extension +property because for all x ∈ M +U(x⊗e1) = (J ◦U0 ◦J−1)(x⊗e1) = (J ◦U0) + +x +0 + + += J + + V0PM x +( 1−PM )x + + = J + + V0x +x−x + + = V0x⊗e1 . +With this lemma at our disposal we are ready to have a different look at (Stinespring) dilations +of quantum maps. +5 + +III. +THE STINESPRING FORM OF QUANTUM MAPS +Given any complex Hilbert spaces H ,K , recall that a (Schrödinger) quantum map is a linear +map Φ between trace classes (Ch. 16 in31) B1(H ), B1(K ) which preserves the trace and is com- +pletely positive, that is, for all n ∈ N the extended map Φ ⊗idn : B1(H ⊗Cn) → B1(K ⊗Cn) +sends positive semi-definite operators to positive semi-definite operators. +Equivalently, Φ is +completely positive if and only if there exists a family (Kj) j∈J ⊂ B(H ,K )—called Kraus op- +erators—such that Φ(·) = ∑j∈J Kj(·)K∗ +j , where the sum converges in trace norm and ∑ j∈J K∗ +j Kj +converges strongly to a bounded operator, cf. Ch. 9, Thm. 2.3 in32. If Φ is additionally trace- +preserving, then ∑j∈J K∗ +j Kj strongly converges to the identity. The collection of all completely +positive and trace-preserving maps (called CPTP or quantum maps) from H to K will be +denoted by CPTP(H ,K ) (and CPTP(H ) := CPTP(H ,H )). +A particularly important ele- +ment in CPTP(H ⊗ K ,H ) is the partial trace trK which is the unique linear map satisfying +tr(trK (A)B) = tr(A(B⊗ +1)) for all A ∈ B1(H ⊗K ), B ∈ B(H ), cf. Def. 2.68 ff. in20. Finally, +D(H ) will denote the set of all states, i.e. all positive semi-definite trace-class operators of trace +one. With all the notation in place let us state and prove our main result: +Theorem 1. Given any Φ ∈ CPTP(H ) there exists a Hilbert space K , a unit vector ψ ∈ K , and +a unitary operator U on H ⊗K such that +Φ ≡ trK +� +U((·)⊗|ψ⟩⟨ψ)U∗� +. +Moreover, +(i) if (Kj) j∈J is any set of Kraus operators of Φ, then one can choose K to be ℓ2(J) ⊗ C2 +and ψ := ej0 ⊗ e1 for any j0 ∈ J. In particular if H is separable, then K can be chosen +separable, as well. +(ii) U can be chosen such that the ancilla qubit is a catalyst, i.e. for all ρ ∈ D(H ) there exists +ω ∈ D(H ⊗ℓ2(J)) such that U(ρ ⊗|ej0⟩⟨ej0|⊗|e1⟩⟨e1|)U∗ = ω ⊗|e1⟩⟨e1|. +Proof. Starting from a set of Kraus operators (Kj) j∈J for Φ as well as an arbitrary (but fixed) +j0 ∈ J we define a map on a (closed) subspace of H ⊗ℓ2(J) via +V0 : H ⊗Cej0 = {x⊗ej0 : x ∈ H } → H ⊗ℓ2(J) +x⊗ej0 �→ ∑ +j∈J +Kjx⊗ej . +6 + +Here ej ∈ ℓ2(J) is the “j-th standard basis vector” ej : J → C, j′ �→ δj j′. Because (Kjx⊗ej) j∈J is +an orthogonal set in the Hilbert space H ⊗ℓ2(J), by Prop. 2.2.5 in30 ∑j∈J Kjx ⊗ej exists if and +only if ∑j∈J ∥Kjx∥2 < ∞; but this holds due to +∑ +j∈JF +∥Kjx∥2 = ∑ +j∈JF +⟨Kjx,Kjx⟩ = +� +x, ∑ +j∈JF +K∗ +j Kjx +� +(4) +for all finite subsets JF ⊆ J and all x ∈ H together with the fact that the Kraus operators satisfy +∑j∈J K∗ +j Kj → +1 in the strong (hence the weak) operator topology. This shows that V0 is well- +defined. Moreover, V0 is an isometry because, again, Prop. 2.2.5 from30 for all x ∈ H yields +∥V0(x⊗ej0)∥2 = +���∑ +j∈J +Kjx⊗ej +��� +2 += ∑ +j∈J +∥Kjx∥2 (4) += ∥x∥2. +Now Lemma 1 comes into play: it lets us extend V0 to a unitary U on H ⊗ℓ2(J) ⊗C2 which by +Eq. (3)—when identifying H ⊗ℓ2(J)⊗C2 ∼= (H ⊗ℓ2(J))×(H ⊗ℓ2(J))—is of the form + + V0( 1⊗|ej0⟩⟨ej0|) +1−V0V ∗ +0 +1⊗( 1−|ej0⟩⟨ej0|) +−V ∗ +0 + + = + +∑j∈J Kj ⊗|ej⟩⟨ej0| +1−(∑j, j′∈J KjK∗ +j′ ⊗|ej⟩⟨ej′|) +1⊗( 1−|ej0⟩⟨ej0|) +−∑ j∈J K∗ +j ⊗|ej0⟩⟨ej| + +. +(5) +Define ψ := ej0 ⊗e1 ∈ ℓ2(J)⊗C2 =: K . Note that if H is separable, then J can be chosen count- +able (Ch. 9, Thm. 2.3 in32) meaning K is separable (this proves (i)). Given x,y ∈ H compute +trK +� +U(|x⟩⟨y|⊗|ψ⟩⟨ψ|)U∗� += trℓ2(J)⊗C2 +� +|U(x⊗ej0 ⊗e1)⟩⟨U(y⊗ej0 ⊗e1)| +� += trℓ2(J)⊗C2 +� +|V0(x⊗ej0)⟩⟨V0(y⊗ej0)|⊗|e1⟩⟨e1| +� += trℓ2(J) +����∑ +j∈J +Kjx⊗ej +�� +∑ +j′∈J +Kj′y⊗ej′ +��� +� += ∑ +j, j′∈J +trℓ2(J) +� +|Kjx⟩⟨Kj′y|⊗|ej⟩⟨ej′| +� += ∑ +j, j′∈J +Kj|x⟩⟨y|K∗ +j′⟨ej′,ej⟩ = ∑ +j∈J +Kj|x⟩⟨y|K∗ +j = Φ(|x⟩⟨y|). +In the second-to-last line we used that the partial trace (just like every quantum map) is con- +tinuous (Prop. 2 in33). Hence this also holds for both Φ and trK (U((·) ⊗ |ψ⟩⟨ψ)U∗), mean- +ing they coincide on all of B1(H ) because we showed that they coincide on the dense subset +span{|x⟩⟨y| : x,y ∈ H } ⊆ B1(H ). The only statement left to prove is the catalyst property (ii); +this follows readily from the extension property in Lemma 1, resp. from Eq. (5). +Note that this construction works for any set of Kraus operators so the smallest ancilla one can +get this way has dimension “twice the Kraus rank”. However, in practice one could also choose a +7 + +“non-minimal” set of Kraus operators – if, e.g., the corresponding unitary is easier to implement +in practice – or use a different (not fully general) construction altogether. +Remark 2. Be aware that the Stinespring form +(i) is not limited to quantum maps with same domain and co-domain: given Φ ∈ CPTP(H ,K ) +one obtains an extension of Theorem 1 to arbitrary quantum maps via the auxilliary map +ρ �→ |ψ′⟩⟨ψ′| ⊗ Φ(trK (ρ)) ∈ CPTP(H ⊗ K ) where ψ′ ∈ H is any unit vector, cf. also +Coro. 1 in33. +(ii) exists equivalently in the Heisenberg picture: given any (Heisenberg) quantum channel Φ∗ +– meaning Φ∗ is any completely positive, unital (i.e. identity preserving), and ultraweakly +continuous map – one has Φ∗ = tr|ψ⟩⟨ψ|(U∗((·) ⊗ +1)U∗) where ψ,U are the same as in +Thm. 1 and tr|ψ⟩⟨ψ| is the partial trace w.r.t. the state |ψ⟩⟨ψ| (Ch. 9, Lemma 1.1 in32). For +more on the Stinespring form in the Heisenberg picture and its relation to Stinespring’s +theorem for C∗-algebras we refer to Coro. 2 ff. in33. +IV. +COMPARING CONSTRUCTIONS: AN EXAMPLE +To better understand the different techniques for generating Stinespring forms let us illustrate +and compare them via a simple example. While the following example will be finite-dimensional +– hence the construction described in the introduction yields the smallest ancilla – such a choice +will clarify how the two constructions intended for infinite dimensions circumvent the problems +discussed in the introduction. +Let Φ ∈ CPTP(n) be given such that {K1,K2} ⊂ Cn×n is a set of Kraus operators for Φ. As +explained in the introduction, the standard construction first collects K1,K2 in a 2n×2n-matrix: + +K1 0 +K2 0 + + +(6) +Due to K∗ +1K1 + K∗ +2K2 = +1 the first n columns form an orthonormal system in C2n which can be +extended to an orthonormal basis. Build U12,U22 ∈ Cn×n from the additional vectors such that +UF := + +K1 U12 +K2 U22 + + +is unitary, and one has Φ ≡ trC2(UF((·)⊗|e1⟩⟨e1|)U∗ +F). +8 + +Next, let us carry out the construction of Hellweg & Kraus (cf. Appendix A): similar to Eq. (6) +one starts with the isometry +A := + +K1 +K2 + + ∈ C2n×n . +(7) +However, instead of completing it to a unitary by adding suitable columns – which cannot be +guaranteed in infinite dimensions – they enlarge the ancilla in order to define +UK := + +0 +A∗ +A −( 1−AA∗) + + = + + + + + +0 +K∗ +1 +K∗ +2 +K1 K1K∗ +1 − +1 +K1K∗ +2 +K2 +K2K∗ +1 +K2K∗ +2 − +1 + + + + +. +This UK satisfies Φ ≡ trC3(UK((·)⊗|e1⟩⟨e1|)U∗ +K). +Finally, the construction presented in our paper also starts with Eq. (7) but then turns A into the +“block matrix” from Eq. (6) as part of the larger matrix +UN := + + + + + + + +K1 0 +1−K1K∗ +1 +−K1K∗ +2 +K2 0 +−K2K∗ +1 +1−K2K∗ +2 +0 +0 +−K∗ +1 +−K∗ +2 +0 +1 +0 +0 + + + + + + + +. +Choosing the ancilla to be C2 ⊗C2 this unitary satisfies Φ ≡ trC2⊗C2(UN((·)⊗|e1⟩⟨e1|)U∗ +N). Note +that a shuffled version of UK “appears” in UN (up to some minus signs which can be neglected) and +the two matrices differ by an identity on a complementary subspace. Thus, while UN is larger than +UK the “effective” dimension (in the sense that part of the space UN acts on is a catalyst w.r.t. UN) +is smaller. This is also showcased here: the first column of the upper left block of UN features only +the Kraus operators. This is why +UN(ρ ⊗|e1⟩⟨e1|)U∗ +N = + +K1ρK∗ +1 K1ρK∗ +2 +K2ρK∗ +1 K2ρK∗ +2 + +⊗|e1⟩⟨e1| = UF(ρ ⊗|e1⟩⟨e1|)U∗ +F ⊗|e1⟩⟨e1| +whereas +UN(ρ ⊗|e1⟩⟨e1|)U∗ +N = + + + + + +0 +0 +0 +0 K1ρK∗ +1 K1ρK∗ +2 +0 K2ρK∗ +1 K2ρK∗ +2 + + + + + = 0⊕UF(ρ ⊗|e1⟩⟨e1|)U∗ +F . +9 + +ACKNOWLEDGMENTS +I would like to thank Jens Eisert for pointing out some references on Stinespring forms of +bosonic channels which I was not aware of yet. This work has been supported by the Einstein +Foundation (Einstein Research Unit on Quantum Devices) and the MATH+ Cluster of Excellence. +Appendix A: Original Proof of Hellwig and Kraus +This appendix will revolve around the following statement, respectively the proof given by +Hellwig and Kraus (originally in16, and in more detail in Sec. 4 in23 or Thm. 2 in24): +Given any complex Hilbert space H and any quantum map Φ on H there exists a +Hilbert space K , a unit vector ψ ∈ K , and a self-adjoint unitary operator U on +H ⊗K such that +Φ ≡ trK +� +U((·)⊗|ψ⟩⟨ψ)U∗� +. +If (Kj) j∈J is a set of Kraus operators of Φ, then one can choose K to be ℓ2(J ∪{s}) +where s is any symbol not in J. +We note that their proof was given for separable Hilbert spaces H but extends without further +ado to arbitrary Hilbert spaces. Their construction goes as follows. Starting from a set of Kraus +operators (Kj) j∈J for Φ (cf. Ch. 9, Thm. 2.3 in32) one first defines the following objects: +• Js := J ∪{s} where s is any symbol not in J +• K := ℓ2(Js) is the Hilbert space of all functions f : Js → C which are square-summable, +i.e. ∑j∈Js | f( j)|2 < ∞, cf. Example 1.7.3 & Example 2.1.12 in30 +• Hs := H =: H j for all j ∈ J +• ι : Hs ⊕ � +j∈J H j → H ⊗ K is the isometric isomorphism defined via xs ⊕ � +j∈J xj �→ +∑j∈Js xj ⊗ej. Hence Hs ⊕ � +j∈J H j ∼= H ⊗K , cf. Remark 2.6.8 in30. +The idea now is to define an isometry A : H → � +j∈J H j, embed it into a unitary operator U0 on +Hs⊕� +j∈J H j, and finally use ι to translate U0 into a unitary operator U on H ⊗K as visualized +10 + +in the following diagram: +H ⊗ℓ2(Js) +H ⊗ℓ2(Js) +Hs ⊕ � +j∈J H j +Hs ⊕ � +j∈J H j +U +ι−1 +U0 +ι +They started by defining A : Hs → � +j∈J H j via Ax := � +j∈J Kjx. One readily verifies that A is an +isometry (so in particular well-defined) because +∥Ax∥2 = ∑ +j∈J +∥Kjx∥2 = ∑ +j∈J +⟨x,K∗ +j Kjx⟩ = ∥x∥2 +as ∑j∈J K∗ +j Kj → +1 in the strong operator topology. With this they defined U0 via34 +U0 : Hs ⊕ +� +j∈J +H j → Hs ⊕ +� +j∈J +H j + +x +y + + �→ + + +A∗y +Ax−( 1−AA∗)y + + = + +0 +A∗ +A −( 1−AA∗) + + + +x +y + +. +Evidently, U0 is a self-adjoint involution; hence U0 is unitary and so is the “translated” operator +U := ι ◦U0 ◦ι−1 on H ⊗K (because ι is a unitary transformation). Note that +U(x⊗es) = (ι ◦U0) + +x +0 + + = ι + + 0 +Ax + + = ι + + +0 +� +j∈J Kjx + + = ∑ +j∈J +Kjx⊗ej +(A1) +for all x ∈ Hs = H . Defining ψ := es ∈ ℓ2(Js) one for all x,y ∈ H finds +trℓ2(Js) +� +U(|x⟩⟨y|⊗|es⟩⟨es|)U∗� += trℓ2(Js) +� +|U(x⊗es)⟩⟨U(y⊗es)| +� +(A1) += trℓ2(J) +����∑ +j∈J +Kjx⊗ej +�� +∑ +j′∈J +Kj′x⊗ej′ +��� +� += ∑ +j, j′∈J +trℓ2(J) +� +|Kjx⟩⟨Kj′x|⊗|ej⟩⟨ej′| +� += ∑ +j, j′∈J +Kj|x⟩⟨y|K∗ +j′⟨ej′,ej⟩ = ∑ +j∈J +Kj|x⟩⟨y|K∗ +j = Φ(|x⟩⟨y|). +A standard continuity argument shows that Φ ≡ trK (U((·)⊗|ψ⟩⟨ψ)U∗) on all of B1(H ). +REFERENCES +1A. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, De Gruyter +Studies in Mathematical Physics 16 (DeGruyter, Berlin, 2012). +11 + +2D. Braun, Dissipative Quantum Chaos and Decoherence (Springer, Berlin, Heidelberg, 2001). +3F. Haake, Quantum Signatures of Chaos, 3rd ed. (Springer, Berlin, Heidelberg, 2010). +4M. Lostaglio, “An Introductory Review of the Resource Theory Approach to Thermodynamics,” +Rep. Prog. Phys. 82, 114001 (2019). +5F. vom Ende, “Which Bath-Hamiltonians Matter for Thermal Operations?” J. Math. Phys. 63, +112202 (2022). +6P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. K. Wilhelm, “Optimal Control of a Qubit +Coupled to a Non-Markovian Environment,” Phys. Rev. Lett. 102, 090401 (2009). +7T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. Glaser, “Optimal Control for Generating +Quantum Gates in Open Dissipative Systems,” J. Phys. B 44, 154013 (2011). +8C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. Ralph, J. Shapiro, and S. Lloyd, +“Gaussian Quantum Information,” Rev. Mod. Phys. 84, 621 (2012). +9L. Lami, K. Sabapathy, and A. Winter, “All Phase-Space Linear Bosonic Channels Are Approx- +imately Gaussian Dilatable,” New J. Phys. 20, 113012 (2018). +10A. Holevo, “One-Mode Quantum Gaussian Channels: Structure and Quantum Capacity,” Probl. +Inf. Transm. 43, 1–11 (2007). +11F. Caruso, J. Eisert, V. Giovannetti, and A. Holevo, “Multi-Mode Bosonic Gaussian Channels,” +New J. Phys. 10, 083030 (2008). +12F. Caruso, J. Eisert, V. Giovannetti, and A. Holevo, “Optimal Unitary Dilation for Bosonic Gaus- +sian Channels,” Phys. Rev. A 84, 022306 (2011). +13M. Shirokov, “Strong Convergence of Quantum Channels: Continuity of the Stinespring Dilation +and Discontinuity of the Unitary Dilation,” J. Math. Phys. 61, 082204 (2020). +14K. Brádler and C. Adami, “Black Holes as Bosonic Gaussian Channels,” Phys. Rev. D 92, +025030 (2015). +15D. Kretschmann, D. Schlingemann, and R. Werner, “The Information-Disturbance Tradeoff and +the Continuity of Stinespring’s Representation,” IEEE T. Inform. Theory 54, 1708–1717 (2008). +16K. Hellweg and K. Kraus, “Operations and Measurements II,” Commun. Math. Phys. 16, 142– +147 (1970). +17K. Kraus, “General State Changes in Quantum Theory,” Ann. Phys. 64, 311–335 (1971). +18F. vom Ende, “Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space,” +(2022), accepted to Open Syst. Inf. Dyn., arXiv:2211.08351. +19W. Stinespring, “Positive Functions on C∗-Algebras,” Proc. Amer. Math. Soc. 6, 211–216 +12 + +(1955). +20T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory: From Uncer- +tainty to Entanglement (Cambridge University Press, Cambridge, 2012). +21However, be aware that the existence of Kraus operators—in particular in infinite dimensions— +is usually proven via Stinespring’s dilation theorem for C∗-algebras, cf. Ch. 9, Thm. 2.3 in32. +22G. Ludwig, Foundations of Quantum Mechanics I (Springer, New York, 1983). +23K. Kraus, “Operations and Effects in the Hilbert Space Formulation of Quantum Theory,” in +Foundations of Quantum Mechanics and Ordered Linear Spaces (Springer, Berlin, 1973) pp. +206–229. +24K. Kraus, States, Effects, and Operations, Lecture Notes in Physics, Vol. 190 (Springer, Berlin, +1983). +25G. Pólya, How to Solve It: A New Aspect of Mathematical Method, 2nd ed. (Princeton University +Press, Princeton, 2004). +26F. Riesz and B. Sz.-Nagy, Functional Analysis, 2nd ed. (Dover, New York, 1990). +27Actually in26 it is proven that + + +T +√ +1− TT ∗ +−√ +1− T∗T +T ∗ + + is a unitary dilation of T. However, +multiplying this from the left with the unitary +1⊕(−1) yields Eq. (2). +28C. Foias and A. Frazho, The Commutant Lifting Approach to Interpolation Problems +(Birkhäuser, Basel, 1990). +29Because M is closed by assumption, it is a Hilbert space itself (Example 11.3 in31) which +implies that PM as well as V ∗ +0 are well-defined (Sec. 2.5 & Thm. 2.4.2 in30). +30R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1: Elemen- +tary Theory (American Mathematical Society, Providence, Rhode Island, 1983). +31R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Graduate Texts in Mathe- +matics (Oxford University Press, Oxford, 1997). +32E. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). +33F. vom Ende and G. Dirr, “Unitary Dilations of Discrete-Time Quantum-Dynamical Semi- +groups,” J. Math. Phys. 60, 122702 (2019). +34Originally, Hellwig and Kraus considered general quantum operations, cf. Sec. 4 in20 meaning +the Kraus operators only need to satisfy ∑j∈J K∗ +j Kj ≤ +1. This made their construction of U0 a bit +more involved (Eq. (4.3) in23); however, as we are only interested in a dilation of quantum maps +we may use the simpler version of U0 (Eq. (5.27) in24). +13 + diff --git a/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/load_file.txt b/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..934ea5923b9794f4b3c49aa04f8be959babe3434 --- /dev/null +++ b/yNE5T4oBgHgl3EQfNQ6I/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf,len=424 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='05488v1 [math-ph] 13 Jan 2023 From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative Construction for Infinite Dimensions Frederik vom Ende Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany (*Electronic mail: frederik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='vom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='ende@fu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='de) (Dated: 16 January 2023) We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map Φ there exists an ancilla Hilbert space K in a pure state |ψ⟩⟨ψ| as well as a unitary operator U on system plus ancilla such that Φ equals trK (U((·) ⊗ |ψ⟩⟨ψ|)U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The main tool of our proof is Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='-Nagy’s dilation theorem ap- plied to isometries defined on a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In our construction, the ancilla consists of a system of dimension the Kraus rank of Φ together with a qubit which, however, only acts as a catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In contrast, the original proof of Hellwig & Kraus given in the 70s yields an ancilla of dimension “Kraus rank plus one”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' We conclude by providing an example which illustrates how the constructions differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' PACS numbers: 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='Tb, 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='Pc, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='-a Keywords: quantum channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' unitary dilation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Kraus operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' open quantum system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' INTRODUCTION It is well known that every completely positive trace-preserving map can be represented by first coupling the original system to an ancilla, followed by applying a unitary operation to the enlarged system, and finally discarding the ancilla, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=', Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='18 in1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This representation – sometimes called Stinespring dilation or Stinespring form of quantum maps – is a natural approach in the analysis of important properties of completely positive maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Due to its direct physical interpretation it can even be applied to certain experimental setups2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The Stinespring form is crucial to many areas of quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' To name just a few: in quantum thermodynamics this is how the allowed operations in the corresponding resource the- ory approach are defined4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in quantum control it is used to emulate arbitrary quantum maps via restricting Markovian evolutions of larger systems6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' and in continuous-variable quantum infor- mation theory8 the channels which have a Stinespring form (where the unitary on the full system is Gaussian) have been shown to approximately coincide with the set of linear bosonic channels9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In- deed, the Stinespring form of bosonic Gaussian channels has been studied a fair bit in the past10–12 as the main information of these infinite-dimensional systems boils down to the underlying classi- cal phase space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' something finite-dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This approach has found use in studying, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='g, the Holevo quantity13 or the black hole quantum information loss problem14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Motivated by the widespread importance of this representation we will take a step back and have a look at how one proves its existence in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Given a linear map Φ on Cn×n, n ∈ N, if Φ ≡ trK � U((·)⊗ω)U∗� (1) for some complex Hilbert space K , some state ω on K , and some unitary U on Cn ⊗K , then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (1) is called a Stinespring form of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' It is an established fact that a linear map has a Stinespring form if and only if it is completely positive and trace-preserving, and this characterization even extends to infinite dimensions16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' To illustrate the common construction – first in the simpler case of finite dimensions – one starts with a set of Kraus operators {Ki}ℓ i=1 of Φ and collects them “in the first column” of a larger matrix U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Then one fills up the rest of U0 such that it becomes unitary: because the Kraus operators satisfy ∑ℓ i=1K∗ i Ki = 1 the “columns” of U0 form an orthonormal system which can then be extended to an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This results in “the” desired unitary U, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='18 & Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='23) in1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' also Appendix A in18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that one can also obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (1) directly from Stinespring’s original dilation theorem for C∗-algebras19 (as done in, 2 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=', Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='18 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in20) which, however, is less explicit in the sense that it translates an abstract object (the isometry V) into another one (the unitary U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Hence our aim is to start from the Kraus operators as they are more relevant from an application point-of-view21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' When trying to generalize the idea stated previously to infinite dimensions one immediately runs into two problems: For general spaces, not every isometry can be extended to a unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The prime example is the left-shift on ℓ2(N) truncated to a subspace: σl,0 : {x ∈ ℓ2(N) : x1 = 0} → ℓ2(N) (0,x1,x2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=') �→ (x1,x2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=') This is an isometry which cannot be extended to a unitary (or even to an isometry) on ℓ2(N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' the reason for this is that σl,0 it is already surjective so there is “no room left” in the co-domain ℓ2(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Of course, this is a purely infinite-dimensional effect because in finite dimensions every isometric endomorphism is automatically unitary by the rank-nullity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Even assuming domain and range of the isometry U0 have the same co-dimension, if this number is infinite, then the above strategy of generating an orthonormal basis relies on the axiom of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This feels counter-intuitive because something as fundamental as describ- ing physical processes via environment interactions should work in any “reasonable” math- ematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For more on the extension of isometric operators defined only on a subspace, as well as the connection to extending symmetric operators refer to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 387 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In their original proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (1) for infinite-dimensional H Hellwig & Kraus16 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='17,23,24) resolved this problem by treating H ⊗K as “dimK -many copies” of H and then simply adding another H to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' As it turns out this is all the extra space one needs in the co-domain to guarantee that an isometry can be extended to something unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For the reader’s convenience we summarize their argument in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In contrast, we want to stay closer to the proof idea from finite dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' defining an isometry on an explicit subspace of H ⊗K via Kraus operators) in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For this we need a result on extensions of isometries on a linear subspace, which is what the next section will be about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Then in Section III we will present the alternative construction of the Stinespring form of arbitrary quantum maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in the spirit of George Pólya “Two proofs are better than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' “It is safe 3 riding at two anchors.”” (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 62 in25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Finally we compare the three methods for constructing a Stinespring form described in this paper by means of a simple example in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' PRELIMINARIES: EXTENSIONS OF ISOMETRIES Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='-Nagy’s dilation theorem is undoubtedly among the most famous results in operator theory as a whole, and in the branch of operator dilations in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' It states that for every contraction T on a Hilbert space H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' ∥T∥op ≤ 1 with ∥·∥op the usual operator norm) the operator \uf8eb \uf8ed T √ 1−TT ∗ √ 1−T ∗T −T ∗ \uf8f6 \uf8f8, (2) on H ×H —where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (2) is clearly a dilation of T—is unitary (Appendix, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 4 in26, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' also27 and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' VI in28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Now assume that we are dealing with an isometry V0 defined on a (closed) subspace M ⊆ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' One way to extend V0 to an operator on all of H is via V ′ := V0PM where PM be the orthogonal projection29 onto M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that V ′ is a contraction (∥V ′∥op ≤ ∥V0∥op∥PM ∥op = ∥PM ∥op ≤ 1) which means that we can apply Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='-Nagy’s dilation theorem to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For this is it crucial to observe that (V ′)∗ coincides with the adjoint V ∗ 0 : H → M of V0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' ⟨V0x,y⟩ = ⟨x,V ∗ 0 y⟩ for all x ∈ M , y ∈ H ) because ⟨x,(V ′)∗y⟩ = ⟨V ′x,y⟩ = ⟨V0(PM x),y⟩ = ⟨PM x,V ∗ 0 y⟩ = ⟨PM x,V ∗ 0 y⟩+⟨( 1−PM )x,V ∗ 0 y⟩ = ⟨x,V ∗ 0 y⟩ for all x,y ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In the second-to-last step we used that V ∗ 0 y ∈ M but ( 1 − PM )x ∈ M ⊥ so the inner product of the two vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' With this as well as the fact that V0 is an isometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' V ∗ 0 V0 = 1) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (2) becomes \uf8eb \uf8ed V ′ � 1−V ′(V ′)∗ � 1−(V ′)∗V ′ −(V ′)∗ \uf8f6 \uf8f8 = \uf8eb \uf8ed V0PM � 1−V0(PMV ∗ 0 ) � 1−V ∗ 0 V0PM −V ∗ 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed V0PM � 1−V0V ∗ 0 √ 1−PM −V ∗ 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Finally, observe that 1−V0V ∗ 0 , 1−PM are orthogonal projections so in particular they are positive semi-definite (note ⟨x,Px⟩ = ⟨x,P2x⟩ = ⟨x,P∗Px⟩ = ⟨Px,Px⟩ = ∥Px∥2 ≥ 0 for all P∗ = P = P2) 4 and thus they are their own square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This yields the unitary dilation \uf8eb \uf8ed V0PM 1−V0V ∗ 0 1−PM −V ∗ 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (3) on H ×H of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' However, the direct product of vector spaces is not as meaningful in a “quantum physics setting” because multipartite quantum systems are described via the tensor product of the respective Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This is why we reformulate our preceding calculations as follows: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Let M ⊆ H be a closed subspace of a complex Hilbert space H , and letV0 : M → H be an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' There exists U : H ⊗ C2 → H ⊗ C2 unitary which extends V0 in the sense that U(x⊗e1) = V0x⊗e1 for all x ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (3) we already saw that U0 : H ×H → H ×H \uf8eb \uf8edx y \uf8f6 \uf8f8 �→ \uf8eb \uf8edV0PM x+( 1−V0V ∗ 0 )y ( 1−PM )x−V ∗ 0 y \uf8f6 \uf8f8 is a unitary dilation of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' All that is left to do is to “translate” U0 into a unitary on H ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For this note that the space H ⊗C2 is isometrically isomorphic to H ×H by means of the map J : H ×H → H ⊗C2, (x,y) �→ x⊗e1 +y⊗e2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' J is a unitary transformation, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='8 in30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Thus we define U : H ⊗C2 → H ⊗C2, x �→ (J ◦U0◦J−1)(x) as visualized in the following commutative diagram: H ⊗C2 H ⊗C2 H ×H H ×H U J−1 U0 J Obviously U is unitary as it is a composition of unitaries, and U satisfies the desired extension property because for all x ∈ M U(x⊗e1) = (J ◦U0 ◦J−1)(x⊗e1) = (J ◦U0) \uf8eb \uf8edx 0 \uf8f6 \uf8f8 = J \uf8eb \uf8ed V0PM x ( 1−PM )x \uf8f6 \uf8f8 = J \uf8eb \uf8ed V0x x−x \uf8f6 \uf8f8 = V0x⊗e1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' With this lemma at our disposal we are ready to have a different look at (Stinespring) dilations of quantum maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' THE STINESPRING FORM OF QUANTUM MAPS Given any complex Hilbert spaces H ,K , recall that a (Schrödinger) quantum map is a linear map Φ between trace classes (Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 16 in31) B1(H ), B1(K ) which preserves the trace and is com- pletely positive, that is, for all n ∈ N the extended map Φ ⊗idn : B1(H ⊗Cn) → B1(K ⊗Cn) sends positive semi-definite operators to positive semi-definite operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Equivalently, Φ is completely positive if and only if there exists a family (Kj) j∈J ⊂ B(H ,K )—called Kraus op- erators—such that Φ(·) = ∑j∈J Kj(·)K∗ j , where the sum converges in trace norm and ∑ j∈J K∗ j Kj converges strongly to a bounded operator, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 in32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' If Φ is additionally trace- preserving, then ∑j∈J K∗ j Kj strongly converges to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The collection of all completely positive and trace-preserving maps (called CPTP or quantum maps) from H to K will be denoted by CPTP(H ,K ) (and CPTP(H ) := CPTP(H ,H )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' A particularly important ele- ment in CPTP(H ⊗ K ,H ) is the partial trace trK which is the unique linear map satisfying tr(trK (A)B) = tr(A(B⊗ 1)) for all A ∈ B1(H ⊗K ), B ∈ B(H ), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='68 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Finally, D(H ) will denote the set of all states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' all positive semi-definite trace-class operators of trace one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' With all the notation in place let us state and prove our main result: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Given any Φ ∈ CPTP(H ) there exists a Hilbert space K , a unit vector ψ ∈ K , and a unitary operator U on H ⊗K such that Φ ≡ trK � U((·)⊗|ψ⟩⟨ψ)U∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Moreover, (i) if (Kj) j∈J is any set of Kraus operators of Φ, then one can choose K to be ℓ2(J) ⊗ C2 and ψ := ej0 ⊗ e1 for any j0 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In particular if H is separable, then K can be chosen separable, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (ii) U can be chosen such that the ancilla qubit is a catalyst, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' for all ρ ∈ D(H ) there exists ω ∈ D(H ⊗ℓ2(J)) such that U(ρ ⊗|ej0⟩⟨ej0|⊗|e1⟩⟨e1|)U∗ = ω ⊗|e1⟩⟨e1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Starting from a set of Kraus operators (Kj) j∈J for Φ as well as an arbitrary (but fixed) j0 ∈ J we define a map on a (closed) subspace of H ⊗ℓ2(J) via V0 : H ⊗Cej0 = {x⊗ej0 : x ∈ H } → H ⊗ℓ2(J) x⊗ej0 �→ ∑ j∈J Kjx⊗ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 6 Here ej ∈ ℓ2(J) is the “j-th standard basis vector” ej : J → C, j′ �→ δj j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Because (Kjx⊗ej) j∈J is an orthogonal set in the Hilbert space H ⊗ℓ2(J), by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='5 in30 ∑j∈J Kjx ⊗ej exists if and only if ∑j∈J ∥Kjx∥2 < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' but this holds due to ∑ j∈JF ∥Kjx∥2 = ∑ j∈JF ⟨Kjx,Kjx⟩ = � x, ∑ j∈JF K∗ j Kjx � (4) for all finite subsets JF ⊆ J and all x ∈ H together with the fact that the Kraus operators satisfy ∑j∈J K∗ j Kj → 1 in the strong (hence the weak) operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This shows that V0 is well- defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Moreover, V0 is an isometry because, again, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='5 from30 for all x ∈ H yields ∥V0(x⊗ej0)∥2 = ���∑ j∈J Kjx⊗ej ��� 2 = ∑ j∈J ∥Kjx∥2 (4) = ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Now Lemma 1 comes into play: it lets us extend V0 to a unitary U on H ⊗ℓ2(J) ⊗C2 which by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (3)—when identifying H ⊗ℓ2(J)⊗C2 ∼= (H ⊗ℓ2(J))×(H ⊗ℓ2(J))—is of the form \uf8eb \uf8ed V0( 1⊗|ej0⟩⟨ej0|) 1−V0V ∗ 0 1⊗( 1−|ej0⟩⟨ej0|) −V ∗ 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed∑j∈J Kj ⊗|ej⟩⟨ej0| 1−(∑j, j′∈J KjK∗ j′ ⊗|ej⟩⟨ej′|) 1⊗( 1−|ej0⟩⟨ej0|) −∑ j∈J K∗ j ⊗|ej0⟩⟨ej| \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (5) Define ψ := ej0 ⊗e1 ∈ ℓ2(J)⊗C2 =: K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that if H is separable, then J can be chosen count- able (Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 in32) meaning K is separable (this proves (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Given x,y ∈ H compute trK � U(|x⟩⟨y|⊗|ψ⟩⟨ψ|)U∗� = trℓ2(J)⊗C2 � |U(x⊗ej0 ⊗e1)⟩⟨U(y⊗ej0 ⊗e1)| � = trℓ2(J)⊗C2 � |V0(x⊗ej0)⟩⟨V0(y⊗ej0)|⊗|e1⟩⟨e1| � = trℓ2(J) ����∑ j∈J Kjx⊗ej �� ∑ j′∈J Kj′y⊗ej′ ��� � = ∑ j, j′∈J trℓ2(J) � |Kjx⟩⟨Kj′y|⊗|ej⟩⟨ej′| � = ∑ j, j′∈J Kj|x⟩⟨y|K∗ j′⟨ej′,ej⟩ = ∑ j∈J Kj|x⟩⟨y|K∗ j = Φ(|x⟩⟨y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' In the second-to-last line we used that the partial trace (just like every quantum map) is con- tinuous (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2 in33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Hence this also holds for both Φ and trK (U((·) ⊗ |ψ⟩⟨ψ)U∗), mean- ing they coincide on all of B1(H ) because we showed that they coincide on the dense subset span{|x⟩⟨y| : x,y ∈ H } ⊆ B1(H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The only statement left to prove is the catalyst property (ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' this follows readily from the extension property in Lemma 1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that this construction works for any set of Kraus operators so the smallest ancilla one can get this way has dimension “twice the Kraus rank”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' However, in practice one could also choose a 7 “non-minimal” set of Kraus operators – if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=', the corresponding unitary is easier to implement in practice – or use a different (not fully general) construction altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Be aware that the Stinespring form (i) is not limited to quantum maps with same domain and co-domain: given Φ ∈ CPTP(H ,K ) one obtains an extension of Theorem 1 to arbitrary quantum maps via the auxilliary map ρ �→ |ψ′⟩⟨ψ′| ⊗ Φ(trK (ρ)) ∈ CPTP(H ⊗ K ) where ψ′ ∈ H is any unit vector, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' also Coro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 1 in33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (ii) exists equivalently in the Heisenberg picture: given any (Heisenberg) quantum channel Φ∗ – meaning Φ∗ is any completely positive, unital (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' identity preserving), and ultraweakly continuous map – one has Φ∗ = tr|ψ⟩⟨ψ|(U∗((·) ⊗ 1)U∗) where ψ,U are the same as in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 1 and tr|ψ⟩⟨ψ| is the partial trace w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' the state |ψ⟩⟨ψ| (Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='1 in32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' For more on the Stinespring form in the Heisenberg picture and its relation to Stinespring’s theorem for C∗-algebras we refer to Coro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' in33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' COMPARING CONSTRUCTIONS: AN EXAMPLE To better understand the different techniques for generating Stinespring forms let us illustrate and compare them via a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' While the following example will be finite-dimensional – hence the construction described in the introduction yields the smallest ancilla – such a choice will clarify how the two constructions intended for infinite dimensions circumvent the problems discussed in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Let Φ ∈ CPTP(n) be given such that {K1,K2} ⊂ Cn×n is a set of Kraus operators for Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' As explained in the introduction, the standard construction first collects K1,K2 in a 2n×2n-matrix: \uf8eb \uf8edK1 0 K2 0 \uf8f6 \uf8f8 (6) Due to K∗ 1K1 + K∗ 2K2 = 1 the first n columns form an orthonormal system in C2n which can be extended to an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Build U12,U22 ∈ Cn×n from the additional vectors such that UF := \uf8eb \uf8edK1 U12 K2 U22 \uf8f6 \uf8f8 is unitary, and one has Φ ≡ trC2(UF((·)⊗|e1⟩⟨e1|)U∗ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 8 Next, let us carry out the construction of Hellweg & Kraus (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Appendix A): similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (6) one starts with the isometry A := \uf8eb \uf8edK1 K2 \uf8f6 \uf8f8 ∈ C2n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (7) However, instead of completing it to a unitary by adding suitable columns – which cannot be guaranteed in infinite dimensions – they enlarge the ancilla in order to define UK := \uf8eb \uf8ed0 A∗ A −( 1−AA∗) \uf8f6 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 K∗ 1 K∗ 2 K1 K1K∗ 1 − 1 K1K∗ 2 K2 K2K∗ 1 K2K∗ 2 − 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This UK satisfies Φ ≡ trC3(UK((·)⊗|e1⟩⟨e1|)U∗ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Finally, the construction presented in our paper also starts with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (7) but then turns A into the “block matrix” from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (6) as part of the larger matrix UN := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed K1 0 1−K1K∗ 1 −K1K∗ 2 K2 0 −K2K∗ 1 1−K2K∗ 2 0 0 −K∗ 1 −K∗ 2 0 1 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Choosing the ancilla to be C2 ⊗C2 this unitary satisfies Φ ≡ trC2⊗C2(UN((·)⊗|e1⟩⟨e1|)U∗ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that a shuffled version of UK “appears” in UN (up to some minus signs which can be neglected) and the two matrices differ by an identity on a complementary subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Thus, while UN is larger than UK the “effective” dimension (in the sense that part of the space UN acts on is a catalyst w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' UN) is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This is also showcased here: the first column of the upper left block of UN features only the Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This is why UN(ρ ⊗|e1⟩⟨e1|)U∗ N = \uf8eb \uf8edK1ρK∗ 1 K1ρK∗ 2 K2ρK∗ 1 K2ρK∗ 2 \uf8f6 \uf8f8⊗|e1⟩⟨e1| = UF(ρ ⊗|e1⟩⟨e1|)U∗ F ⊗|e1⟩⟨e1| whereas UN(ρ ⊗|e1⟩⟨e1|)U∗ N = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 K1ρK∗ 1 K1ρK∗ 2 0 K2ρK∗ 1 K2ρK∗ 2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = 0⊕UF(ρ ⊗|e1⟩⟨e1|)U∗ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9 ACKNOWLEDGMENTS I would like to thank Jens Eisert for pointing out some references on Stinespring forms of bosonic channels which I was not aware of yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This work has been supported by the Einstein Foundation (Einstein Research Unit on Quantum Devices) and the MATH+ Cluster of Excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Appendix A: Original Proof of Hellwig and Kraus This appendix will revolve around the following statement, respectively the proof given by Hellwig and Kraus (originally in16, and in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 4 in23 or Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2 in24): Given any complex Hilbert space H and any quantum map Φ on H there exists a Hilbert space K , a unit vector ψ ∈ K , and a self-adjoint unitary operator U on H ⊗K such that Φ ≡ trK � U((·)⊗|ψ⟩⟨ψ)U∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' If (Kj) j∈J is a set of Kraus operators of Φ, then one can choose K to be ℓ2(J ∪{s}) where s is any symbol not in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' We note that their proof was given for separable Hilbert spaces H but extends without further ado to arbitrary Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Their construction goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Starting from a set of Kraus operators (Kj) j∈J for Φ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 in32) one first defines the following objects: Js := J ∪{s} where s is any symbol not in J K := ℓ2(Js) is the Hilbert space of all functions f : Js → C which are square-summable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' ∑j∈Js | f( j)|2 < ∞, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 & Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='12 in30 Hs := H =: H j for all j ∈ J ι : Hs ⊕ � j∈J H j → H ⊗ K is the isometric isomorphism defined via xs ⊕ � j∈J xj �→ ∑j∈Js xj ⊗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Hence Hs ⊕ � j∈J H j ∼= H ⊗K , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='8 in30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' The idea now is to define an isometry A : H → � j∈J H j, embed it into a unitary operator U0 on Hs⊕� j∈J H j, and finally use ι to translate U0 into a unitary operator U on H ⊗K as visualized 10 in the following diagram: H ⊗ℓ2(Js) H ⊗ℓ2(Js) Hs ⊕ � j∈J H j Hs ⊕ � j∈J H j U ι−1 U0 ι They started by defining A : Hs → � j∈J H j via Ax := � j∈J Kjx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' One readily verifies that A is an isometry (so in particular well-defined) because ∥Ax∥2 = ∑ j∈J ∥Kjx∥2 = ∑ j∈J ⟨x,K∗ j Kjx⟩ = ∥x∥2 as ∑j∈J K∗ j Kj → 1 in the strong operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' With this they defined U0 via34 U0 : Hs ⊕ � j∈J H j → Hs ⊕ � j∈J H j \uf8eb \uf8edx y \uf8f6 \uf8f8 �→ \uf8eb \uf8ed A∗y Ax−( 1−AA∗)y \uf8f6 \uf8f8 = \uf8eb \uf8ed0 A∗ A −( 1−AA∗) \uf8f6 \uf8f8 \uf8eb \uf8edx y \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Evidently, U0 is a self-adjoint involution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' hence U0 is unitary and so is the “translated” operator U := ι ◦U0 ◦ι−1 on H ⊗K (because ι is a unitary transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Note that U(x⊗es) = (ι ◦U0) \uf8eb \uf8edx 0 \uf8f6 \uf8f8 = ι \uf8eb \uf8ed 0 Ax \uf8f6 \uf8f8 = ι \uf8eb \uf8ed 0 � j∈J Kjx \uf8f6 \uf8f8 = ∑ j∈J Kjx⊗ej (A1) for all x ∈ Hs = H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Defining ψ := es ∈ ℓ2(Js) one for all x,y ∈ H finds trℓ2(Js) � U(|x⟩⟨y|⊗|es⟩⟨es|)U∗� = trℓ2(Js) � |U(x⊗es)⟩⟨U(y⊗es)| � (A1) = trℓ2(J) ����∑ j∈J Kjx⊗ej �� ∑ j′∈J Kj′x⊗ej′ ��� � = ∑ j, j′∈J trℓ2(J) � |Kjx⟩⟨Kj′x|⊗|ej⟩⟨ej′| � = ∑ j, j′∈J Kj|x⟩⟨y|K∗ j′⟨ej′,ej⟩ = ∑ j∈J Kj|x⟩⟨y|K∗ j = Φ(|x⟩⟨y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' A standard continuity argument shows that Φ ≡ trK (U((·)⊗|ψ⟩⟨ψ)U∗) on all of B1(H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' REFERENCES 1A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' vom Ende, “Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space,” (2022), accepted to Open Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=', arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='08351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 19W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Stinespring, “Positive Functions on C∗-Algebras,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 6, 211–216 12 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 20T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Heinosaari and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ziman, The Mathematical Language of Quantum Theory: From Uncer- tainty to Entanglement (Cambridge University Press, Cambridge, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 21However, be aware that the existence of Kraus operators—in particular in infinite dimensions— is usually proven via Stinespring’s dilation theorem for C∗-algebras, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 in32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 22G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ludwig, Foundations of Quantum Mechanics I (Springer, New York, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 23K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Kraus, “Operations and Effects in the Hilbert Space Formulation of Quantum Theory,” in Foundations of Quantum Mechanics and Ordered Linear Spaces (Springer, Berlin, 1973) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 206–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 24K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Kraus, States, Effects, and Operations, Lecture Notes in Physics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 190 (Springer, Berlin, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 25G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Pólya, How to Solve It: A New Aspect of Mathematical Method, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (Princeton University Press, Princeton, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 26F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Riesz and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='-Nagy, Functional Analysis, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (Dover, New York, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 27Actually in26 it is proven that \uf8eb \uf8ed T √ 1− TT ∗ −√ 1− T∗T T ∗ \uf8f6 \uf8f8 is a unitary dilation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' However, multiplying this from the left with the unitary 1⊕(−1) yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 28C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Foias and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Frazho, The Commutant Lifting Approach to Interpolation Problems (Birkhäuser, Basel, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 29Because M is closed by assumption, it is a Hilbert space itself (Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3 in31) which implies that PM as well as V ∗ 0 are well-defined (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='5 & Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='2 in30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 30R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Kadison and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Ringrose, Fundamentals of the Theory of Operator Algebras, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 1: Elemen- tary Theory (American Mathematical Society, Providence, Rhode Island, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 31R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Meise and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Vogt, Introduction to Functional Analysis, Oxford Graduate Texts in Mathe- matics (Oxford University Press, Oxford, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 32E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 33F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' vom Ende and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Dirr, “Unitary Dilations of Discrete-Time Quantum-Dynamical Semi- groups,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 60, 122702 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 34Originally, Hellwig and Kraus considered general quantum operations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' 4 in20 meaning the Kraus operators only need to satisfy ∑j∈J K∗ j Kj ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' This made their construction of U0 a bit more involved (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content='3) in23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE5T4oBgHgl3EQfNQ6I/content/2301.05488v1.pdf'} +page_content=' however, as we are only 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Let φ be an L2-normalized Hecke–Maaß cusp form for PGLn(Z[i]) on the locally symmetric space +X := PGLn(Z[i])\PGLn(C)/PUn. If Ω is a compact subset of X, then we prove the bound ∥φ|Ω∥∞ ≪Ω +λn(n−1)/4−δ +φ +for some δ > 0 depending only on n, where λφ is the Laplace eigenvalue of φ. +1. Introduction +Given a Riemannian locally symmetric space X = Γ\S, it is a classical analytic problem to give pointwise +bounds for (L2-normalized) eigenfunctions F of the algebra of invariant differential operators D(S), uniformly +in S in terms of the Laplace eigenvalue λF of F. If X is compact, then Sarnak [Sar] proved the baseline +bound +(1) +∥F∥∞ ≪X λ +dim(X)−rk(X) +4 +F +. +The exponent on the right-hand side is known to be sharp in general. It is also known in some special cases +that if X is not compact, then ∥F∥∞ might be considerably larger (see [BT20]), but (1) still holds if F is +retsriced to compact subsets of X. The sup-norm problem in the theory of automorphic forms asks if the +exponent of (1) can be strengthened if X is an arithmetic manifold and F is an eigenfunction not only of +D(S) but of the full Hecke algebra of X. +An important motivation comes from quantum mechanics. Classical mechanics interprets a freely moving +particle as a geodesic flow. The quantum mechanical interpretation of the same object is an L2-normalized +linear combination of eigenstates. Since the geodesic flow is ergodic with respect to the Liouville measure +on the tangent space, the correspondence principle of quantum mechanics suggests that the masses of the +eigenstates reproduce the invariant measure in the high-energy limit. Bounds on the sup-norm (or in general, +the Lp norm for any p > 2) of eigenstates control their mass concentration, and hence are in connection with +the quantum unique ergodicity conjecture of Rudnick and Sarnak [RS94]. Other important connections of +sup-norm bounds are towards the multiplicity problem, nodal domains of automorphic forms and bounds +for L-functions, see e.g. [Sar], [GRS13], [BH10]. +In improving (1), there are (at least) two independent directions of research: one is to find as strong power- +savings as possible among special circumstances, typically in rank one (see e.g. [IS95] and [BHMM20]), the +other one is to find any power-saving in higher rank or among as general circumstances as possible (see +e.g. [BP16], [BM15], [BM16]). We pick up the thread in the second theme, where the current limitation +of our knowledge is an unpublished a manuscipt [Mar], which proves power-saving for a wide class of sym- +metric spaces, more specifically, for arithmetic quotients of quasi-split groups with the exception of the type +SU(n, n − 1). (We note that even though [Mar] is not peer-reviewed, it is widely accepted in the community +as being correct.) +In these notes, we introduce a new method, based and improved on that of [BM16], to tackle the sup-norm +problem. As a test application, we prove a saving over (1) in the case of the locally symmetric space +X := Γ\G/K, +Γ := PGLn(Z[i]), +G := PGLn(C), +K := PUn. +Admittedly, our result follows from the main result of [Mar] (see especially [Mar, Corollary 1.4]), but we +strongly believe that the novelty of the method deserves attention, and might have the potential to address +the type SU(n, n − 1), the exceptional case in Marshall’s work. +2020 Mathematics Subject Classification. 11F55, 11F72, 11D75. +Key words and phrases. automorphic forms in higher rank, sup-norm problem, Hecke operators, trace formula, diophantine +approximation, amplification. +1 + +All along the paper, we think of n ⩾ 2 as being fixed, in particular, all implied constants below are allowed +to depend on n. +Before stating our main result, we fix some notations. Since it is convenient to work with matrices instead +of their projectivization, we shall realize elements of G as the rightmost nonzero entry of the bottom row is +1. We will also talk about points of X or G/K as matrices, by which we mean any matrix which represents +them. +Let A stand for the diagonal subgroup of G consisting of positive real entries and let N be the +upper-triangular unipotent subgroup. Then the Iwasawa decomposition reads as G = NAK. We write a for +the Lie algebra of A, a∗ for its dual and a∗ +C for the complexification of a∗. Fixing bases in a∗ and a∗ +C, we +may view them as appropriate subsets of Rn and Cn, respectively. Let W stand for the Weyl group, Σ for +the set of roots and Σ+ for the set of positive roots corresponding to N. For α ∈ Σ, let m(α) be the real +dimension of the corresponding root space. For λ ∈ a∗, define +D(λ) := +� +α∈Σ+ +(1 + |⟨α, λ⟩|)m(α), +where ⟨·, ·⟩ is induced by the Killing form. +In our special situation, define, for 1 ⩽ j ⩽ n, the function ej on a as ej(diag(a1, . . . , an)) = aj, and then +a set of positive roots is given by ej − ek with 1 ⩽ j < k ⩽ n. The corresponding root space is spanned +by the matrices Ejk and iEjk, where Ejk is the matrix which 1 at position j, k and otherwise zero, hence +m(ej − ek) = 2. +A Hecke–Maaß cusp form φ on X comes with archimedean Langlands parameters µφ = (µ1, . . . , µn) ∈ +a∗ +C/W ⊂ Cn/W such that µ1 + . . . + µn = 0 and {µ1, . . . , µn} = {µ1, . . . , µn}. In our parametrization, +(µ1, . . . , µn) ∈ Rn corresponds to the tempered spectrum, and we have that ℑµ1, . . . , ℑµn = O(1). Let µ∗ +φ +stand for ℜµφ. Since for the Laplace eigenvalue λφ of φ, +λφ ≍ 1 + ∥µ1∥2 + . . . + ∥µn∥2 +holds, we have +D(µ∗ +φ) ≪ λ +n(n−1) +2 +φ +. +The main result of this paper is the following. +Theorem 1. For any n ⩾ 2, there exists some δ = δ(n) > 0 with the following property. For any L2- +normalized Hecke–Maaß cusp form φ on X and any compact Ω ⊂ X, +∥φ|Ω∥∞ ≪Ω D(µ∗ +φ) +1 +2 −δ. +In particular, +∥φ|Ω∥∞ ≪Ω λ +n(n−1) +4 +(1−2δ) +φ +. +Since n(n − 1)/4 = (dim(X) − rk(X))/4, this is a saving over (1). We note that the case n = 2 was solved +in [BHM16]. +Our method closely follows the one introduced in [BM16], however, the counting problem in this situation +is more challenging (because of the too large maximal compact subgroup of G, as it will be briefly exposed +below). This requires an improvement in the counting techniques, which is the heart of this paper (vaguely +speaking, a combination of Lemma 4 and Lemma 8). As a by-product, there is no need of Linnik type +theorems about small primes in arithmetic progressions (i.e. zero repulsion of L-functions) any more. +We remark that with some work, the jungle of O(1)’s below (particularly in Section 4) can be made +explicit, hence an explicit subconvexity saving is available, see the work [Gil20] over the rational field (in +fact, now it is easier, since the reference to Linnik type theorems is removed). We also note that the implied +constant (which depends on Ω) can be also made fully explicit, so the method in principle is effective, e.g. +we do not need any reference to Siegel zeroes. +Acknowledgements. We thank Valentin Blomer for useful discussions about the topic of the paper. We +also thank Gergely Harcos and Vitezslav Kala for discussions about some closely related questions. +The research towards this work was supported by NKFIH (National Research, Development and In- +novation Office) Grants KKP 133819 (PM), FK 135218 (PM), FK 127906 (GZ), K 135885 (GZ), ELKH +(E¨otv¨os Lor´and Research Network) Grant SA-71/2021 (PM & GZ), and the MTA R´enyi Int´ezet Lend¨ulet +Automorphic Research Group (PM & GZ). +2 + +2. Reduction to a counting problem +In this section, we reduce the problem to a matrix counting problem, following the lines of [BM15] and +[BM16], where PGLn(Z)\PGLn(R)/POn was treated. Since the spherical Hecke algebra is isomorphic in +that case to ours at the archimedean and all relevant non-archimedean places, almost everything follows +verbatim, so we give only a brief exposition, with some emphasis on the small difference coming from the +fact that the maximal compact subgroup is the orthogonal group in the real case and the unitary group in +the complex case. +Let fµφ : K\G/K → C be the spherical function constructed in [BM16, pp.1276–1277] (here we utilize +that A, a∗, a∗ +C are the same for PGLn(R) and PGLn(C)). Denoting by ˜fµφ its spherical transform, for any +x, y ∈ G, we have the pre-trace formula +(2) +� +˜fµφ(µ̟)F̟(x)F̟(y) d̟ = +� +γ∈Γ +fµφ(x−1γy), +where the integral on the left is meant over the full spectrum of Γ\G. +The idea of amplification in the current setup can be summarized as follows. Assume we want to estimate +our form φ at a point g ∈ Ω. Then in (2), we set x := g and we take y of the form γg, where γ runs through +right coset representatives corresponding to certain Hecke operators. An appropriate weighted sum of these +results in a positive operator on L2(X), hence from the left-hand side, all but the one term corresponding to +F̟ = φ can be dropped (for this positivity argument, see also [BHMM20, Section 3]). All in all, we arrive +at a bound of the form +C|φ(g)|2 ⩽ +� +γ∈Γ′ +fµφ(g−1γg) +with Γ′ being a finite set of matrices arising from the Hecke operators utilized, and C is a positive constant +coming from ˜fµφ and the corresponding Hecke eigenvalues. Then, using that fµφ decays controllably away +from K (see [BP16, Theorem 2]), to bound |φ(g)|, we essentially have to count those γ ∈ Γ′ for which g−1γg +is close to K. +To be a little more concrete, let L ⩾ 2 be a parameter, and assume that P is a set of primes π lying above +distinct split rational primes such that L ⩽ N(π) < 2L with N standing for the norm. Then the discussion +of [BM15, Sections 4, 6] (which in fact uses [BP16, Theorem 2], and see also [BM16, Section 2]) leads to, for +any number M ⩾ 1, +(3) +|φ(g)|2 ≪M,Ω D(µ∗ +φ) · + + 1 +#P + D(µ∗ +φ)−κLK + +n +� +ν=1 +1 +(#P)2 +� +π,π′∈P +#S(Q, πν, π′ν, M) +Lν(n−1) + + , +g ∈ Ω, +where K is a fixed number depending only on M and n, κ > 0 is also a fixed number depending only on n; +and +S(Q, πν, π′ν, M) := +� +γ ∈ SLn(Z[i]) · diag(1, πν, . . . , πν, πνπ′ν) · SLn(Z[i]) : +∥| det γ|− 2 +n · γ∗Qγ − Q∥∞ ⩽ max +� +N(π)−M, N(π′)−M�� +(4) +with Q := | det g|2/n · (g∗)−1g−1, and by ∥ · ∥∞ applied to a matrix, we mean its largest entry in absolute +value. Note that in [BM15] and [BM16], the transpose of γ and g are taken instead of their adjoint. +By the prime number theorem, we can choose P to satisfy #P ≫ε L1−ε for any ε > 0. Therefore, if we +were able to prove that +#S(Q, πν, π′ν, M) ≪Ω,M Lν(n−1)−η +holds for every 1 ⩽ ν ⩽ n with some η > 0, then (3) would imply Theorem 1 by setting L to be a very small +positive power of D(µ∗ +φ). We prove it in a weaker form, which still suffices for Theorem 1. +Before formulating this weaker statement, we fix some notation. Introduce the notation Sn for the real +vector space of self-adjoint matrices in Cn×n and Pn ⊂ Sn for the open convex cone of positive definite +matrices. Given a compact set Ω ⊂ X as in Theorem 1, we introduce +Ω′ := +��1 +2, 2 +� +· | det g| +2 +n · (g∗)−1g−1 : g ∈ Ω +� +. +3 + +Then Ω′ is a compact subset of Pn in the subspace topology. +Proposition 1. Let ε > 0 be arbitrary. There exist positive numbers α(ε) ⩾ 1 and M(ε) ⩾ 1 (both depending +only on n and ε) with the following properties. For any compact Ω ⊂ X, there exists a constant L(Ω) ⩾ 2 +(depending only on n, α(ε), M(ε) and Ω) such that the following holds. For any L0 ⩾ L(Ω), there exists some +L0 ⩽ L ⩽ Lα(ε) +0 +such that for any prime π lying above a split rational prime and satisfying L ⩽ N(π) < 2L, +we have +(5) +#S(Q, πν, πν, M(ε)) ≪Ω,ε Lν(n−1)+ε, +Q ∈ Ω′, 1 ⩽ ν ⩽ n; +and for any two distinct primes π, π′ lying above distinct split rational primes, we have +(6) +#S(Q, πν, π′ν, M(ε)) = 0, +Q ∈ Ω′, 1 ⩽ ν ⩽ n. +This still suffices for the proof of Theorem 1. Indeed, we apply Proposition 1 with any 0 < ε < 1/2. +There are implied numbers α := α(ε) ⩾ 1 and M := M(ε) ⩾ 1, and together with the further input Ω, +one more number L(Ω) by Proposition 1. Then take L0 := D(µ∗ +φ)ω, where ω > 0 is a fixed constant to be +specified later. By Proposition 1, if L0 ⩾ L(Ω), we can find some with D(µ∗ +φ)ω ⩽ L ⩽ D(µ∗ +φ)αω such that +the countings (5) and (6) hold for any primes L ⩽ N(π), N(π′) < 2L. Then in every term of (3), we get a +power-saving, as soon as ω > 0 in the beginning is chosen sufficiently small. Now we return to the condition +L0 ⩾ L(Ω). Apart from a finite set of Hecke–Maaß cusp forms depending only on Ω, this is indeed satisfied +by L0 = D(µ∗ +φ)ω. The finitely many exceptional forms are treated then by adjusting the implied constant +(note that the implied constant might depend on Ω, only the saving in the exponent must be absolute). +The rest of the paper is hence devoted to the proof of Proposition 1. +We conclude this section by illustrating why this counting problem is harder for PGLn(Z[i]) than for +PGLn(Z) in the special case when Q is the unit matrix. Then in S(. . . ), we have a condition on the Smith +normal form which is of the same complexity in both cases. +However, the other condition is that γ is +projectively equivalent to an orthonormal matrix, which intuitively happens more often over C than over R, +since +dim(PGLn(C)) = 2n2 − 2, +dim(PUn) = n2 − 1 = dim(PGLn(C)) +2 +, +while +dim(PGLn(R)) = n2 − 1, +dim(POn) = n2 − n +2 +< dim(PGLn(R)) +2 +, +i.e. PUn is a “thicker” subgroup of PGLn(C) than POn of PGLn(R). +3. Counting techniques in a special case +In this section, we introduce counting techniques for a situation which is very special in many different +aspects. First, we will assume that Q is diagonal, moreover, its entries belong to the base field Q(i). Then +these together imply that the diagonal entries are rational, i.e. Q = diag(q1, . . . , qn) with q1, . . . , qn ∈ Q. +Then qj ≍Ω 1 for all 1 ⩽ j ⩽ n, and we also assume that the numerator and the denominator of qj for all +1 ⩽ j ⩽ n (in their simplest form) are both coprime to π, π′. Our final simplification is that we allow no +error term in (4), which we will denote by writing ∞ in place of M. +Our convention will be that vectors, i.e. elements of Z[i]n, Cn, etc. are always meant as column vectors. +We also introduce the notation vπ(q), for any prime π ∈ Z[i] and any q ∈ Q(i), which denotes the π-valuation +of q. Occasionally, we may use this notation for vectors or matrices with entries from Q(i), then it means +the minimal π-adic valuation attained by the entries. +Lemma 1. Let π ∈ Z[i] be a prime lying above a split rational prime p = ππ, ρ ∈ N, and let A ∈ Q(i)n×n +be a self-adjoint matrix such that vπ(A) ⩾ 0. Let x = (ξ1, . . . , ξn)t, y = (υ1, . . . , υn)t ∈ Z[i]n be vectors +satisfying vπ(x) = vπ(y) = 0 such that for any 1 ⩽ j < k ⩽ n, πρ | (ξjυk − ξkυj). Then the following +statements hold. +(a) For some a ∈ Z coprime to p, we have y ≡ ax mod πρ, and this a is well-defined modulo pρ. +(b) Choosing a ∈ Z with this property in (a), there also exists a b ∈ Z (unique modulo pρ) with b ≡ ai mod πρ, +so we further have +(7) +2x∗Ay ≡ (a − bi)x∗Ax + (a′ − b′i)y∗Ay +mod pρ, +4 + +where a′ ∈ Z (resp. b′ ∈ Z) stands for the multiplicative inverse of a (resp. of b) modulo pρ. +(c) We have πρ | a′ − b′i and πρ | a − bi for the integers a, b, a′, b′ defined above. +Proof. This is a variant of [BM16, Lemma 3], but the proof in this case is a little more computational. +Without loss of generality, we may assume that π ∤ ξn. Then π ∤ υn, for if not, then for some 1 ⩽ j ⩽ n − 1, +π ∤ υj, and then π ∤ (ξnυj − ξjυn). Then a ≡ υnξ−1 +n +mod πρ does the job, since υj ≡ ξjυnξ−1 +n +≡ aξj mod πρ +for any 1 ⩽ j ⩽ n − 1. Also, a can be chosen in Z uniquely modulo pρ since 0, 1, . . . , pρ − 1 is a set of +representatives modulo πρ (i.e. we have the isomorphism Z/(pρ) ∼= Z[i]/(πρ)). The proof of (a) is complete. +As for (b), fix any representative a ∈ Z and let b ∈ Z be such that b ≡ ai mod πρ. Then πρ divides all +the entries in y − ax whence πρ divides all the entries in (y − ax)∗. So we deduce +0 ≡ (y − ax)∗A(y − ax) = y∗Ay + a2x∗Ax − ax∗Ay − ay∗Ax +mod pρ = (ππ)ρ +hence +(8) +x∗Ay + y∗Ax ≡ ax∗Ax + a′y∗Ay +mod pρ +where a′ ∈ Z is a multiplicative inverse of a modulo pρ. Similarly, πρ divides all the entries in iy − bx and +πρ divides all the entries in (iy − bx)∗ as ai ≡ b mod πρ. So we compute +0 ≡ (iy − bx)∗A(iy − bx) = y∗Ay + b2x∗Ax − bix∗Ay + biy∗Ax +mod pρ +whence +(9) +x∗Ay − y∗Ax ≡ −bix∗Ax − b′iy∗Ay +mod pρ +where b′ ∈ Z is a multiplicative inverse of b modulo pρ. The statement follows by adding equations (8) and +(9). +For part (c), we compute πρ | i(b − ai) = ib + a whence πρ | a + bi = a − bi. Similarly, πρ | a′b′(b − ai) ≡ +a′ − b′i. +□ +Lemma 2. Let A ∈ Ω′, and assume that x1, . . . , xk ∈ Z[i]n are linearly independent vectors for some +0 ⩽ k ⩽ n − 1. Then for any real number β ⩾ 2, +#{y ∈ Z[i]n : y∗Ay = β2, and x∗ +jAy = 0 for any j = 1, . . . , k} ≪Ω,ε β2(n−k−1)+ε +for any ε > 0. +Proof. See [BM15, Corollary 5.3]. +□ +We slightly extend the notation S(Q, πν, π′ν, ∞) for π, π′ ∤ m ∈ Z[i] as +Sm(Q, πν, π′ν, ∞) := +� +γ ∈ SLn(Z[i]π,π′) · diag(1, πν, . . . , πν, πνπ′ν) · SLn(Z[i]π,π′) : +mγ ∈ SLn(Z[i]), | det γ|− 2 +n · γ∗Qγ = Q +� +, +where by Z[i]π,π′, we mean the ring of elements a ∈ Q(i) which satisfy vπ(a), vπ′(a) ⩾ 0. +Lemma 3. Let π ∈ Z[i] be a prime lying above a split rational prime, and π ∤ m ∈ Z[i]. +Let Q = +diag(q1, . . . , qn) with qj ∈ Q, qj ≍Ω 1 for all 1 ⩽ j ⩽ n. Then for any 1 ⩽ ν ⩽ n, +#Sm(Q, πν, πν, ∞) ≪Ω,ε |m|2n2−2+ε den(Q) +(2n−1)(n−1) +2 +N(π)ν(n−1)+ε. +for any ε > 0. Here, den(Q) denotes the least common multiple of the denominator of the diagonal entries +of Q (written in simplest form). +Proof. By the definition of Sm(Q, πν, πν, ∞), any matrix counted there must have a column not completely +divisible by π, and we may assume that it is the first column γ1. Since (mγ1)∗Q(mγ1) = |m|2N(π)ν, we +have, by Lemma 2 that the number of possible mγ1’s is Oε(|m|2(n−1)+εN(π)ν(n−1)+ε). Now it suffices to +prove that fixing γ1, there are OΩ(m2n(n−1) den(Q)(2n−1)(n−1)/2) ways to finish the matrix. +We group the possible second columns γ2 according to their π-valuation. +When vπ(γ2) ⩾ ν, then +mγ2/πν ∈ Z[i]n and ∥mγ2/πν∥ ≍Ω ∥mγ2/πν∥Q = |m|, hence there are OΩ(|m|2n) such choices for such +a γ2. Now fix 0 ⩽ µ < ν, and then it suffices prove that the number of possible γ2’s satisfying vπ(γ2) = µ is +OΩ(|m|2n den(Q)(2n−1)/2), since the same can be repeated for all further columns. +5 + +Let x, y be second columns after the first column γ1 such that vπ(x) = vπ(y) = µ. Consider then the +vectors x′ := mx/πµ and y′ := my/πµ. By Lemma 1(a), x′ and y′ are both multiples of mγ1 modulo +πν−µ, and then by transitivity, multiples of each other modulo πµ−ν. Introduce then Q′ := den(Q)Q. With +this notation, since x′∗Q′x′ = y′∗Q′y′ = |m|2q2 den(Q)πν−µπν−µ = |m|2q2 den(Q)N(π)ν−µ, (7) gives that +x′∗Q′y′ is divisible by N(π)ν−µ. On the other hand, viewing Cn as R2n, we obtain, for the Q-angle of x and +y that +∢Q(x, y) = arccos +ℜ(x∗Qy) +� +(x∗Qx)(y∗Qy) += arccos +ℜ(x′∗Q′y′) +� +(x′∗Q′x′)(y′∗Q′y′) += arccos +ℜ(x′∗Q′y′) +|m|2q2 den(Q)N(π)µ−ν . +Since the numerator in the rightmost expression is divisible by N(πµ−ν), this continues, for some ℓ ∈ +{0, 1, 2, . . .}, as +∢Q(x, y) = arccos +� +1 − +ℓ +|m|2q2 den(Q) +� +≫Ω |m|−1 den(Q)−1/2, +if ℓ ̸= 0, i.e. x ̸= y. +This means that the number of possible second columns is OΩ(|m|2n−1 den(Q)(2n−1)/2) (this is rather ele- +mentary, see also [BM16, Lemma 4]). The proof is complete. +□ +Lemma 4. Let π, π′ ∈ Z[i] be distinct primes lying above distinct split rational primes, and let m ∈ Z[i] +such that vπ(m) = vπ′(m) = 0. Let further Q = diag(q1, . . . , qn) such that qj ∈ Q and vπ(qj) = vπ′(qj) = 0 +for all 1 ⩽ j ⩽ n. Then for any 1 ⩽ ν ⩽ n, +#Sm(Q, πν, π′ν, ∞) = 0. +Proof. First observe that if 1 ⩽ ν ⩽ n − 1, then +| det γ| +2 +n = (ππ) +ν(n−1) +n +� +π′π′� ν +n /∈ Q, +hence | det γ|2/nγ∗Qγ ̸= Q, implying #Sm(Q, πν, π′ν, ∞) = 0 in this case, so from now on, we assume ν = n. +Any γ counted in #Sm(Q, πn, π′n, ∞) must have a column, say, the first one γ1 such that vπ(γ1) = 0. +Now we prove that any choice of γ2 leads to a contradiction. Let µ := vπ(γ2) ⩾ 0. Then we apply Lemma 1, +in particular, (7) for γ1 and γ′ +2 := γ2/πµ, with the notation p = ππ, a, b ∈ Z coprime to p such that +1 ⩽ a, b, a′, b′ ⩽ pn−µ − 1, γ′ +2 ≡ aγ1 mod πn−µ, b ≡ ai mod πn−µ, a′, b′ are the multiplicative inverse of a, b +modulo pn−µ, respectively. We infer +(10) +0 = π−µγ∗ +1Qγ2 = γ∗ +1Qγ′ +2 ≡ (a − bi)pn−1π′π′q1 + (a′ − b′i)pn−1−µπ′π′q2 +mod pn−µ, +which is immediately a contradiction for µ ⩾ 1, since the second term in the rightmost expression is not +divisible by pn−µ, while the first one and the sum are. +Now assume µ = 0. By Lemma 1(c) we have π | a − bi and π | a′ − b′i. So in (10) the first term on the +right-hand side is not divisible by πn (since π ∤ a−bi as p ∤ a−bi), while the second one and the sum are. +□ +4. Exchanging matrices +Recall that we want to prove the countings (5) and (6) for N(π), N(π′) ∈ [L, 2L) with an appropriate +choice of L ⩾ L0 not exceeding a fixed power of L0. Our strategy is to switch Q for other matrices with +better and better arithmetic properties. Note that a priori, Q is a point on a real manifold with entries that +might be highly transcendental. We first informally describe these switches. First we will write Q1 in place +of Q in order to guarantee that +S(Q, πν, π′ν, M) ⊆ S(Q1, πν, π′ν, ∞) +for every admissible choice of π, π′, ν satisfying that N(π), N(π′) are between two fixed powers of L0. This +new Q1 will have entries in a number field K ⊇ Q(i). Using this Q1, we will be able to show that in an +appropriate subinterval (again, the norms are between two fixed powers of L0), all the γ’s correspond to +π = π′ or ν = n. +Secondly, in this subinterval, we will find a Q2, this time satisfying that +S(Q, πν, π′ν, M) ⊆ S(Q2, πν, π′ν, ∞) +for every admissible choice of π, π′, ν satisfying that N(π), N(π′) fall into an even shorter subinterval (again, +between two powers of L0). This new Q2 will have entries in Q(i), and we will have a control on their height. +6 + +Finally, we will diagonalize Q2 into Q3, again, with a control on the height of the entries, which necessarily +will be rational. This diagonalization process will affect the counted γ’s themselves, but not their number, +and we will have a good control on the possible denominators of the newly counted γ’s, i.e. +#S(Q, πν, π′ν, M) ⩽ #S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞) +for π, π′, ν as above. Then to the rightmost count here, we will apply Lemmata 3–4, which in fact will be of +the quality of (5) and (6). +Now we carry out this plan in detail. First we formulate a statement on effective computability. The field +of algebraic numbers is denoted by Q. Given an algebraic number a, we define the complexity of a as +comp(a) := +inf +a= b +c +b,c∈OQ(a) +� +max +σ∈HomQ(Q(a),C) |σ(b)| + +max +σ′∈HomQ(Q(a),C) |σ′(c)| +� ++ 1 +with OQ(a) standing for the ring of integers of Q(a). +Lemma 5. Let a1, . . . , am be algebraic numbers and put K := Q(a1, . . . , am). Assume that f : Q +m → Q is a +function of m variables that is computed altogether by t additions, subtractions, multiplications and divisions. +Then we have +comp(f(a1, . . . , am)) ⩽ +� +max +1⩽j⩽m(comp(aj)) +�Om,t,deg(K:Q)(1) +. +Proof. See the first paragraph of the proof of [BM16, Lemma 5]. +□ +Include Ω′ in some Ω1 which is still a compact subset of Pn in such a way that Ω′ ⊂ int(Ω1). We make +this inclusion in a well-defined way, say, Ω1 is the set of self-adjoint, positive definite matrices of eigenvalues +between a/2 and 2b, where a, b are respectively the smallest and the largest eigenvalues of matrices in Ω′. +Then dist(Ω′, Pn \ Ω1) ≫Ω 1, where by the distance of matrices, we mean the one induced by the maximum +of the entrywise distance. +Given any matrix γ ∈ GLn(C), we define the following linear transformation Bγ : Sn → Sn: +Bγ(A) := γ∗Aγ − | det(γ)| +2 +n A, +A ∈ Sn. +For any 0 < C1 < C2, we introduce the notation P(C1, C2) for the set of primes π ∈ Z[i] satisfying that +ℜπ, ℑπ > 0 and C1 ⩽ N(π) < C2 (then in particular, elements of P(C1, C2) lie above distinct split rational +primes). Another notation we introduce is T ⩾ 1, a fixed number depending only on n, which exceeds all +the implicit constants in O(1)’s ever mentioned in the paper. +Lemma 6. Let D ⩾ 2 and E ⩾ 2 be arbitrary. Then there exist some M ⩾ 2 (depending on n, D, E) and +L1(Ω) ⩾ 2 (depending on n, D, E, M, Ω) such that the following holds. For any Q ∈ Ω′, there exists some +1 ⩽ j ⩽ n2 + 1 satisfying that +S(Q, πν, π′ν, M) = ∅ +for all π, π′ ∈ P(2L(DE)j +0 +, 2L(DE)j+1 +0 +), unless π = π′ or ν = n. +Proof. Fix Q ∈ Ω′. For any L0 ⩾ 2, consider the subspaces (for the moment, with some unspecified M ⩾ 2) +Hj := +� +γ∈S(Q,πν,π′ν,M) +π,π′∈P(L0,2L(DE)j +0 +) +1⩽ν⩽n +ker Bγ, +j = 1, . . . , n2 + 2. +Then Sn ⊇ H1 ⊇ . . . ⊇ Hn2+2 ⊇ {0} and dim(Sn) = n2 imply that Hj = Hj+1 for some 1 ⩽ j ⩽ n2 + 1. Fix +the smallest such j. Since dim Sn −dim Hj ⩽ n2, we in fact obtain Hj by intersecting only n2 many ker Bγ’s, +say, Hj = ∩t +ℓ=1 ker Bγℓ with some t ⩽ n2. Since each entry of each such γℓ has complexity OΩ((L(DE)j +0 +)O(1)), +Hj is defined via a system of linear equations with entries from K = Q(i, π1/n +1 +, . . . , π1/n +2n2 ) of complexity +OΩ((L(DE)j +0 +)O(1)) by Lemma 5 (since the linear system defining Hj can be computed in O(1) steps from the +used γ’s). +7 + +By assumption, dist(Q, ker Bγ) ≪ L−M +0 +, where by dist, we mean the distance in Sn as a real vector space +of dimension n2. We claim that this implies that dist(Q, Hj) = OΩ((L(DE)j +0 +)O(1)L−M +0 +). Indeed, take basis +matrices V1, . . . , Vm of the linear span of the (ker Bγℓ)⊥’s such that each Vℓ′ is in one of the ker Bγℓ’s, its +entries are in K of complexity OΩ((L(DE)j +0 +)O(1)) (this can be done, because such a basis can be computed +from the γℓ’s, and then Lemma 5 applies). +Such a basis can be orthogonalized by Gram–Schmidt into +V ′ +1, . . . , V ′ +m, and then +dist(Q, Hj) = ∥ projH⊥ +j (Q)∥ = +����� +m +� +ℓ′=1 +⟨Q, V ′ +ℓ′⟩ +⟨V ′ +ℓ′, V ′ +ℓ′⟩V ′ +ℓ′ +����� = +����� +m +� +ℓ′=1 +�m +ℓ′′=1 Uℓ′ℓ′′⟨Q, Vℓ′′⟩ +⟨V ′ +ℓ′, V ′ +ℓ′⟩ +V ′ +ℓ′ +����� , +where U is the matrix standing for the Gram–Schmidt process. Since ⟨Q, Vℓ′′⟩ ≪ L−M +0 +(because for every +ℓ′′, we have Vℓ′′ ∈ ker Bγℓ for some ℓ), and all the coefficients (including the entries of U, by Lemma 5) are of +complexity OΩ((L(DE)j +0 +)O(1)), we indeed see that dist(Q, Hj) = OΩ((L(DE)j +0 +)O(1)L−M +0 +), applying Lemma 5 +again. +Now if M is large enough, say, +(11) +M ⩾ T · (DE)n2+2 + 1, +where recall that T ⩾ 1 is an upper bound on all the implicit constants in O(1)’s ever mentioned in the +paper, then dist(Q, Hj) = OΩ(L−1 +0 ) (since j ⩽ n2 + 1, the exponent n2 + 2 seems to be an overkill, but for a +later reference, it is better to force M even a slightly larger). Then with the convenient choice of L1(Ω) ⩾ 2, +if L0 ⩾ L1(Ω), then Hj ∩ int Ω1 ̸= ∅, in particular, Hj ̸= {0}. Further, since Hj is defined over K, we may +find and fix some 0 ̸= Q1 ∈ Hj with entries in K. +By the definition of Hj and its choice Hj = Hj+1, we have that for any π, π′ ∈ P(L0, 2L(DE)j+1 +0 +) and any +1 ⩽ ν ⩽ n, +S(Q, πν, π′ν, M) ⊆ S(Q1, πν, π′ν, ∞). +Assume that γ ∈ S(Q, πν, π′ν, M) for some π, π′ ∈ P(2L(DE)j +0 +, 2L(DE)j+1 +0 +). Since Q1 has a nonzero entry, +say, Qrs, this implies via the last display that +γ∗ +rQ1γs = | det(γ)| +2 +n Qrs, +where γr, γs stand for the rth and sth column of γ, respectively. Here, the left-hand side is in K, so is the +right-hand side, which implies that | det(γ)|2/n = (ππ)ν(n−1)/n(π′π′)ν/n ∈ K. By the independence of roots +(a theorem of Besicovitch [Bes40]) and that K is defined by nth roots of primes of norm less than 2L(DE)j, +while primes in the definition of Hj+1 have norms at least 2L(DE)j, we see that π = π′ or ν = n, and the +proof is complete. +□ +Lemma 7. Let D ⩾ 2 be arbitrary, and assume that E > Dn2+1. Let M, L1(Ω) ⩾ 2 be given by Lemma 6 +and (11). For any Q ∈ Ω′, and for the corresponding j ⩽ n2 +1 given by Lemma 6, there exists a self-adjoint +matrix Q2 with entries in Q(i) of complexity OΩ((LDk(DE)j +0 +)O(1)) for some k ∈ {0, . . . , n2} such that for any +π, π′ ∈ P(2LDk(DE)j +0 +, 2LDk+1(DE)j +0 +) and any 1 ⩽ ν ⩽ n, +S(Q, πν, π′ν, M) ⊆ S(Q2, πν, π′ν, ∞). +Proof. Define the subspaces +H′ +k := +� +γ∈S(Q,πν,π′ν,M) +π,π′∈P(2LDk(DE)j +0 +,2LDk+1(DE)j +0 +) +1⩽ν⩽n +ker Bγ, +k = 0, . . . , n2 + 1. +Then Sn ⊇ H′ +0 ⊇ . . . ⊇ H′ +n2+1 ⊇ {0} and dim(Sn) = n2 imply that H′ +k = H′ +k+1 for some 1 ⩽ k ⩽ n2. Fix +the smallest such k. +By our choice Dn2+1 < E we have 2L(DE)j +0 +⩽ 2LDk(DE)j +0 +< 2LDk+1(DE)j +0 +< 2L(DE)j+1 +0 +, so we may apply +Lemma 6 to deduce S(Q, πν, π′ν, M) = ∅ unless π = π′ or ν = n. In particular, we have | det(γ)|2/n = +(ππ)ν(n−1)/n(π′π′)ν/n ∈ Z for all γ ∈ S(Q, πν, π′ν, M) in the definition of H′ +k. Therefore all linear maps Bγ +8 + +in the definition of H′ +k are defined over Q(i). As in the proof of Lemma 6, we find a matrix Q2 ∈ int Ω1 ∩ H′ +k +with entries in Q(i) of complexity OΩ((LDk(DE)j +0 +)O(1)) by Lemma 5 such that the conclusion holds. (Note +that this is the point where we use the slightly stronger condition (11) put on M that it exceeds not only +(DE)j · O(1), but Dk(DE)j · O(1).) +□ +Lemma 8. Let D ⩾ 2 be arbitrary, and let E be as in Lemma 7, then M, L1(Ω) as given by Lemma 6. For +any Q ∈ Ω1, let j, k be the numbers given also by Lemmata 6–7. There exists some m ∈ Z[i] of complexity +OΩ((LDk(DE)j +0 +)O(1)) and a matrix U ∈ SLn(Z[i, 1 +m]) with the following properties: +(a) All entries of U and U −1 have complexity OΩ((LDk(DE)j +0 +)O(1)). +(b) Q3 := U ∗Q2U is diagonal with entries in Q of complexity OΩ((LDk(DE)j +0 +)O(1)) and lies in a compact +subset Ω2 ⊂ Pn depending only on Ω. +(c) For any π, π′ ∈ P(2LDk(DE)j +0 +, 2LDk+1(DE)j +0 +) not dividing m, and any 1 ⩽ ν ⩽ n, we have +#S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞). +Proof. We construct U in a way such that Q3 is a Gram–Schmidt orthogonalized form of Q2, i.e. +Q3 = U ∗Q2U, +where U is an +�n +2 +� +long composition of elementary base changes idn +uEℓ1ℓ2 for ℓ1 ̸= ℓ2, where u is chosen to +eliminate the entry at position ℓ1, ℓ2 (note that U is not canonically determined, since we can eliminate the +elements in any order, fix one once for all). Noting that each entry of Q2 has complexity OΩ((LDk(DE)j +0 +)O(1)), +this proves (a) by Lemma 5. +As for (b), first we claim that in a Gram–Schmidt step applied to a positive definite matrix, one diagonal +entry decreases in absolute value, and all the others remain the same. It suffices to check this for 2 × 2 +blocks, where we see the calculation +� 1 +0 +− b +a +1 +� � a +b +−b +c +� � +1 +− b +a +0 +1 +� += +� +a +0 +0 +c − |b|2 +a +� +. +Here, 0 < c − |b|2/a < c, so the claim is proven. +As a result, when we diagonalize any element of Ω1, the resulting diagonal matrix cannot have entry larger +than the largest eigenvalue in Ω1, say, λ. But then the smallest eigenvalue cannot be smaller than ∆/λn−1, +where ∆ is the smallest determinant attained in Ω1. Therefore, any Gram–Schmidt process applied to any +element of Ω1 leads to positive definite matrices with eigevalues between ∆/λn−1 and λ. The set of such +matrices is a good choice for Ω2. +The rationality of Q3 is obvious, since it is self-adjoint and with diagonal entries a priori in Q(i). Then +observe that the matrix Q3 is computed in O(1) many steps from Q2, which verifies via Lemma 5 that the +entries of Q3 are indeed of complexity OΩ((LDk(DE)j +0 +)O(1)). This proves (b). +Also, put m := den(U −1) den(U) (which is den(U)2, since det(U) = 1). Then m can be computed in +O(1) steps, and referring to Lemma 5, we see that the complexity of m is OΩ((LDk(DE)j +0 +)O(1)) as claimed. +In particular, we have mU −1γU ∈ Z[i]n×n for any γ ∈ SLn(Z[i]). +Finally, if π, π′ ∤ m, then the base change given by U does not alter the π- and π′-parts of the Smith +normal form. Hence if γ ∈ S(Q2, πν, π′ν, ∞), then +| det(γ)| +2 +n Q3 = | det(γ)| +2 +n U ∗Q2U = U ∗γ∗Q2γU = (U −1γU)∗U ∗Q2U(U −1γU) = (U −1γU)∗Q3(U −1γU) +shows that U −1γU ∈ Sm(Q3, πν, π′ν, ∞). Since the U-conjugation is a bijection, we obtain (c). +□ +5. The endgame +Let ε > 0 be given as in the input of Proposition 1. Choose then D ⩾ 2 such that Dε/2 > T + 1, where +recall that T ⩾ 1 is an upper bound on all implicit constants in O(1)’s ever mentioned in the paper. Let +then E ⩾ 2 as needed in Lemma 7, and then M ⩾ 2 as implied by Lemma 6. This M will be the M(ε) of +Proposition 1. +9 + +Now let Ω, hence Ω′, Ω1, Ω2 be given. Let L(Ω) be large enough such that on the other hand L(Ω) ⩾ +L1(Ω) of Lemmata 6–7, and on the other hand that for any L0 ⩾ L(Ω), and any m implied by Lemma 8 +(for any possible 1 ⩽ j ⩽ n2 + 1 and 0 ⩽ k ⩽ n2), N(m) < LDk+1(DE)j +0 +, which can be achieved, since +N(m) = OΩ(LDk(DE)j +0 +). This in particular implies π ∤ m for any π ∈ P(LDk+1(DE)j +0 +, 2LDk+1(DE)j +0 +). Note +that at this point none of j, k, m is fixed, but we have the claimed bounds and the non-divisibility relations +on them. +Let then Q ∈ Ω′ and L0 ⩾ L(Ω) be arbitrary. Then there exist some 1 ⩽ j ⩽ n2 + 1 given by Lemma 6, +0 ⩽ k ⩽ n2 and Q2 given by Lemma 7, Q3 and m ∈ Z[i] given by Lemma 8 with the properties given there. +Let then L := LDk+1(DE)j +0 +, which satisfies the magnitude requirement of Proposition 1 that L0 ⩽ L ⩽ Lα(ε) +0 +, +where α(ε) is a constant depending only on ε (we can take α(ε) = Dn2+1(DE)n2+1). For any π, π′ ∈ P(L, 2L) +and any 1 ⩽ ν ⩽ n, we have, combining Lemmata 7–8, that +#S(Q, πν, π′ν, M) ⩽ #S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞). +Therefore, it suffices to estimate the rightmost expression from above. We apply Lemma 4 for π ̸= π′ to +see the count is 0, which proves (6). When π = π′, then we apply Lemma 3 (with ε/2 written in place of ε +there) to see it is +OΩ((LDk(DE)j +0 +)O(1)) · OΩ,ε((LDk+1(DE)j +0 +)ν(n−1)+ ε +2 ) = OΩ,ε(Lν(n−1)+ε), +where the last bound holds by the choice of D and by N(π) < 2L. This proves (5). +The proof of Proposition 1 and hence that of Theorem 1 are complete. +References +[Bes40] +A. S. Besicovitch. On the linear independence of fractional powers of integers. J. London Math. Soc., 15:3–6, 1940. +[BH10] +V. Blomer and G. Harcos. Twisted L-functions over number fields and Hilbert’s eleventh problem. Geom. Funct. +Anal., 20(1):1–52, 2010. +[BHM16] +V. Blomer, G. Harcos, and D. Mili´cevi´c. Bounds for eigenforms on arithmetic hyperbolic 3-manifolds. Duke Math. +J., 165(4):625–659, 2016. +[BHMM20] V. Blomer, G. Harcos, P. Maga, and D. Mili´cevi´c. The sup-norm problem for GL(2) over number fields. J. Eur. +Math. Soc. (JEMS), 22(1):1–53, 2020. +[BM15] +V. Blomer and P. Maga. The sup-norm problem for PGL(4). Int. Math. Res. Not. IMRN, (14):5311–5332, 2015. +[BM16] +V. Blomer and P. Maga. Subconvexity for sup-norms of cusp forms on PGL(n). Selecta Math. (N.S.), 22(3):1269– +1287, 2016. +[BP16] +V. Blomer and A. Pohl. The sup-norm problem on the Siegel modular space of rank two. Amer. J. Math., 138(4):999– +1027, 2016. +[BT20] +F. Brumley and N. Templier. Large values of cusp forms on GLn. Selecta Math. (N.S.), 26(4):Paper No. 63, 71, +2020. +[Gil20] +N. Gillman. Explicit subconvexity savings for sup-norms of cusp forms on PGLn(R). J. Number Theory, 206:46–61, +2020. +[GRS13] +A. Ghosh, A. Reznikov, and P. Sarnak. Nodal domains of Maass forms I. Geom. Funct. Anal., 23(5):1515–1568, +2013. +[IS95] +H. Iwaniec and P. Sarnak. L∞ norms of eigenfunctions of arithmetic surfaces. Ann. of Math. (2), 141(2):301–320, +1995. +[Mar] +S. Marshall. Upper bounds for Maass forms on semisimple groups. Available at https://arxiv.org/abs/1405.7033. +[RS94] +Z. Rudnick and P. Sarnak. The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys., +161(1):195–213, 1994. +[Sar] +P. Sarnak. Letter to Morawetz. Available at http://publications.ias.edu/sites/default/files/Sarnak_Letter_to_Morawetz.pdf. +Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest H-1364, Hungary +Email address: magapeter@gmail.com, gergely.zabradi@ttk.elte.hu +E¨otv¨os Lor´and University, Institute of Mathematics, P´azm´any P´eter s´et´any 1/C, Budapest H-1117, Hungary +Email address: gergely.zabradi@ttk.elte.hu +10 + diff --git a/zdE3T4oBgHgl3EQfPwkv/content/tmp_files/load_file.txt b/zdE3T4oBgHgl3EQfPwkv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..544365fdfc7c4ebb57559064555764748e32af0c --- /dev/null +++ b/zdE3T4oBgHgl3EQfPwkv/content/tmp_files/load_file.txt @@ -0,0 +1,529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf,len=528 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='04405v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='NT] 11 Jan 2023 THE SUP-NORM PROBLEM FOR AUTOMORPHIC CUSP FORMS OF PGL(n, Z[i]) P´ETER MAGA AND GERGELY Z´ABR´ADI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let φ be an L2-normalized Hecke–Maaß cusp form for PGLn(Z[i]) on the locally symmetric space X := PGLn(Z[i])\\PGLn(C)/PUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' If Ω is a compact subset of X, then we prove the bound ∥φ|Ω∥∞ ≪Ω λn(n−1)/4−δ φ for some δ > 0 depending only on n, where λφ is the Laplace eigenvalue of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Introduction Given a Riemannian locally symmetric space X = Γ\\S, it is a classical analytic problem to give pointwise bounds for (L2-normalized) eigenfunctions F of the algebra of invariant differential operators D(S), uniformly in S in terms of the Laplace eigenvalue λF of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' If X is compact, then Sarnak [Sar] proved the baseline bound (1) ∥F∥∞ ≪X λ dim(X)−rk(X) 4 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The exponent on the right-hand side is known to be sharp in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' It is also known in some special cases that if X is not compact, then ∥F∥∞ might be considerably larger (see [BT20]), but (1) still holds if F is retsriced to compact subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The sup-norm problem in the theory of automorphic forms asks if the exponent of (1) can be strengthened if X is an arithmetic manifold and F is an eigenfunction not only of D(S) but of the full Hecke algebra of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' An important motivation comes from quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Classical mechanics interprets a freely moving particle as a geodesic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The quantum mechanical interpretation of the same object is an L2-normalized linear combination of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since the geodesic flow is ergodic with respect to the Liouville measure on the tangent space, the correspondence principle of quantum mechanics suggests that the masses of the eigenstates reproduce the invariant measure in the high-energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Bounds on the sup-norm (or in general, the Lp norm for any p > 2) of eigenstates control their mass concentration, and hence are in connection with the quantum unique ergodicity conjecture of Rudnick and Sarnak [RS94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Other important connections of sup-norm bounds are towards the multiplicity problem, nodal domains of automorphic forms and bounds for L-functions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' [Sar], [GRS13], [BH10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In improving (1), there are (at least) two independent directions of research: one is to find as strong power- savings as possible among special circumstances, typically in rank one (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' [IS95] and [BHMM20]), the other one is to find any power-saving in higher rank or among as general circumstances as possible (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' [BP16], [BM15], [BM16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We pick up the thread in the second theme, where the current limitation of our knowledge is an unpublished a manuscipt [Mar], which proves power-saving for a wide class of sym- metric spaces, more specifically, for arithmetic quotients of quasi-split groups with the exception of the type SU(n, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (We note that even though [Mar] is not peer-reviewed, it is widely accepted in the community as being correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=') In these notes, we introduce a new method, based and improved on that of [BM16], to tackle the sup-norm problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As a test application, we prove a saving over (1) in the case of the locally symmetric space X := Γ\\G/K, Γ := PGLn(Z[i]), G := PGLn(C), K := PUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Admittedly, our result follows from the main result of [Mar] (see especially [Mar, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='4]), but we strongly believe that the novelty of the method deserves attention, and might have the potential to address the type SU(n, n − 1), the exceptional case in Marshall’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 11F55, 11F72, 11D75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' automorphic forms in higher rank, sup-norm problem, Hecke operators, trace formula, diophantine approximation, amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 1 All along the paper, we think of n ⩾ 2 as being fixed, in particular, all implied constants below are allowed to depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Before stating our main result, we fix some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since it is convenient to work with matrices instead of their projectivization, we shall realize elements of G as the rightmost nonzero entry of the bottom row is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We will also talk about points of X or G/K as matrices, by which we mean any matrix which represents them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let A stand for the diagonal subgroup of G consisting of positive real entries and let N be the upper-triangular unipotent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then the Iwasawa decomposition reads as G = NAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We write a for the Lie algebra of A, a∗ for its dual and a∗ C for the complexification of a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Fixing bases in a∗ and a∗ C, we may view them as appropriate subsets of Rn and Cn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let W stand for the Weyl group, Σ for the set of roots and Σ+ for the set of positive roots corresponding to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For α ∈ Σ, let m(α) be the real dimension of the corresponding root space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For λ ∈ a∗, define D(λ) := � α∈Σ+ (1 + |⟨α, λ⟩|)m(α), where ⟨·, ·⟩ is induced by the Killing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In our special situation, define, for 1 ⩽ j ⩽ n, the function ej on a as ej(diag(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , an)) = aj, and then a set of positive roots is given by ej − ek with 1 ⩽ j < k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The corresponding root space is spanned by the matrices Ejk and iEjk, where Ejk is the matrix which 1 at position j, k and otherwise zero, hence m(ej − ek) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' A Hecke–Maaß cusp form φ on X comes with archimedean Langlands parameters µφ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , µn) ∈ a∗ C/W ⊂ Cn/W such that µ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' + µn = 0 and {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , µn} = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , µn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In our parametrization, (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , µn) ∈ Rn corresponds to the tempered spectrum, and we have that ℑµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , ℑµn = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let µ∗ φ stand for ℜµφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since for the Laplace eigenvalue λφ of φ, λφ ≍ 1 + ∥µ1∥2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' + ∥µn∥2 holds, we have D(µ∗ φ) ≪ λ n(n−1) 2 φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The main result of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any n ⩾ 2, there exists some δ = δ(n) > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any L2- normalized Hecke–Maaß cusp form φ on X and any compact Ω ⊂ X, ∥φ|Ω∥∞ ≪Ω D(µ∗ φ) 1 2 −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In particular, ∥φ|Ω∥∞ ≪Ω λ n(n−1) 4 (1−2δ) φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since n(n − 1)/4 = (dim(X) − rk(X))/4, this is a saving over (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We note that the case n = 2 was solved in [BHM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Our method closely follows the one introduced in [BM16], however, the counting problem in this situation is more challenging (because of the too large maximal compact subgroup of G, as it will be briefly exposed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This requires an improvement in the counting techniques, which is the heart of this paper (vaguely speaking, a combination of Lemma 4 and Lemma 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As a by-product, there is no need of Linnik type theorems about small primes in arithmetic progressions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' zero repulsion of L-functions) any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We remark that with some work, the jungle of O(1)’s below (particularly in Section 4) can be made explicit, hence an explicit subconvexity saving is available, see the work [Gil20] over the rational field (in fact, now it is easier, since the reference to Linnik type theorems is removed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We also note that the implied constant (which depends on Ω) can be also made fully explicit, so the method in principle is effective, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' we do not need any reference to Siegel zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We thank Valentin Blomer for useful discussions about the topic of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We also thank Gergely Harcos and Vitezslav Kala for discussions about some closely related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The research towards this work was supported by NKFIH (National Research, Development and In- novation Office) Grants KKP 133819 (PM), FK 135218 (PM), FK 127906 (GZ), K 135885 (GZ), ELKH (E¨otv¨os Lor´and Research Network) Grant SA-71/2021 (PM & GZ), and the MTA R´enyi Int´ezet Lend¨ulet Automorphic Research Group (PM & GZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Reduction to a counting problem In this section, we reduce the problem to a matrix counting problem, following the lines of [BM15] and [BM16], where PGLn(Z)\\PGLn(R)/POn was treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since the spherical Hecke algebra is isomorphic in that case to ours at the archimedean and all relevant non-archimedean places, almost everything follows verbatim, so we give only a brief exposition, with some emphasis on the small difference coming from the fact that the maximal compact subgroup is the orthogonal group in the real case and the unitary group in the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let fµφ : K\\G/K → C be the spherical function constructed in [BM16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='1276–1277] (here we utilize that A, a∗, a∗ C are the same for PGLn(R) and PGLn(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Denoting by ˜fµφ its spherical transform, for any x, y ∈ G, we have the pre-trace formula (2) � ˜fµφ(µ̟)F̟(x)F̟(y) d̟ = � γ∈Γ fµφ(x−1γy), where the integral on the left is meant over the full spectrum of Γ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The idea of amplification in the current setup can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Assume we want to estimate our form φ at a point g ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then in (2), we set x := g and we take y of the form γg, where γ runs through right coset representatives corresponding to certain Hecke operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' An appropriate weighted sum of these results in a positive operator on L2(X), hence from the left-hand side, all but the one term corresponding to F̟ = φ can be dropped (for this positivity argument, see also [BHMM20, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' All in all, we arrive at a bound of the form C|φ(g)|2 ⩽ � γ∈Γ′ fµφ(g−1γg) with Γ′ being a finite set of matrices arising from the Hecke operators utilized, and C is a positive constant coming from ˜fµφ and the corresponding Hecke eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then, using that fµφ decays controllably away from K (see [BP16, Theorem 2]), to bound |φ(g)|, we essentially have to count those γ ∈ Γ′ for which g−1γg is close to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' To be a little more concrete, let L ⩾ 2 be a parameter, and assume that P is a set of primes π lying above distinct split rational primes such that L ⩽ N(π) < 2L with N standing for the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then the discussion of [BM15, Sections 4, 6] (which in fact uses [BP16, Theorem 2], and see also [BM16, Section 2]) leads to, for any number M ⩾ 1, (3) |φ(g)|2 ≪M,Ω D(µ∗ φ) · \uf8eb \uf8ed 1 #P + D(µ∗ φ)−κLK + n � ν=1 1 (#P)2 � π,π′∈P #S(Q, πν, π′ν, M) Lν(n−1) \uf8f6 \uf8f8 , g ∈ Ω, where K is a fixed number depending only on M and n, κ > 0 is also a fixed number depending only on n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' and S(Q, πν, π′ν, M) := � γ ∈ SLn(Z[i]) · diag(1, πν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , πν, πνπ′ν) · SLn(Z[i]) : ∥| det γ|− 2 n · γ∗Qγ − Q∥∞ ⩽ max � N(π)−M, N(π′)−M�� (4) with Q := | det g|2/n · (g∗)−1g−1, and by ∥ · ∥∞ applied to a matrix, we mean its largest entry in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Note that in [BM15] and [BM16], the transpose of γ and g are taken instead of their adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By the prime number theorem, we can choose P to satisfy #P ≫ε L1−ε for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Therefore, if we were able to prove that #S(Q, πν, π′ν, M) ≪Ω,M Lν(n−1)−η holds for every 1 ⩽ ν ⩽ n with some η > 0, then (3) would imply Theorem 1 by setting L to be a very small positive power of D(µ∗ φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We prove it in a weaker form, which still suffices for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Before formulating this weaker statement, we fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Introduce the notation Sn for the real vector space of self-adjoint matrices in Cn×n and Pn ⊂ Sn for the open convex cone of positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Given a compact set Ω ⊂ X as in Theorem 1, we introduce Ω′ := ��1 2, 2 � | det g| 2 n · (g∗)−1g−1 : g ∈ Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 3 Then Ω′ is a compact subset of Pn in the subspace topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let ε > 0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' There exist positive numbers α(ε) ⩾ 1 and M(ε) ⩾ 1 (both depending only on n and ε) with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any compact Ω ⊂ X, there exists a constant L(Ω) ⩾ 2 (depending only on n, α(ε), M(ε) and Ω) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any L0 ⩾ L(Ω), there exists some L0 ⩽ L ⩽ Lα(ε) 0 such that for any prime π lying above a split rational prime and satisfying L ⩽ N(π) < 2L, we have (5) #S(Q, πν, πν, M(ε)) ≪Ω,ε Lν(n−1)+ε, Q ∈ Ω′, 1 ⩽ ν ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' and for any two distinct primes π, π′ lying above distinct split rational primes, we have (6) #S(Q, πν, π′ν, M(ε)) = 0, Q ∈ Ω′, 1 ⩽ ν ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This still suffices for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Indeed, we apply Proposition 1 with any 0 < ε < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' There are implied numbers α := α(ε) ⩾ 1 and M := M(ε) ⩾ 1, and together with the further input Ω, one more number L(Ω) by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then take L0 := D(µ∗ φ)ω, where ω > 0 is a fixed constant to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By Proposition 1, if L0 ⩾ L(Ω), we can find some with D(µ∗ φ)ω ⩽ L ⩽ D(µ∗ φ)αω such that the countings (5) and (6) hold for any primes L ⩽ N(π), N(π′) < 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then in every term of (3), we get a power-saving, as soon as ω > 0 in the beginning is chosen sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now we return to the condition L0 ⩾ L(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Apart from a finite set of Hecke–Maaß cusp forms depending only on Ω, this is indeed satisfied by L0 = D(µ∗ φ)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The finitely many exceptional forms are treated then by adjusting the implied constant (note that the implied constant might depend on Ω, only the saving in the exponent must be absolute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The rest of the paper is hence devoted to the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We conclude this section by illustrating why this counting problem is harder for PGLn(Z[i]) than for PGLn(Z) in the special case when Q is the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then in S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' ), we have a condition on the Smith normal form which is of the same complexity in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' However, the other condition is that γ is projectively equivalent to an orthonormal matrix, which intuitively happens more often over C than over R, since dim(PGLn(C)) = 2n2 − 2, dim(PUn) = n2 − 1 = dim(PGLn(C)) 2 , while dim(PGLn(R)) = n2 − 1, dim(POn) = n2 − n 2 < dim(PGLn(R)) 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' PUn is a “thicker” subgroup of PGLn(C) than POn of PGLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Counting techniques in a special case In this section, we introduce counting techniques for a situation which is very special in many different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' First, we will assume that Q is diagonal, moreover, its entries belong to the base field Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then these together imply that the diagonal entries are rational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Q = diag(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , qn) with q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , qn ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then qj ≍Ω 1 for all 1 ⩽ j ⩽ n, and we also assume that the numerator and the denominator of qj for all 1 ⩽ j ⩽ n (in their simplest form) are both coprime to π, π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Our final simplification is that we allow no error term in (4), which we will denote by writing ∞ in place of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Our convention will be that vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' elements of Z[i]n, Cn, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' are always meant as column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We also introduce the notation vπ(q), for any prime π ∈ Z[i] and any q ∈ Q(i), which denotes the π-valuation of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Occasionally, we may use this notation for vectors or matrices with entries from Q(i), then it means the minimal π-adic valuation attained by the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let π ∈ Z[i] be a prime lying above a split rational prime p = ππ, ρ ∈ N, and let A ∈ Q(i)n×n be a self-adjoint matrix such that vπ(A) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let x = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , ξn)t, y = (υ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , υn)t ∈ Z[i]n be vectors satisfying vπ(x) = vπ(y) = 0 such that for any 1 ⩽ j < k ⩽ n, πρ | (ξjυk − ξkυj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (a) For some a ∈ Z coprime to p, we have y ≡ ax mod πρ, and this a is well-defined modulo pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (b) Choosing a ∈ Z with this property in (a), there also exists a b ∈ Z (unique modulo pρ) with b ≡ ai mod πρ, so we further have (7) 2x∗Ay ≡ (a − bi)x∗Ax + (a′ − b′i)y∗Ay mod pρ, 4 where a′ ∈ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' b′ ∈ Z) stands for the multiplicative inverse of a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' of b) modulo pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (c) We have πρ | a′ − b′i and πρ | a − bi for the integers a, b, a′, b′ defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This is a variant of [BM16, Lemma 3], but the proof in this case is a little more computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Without loss of generality, we may assume that π ∤ ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then π ∤ υn, for if not, then for some 1 ⩽ j ⩽ n − 1, π ∤ υj, and then π ∤ (ξnυj − ξjυn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then a ≡ υnξ−1 n mod πρ does the job, since υj ≡ ξjυnξ−1 n ≡ aξj mod πρ for any 1 ⩽ j ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Also, a can be chosen in Z uniquely modulo pρ since 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , pρ − 1 is a set of representatives modulo πρ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' we have the isomorphism Z/(pρ) ∼= Z[i]/(πρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The proof of (a) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As for (b), fix any representative a ∈ Z and let b ∈ Z be such that b ≡ ai mod πρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then πρ divides all the entries in y − ax whence πρ divides all the entries in (y − ax)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' So we deduce 0 ≡ (y − ax)∗A(y − ax) = y∗Ay + a2x∗Ax − ax∗Ay − ay∗Ax mod pρ = (ππ)ρ hence (8) x∗Ay + y∗Ax ≡ ax∗Ax + a′y∗Ay mod pρ where a′ ∈ Z is a multiplicative inverse of a modulo pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Similarly, πρ divides all the entries in iy − bx and πρ divides all the entries in (iy − bx)∗ as ai ≡ b mod πρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' So we compute 0 ≡ (iy − bx)∗A(iy − bx) = y∗Ay + b2x∗Ax − bix∗Ay + biy∗Ax mod pρ whence (9) x∗Ay − y∗Ax ≡ −bix∗Ax − b′iy∗Ay mod pρ where b′ ∈ Z is a multiplicative inverse of b modulo pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The statement follows by adding equations (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For part (c), we compute πρ | i(b − ai) = ib + a whence πρ | a + bi = a − bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Similarly, πρ | a′b′(b − ai) ≡ a′ − b′i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let A ∈ Ω′, and assume that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , xk ∈ Z[i]n are linearly independent vectors for some 0 ⩽ k ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then for any real number β ⩾ 2, #{y ∈ Z[i]n : y∗Ay = β2, and x∗ jAy = 0 for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , k} ≪Ω,ε β2(n−k−1)+ε for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' See [BM15, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ We slightly extend the notation S(Q, πν, π′ν, ∞) for π, π′ ∤ m ∈ Z[i] as Sm(Q, πν, π′ν, ∞) := � γ ∈ SLn(Z[i]π,π′) · diag(1, πν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , πν, πνπ′ν) · SLn(Z[i]π,π′) : mγ ∈ SLn(Z[i]), | det γ|− 2 n · γ∗Qγ = Q � , where by Z[i]π,π′, we mean the ring of elements a ∈ Q(i) which satisfy vπ(a), vπ′(a) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let π ∈ Z[i] be a prime lying above a split rational prime, and π ∤ m ∈ Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let Q = diag(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , qn) with qj ∈ Q, qj ≍Ω 1 for all 1 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then for any 1 ⩽ ν ⩽ n, #Sm(Q, πν, πν, ∞) ≪Ω,ε |m|2n2−2+ε den(Q) (2n−1)(n−1) 2 N(π)ν(n−1)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Here, den(Q) denotes the least common multiple of the denominator of the diagonal entries of Q (written in simplest form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By the definition of Sm(Q, πν, πν, ∞), any matrix counted there must have a column not completely divisible by π, and we may assume that it is the first column γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since (mγ1)∗Q(mγ1) = |m|2N(π)ν, we have, by Lemma 2 that the number of possible mγ1’s is Oε(|m|2(n−1)+εN(π)ν(n−1)+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now it suffices to prove that fixing γ1, there are OΩ(m2n(n−1) den(Q)(2n−1)(n−1)/2) ways to finish the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We group the possible second columns γ2 according to their π-valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' When vπ(γ2) ⩾ ν, then mγ2/πν ∈ Z[i]n and ∥mγ2/πν∥ ≍Ω ∥mγ2/πν∥Q = |m|, hence there are OΩ(|m|2n) such choices for such a γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now fix 0 ⩽ µ < ν, and then it suffices prove that the number of possible γ2’s satisfying vπ(γ2) = µ is OΩ(|m|2n den(Q)(2n−1)/2), since the same can be repeated for all further columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 5 Let x, y be second columns after the first column γ1 such that vπ(x) = vπ(y) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Consider then the vectors x′ := mx/πµ and y′ := my/πµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By Lemma 1(a), x′ and y′ are both multiples of mγ1 modulo πν−µ, and then by transitivity, multiples of each other modulo πµ−ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Introduce then Q′ := den(Q)Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' With this notation, since x′∗Q′x′ = y′∗Q′y′ = |m|2q2 den(Q)πν−µπν−µ = |m|2q2 den(Q)N(π)ν−µ, (7) gives that x′∗Q′y′ is divisible by N(π)ν−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' On the other hand, viewing Cn as R2n, we obtain, for the Q-angle of x and y that ∢Q(x, y) = arccos ℜ(x∗Qy) � (x∗Qx)(y∗Qy) = arccos ℜ(x′∗Q′y′) � (x′∗Q′x′)(y′∗Q′y′) = arccos ℜ(x′∗Q′y′) |m|2q2 den(Q)N(π)µ−ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since the numerator in the rightmost expression is divisible by N(πµ−ν), this continues, for some ℓ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' }, as ∢Q(x, y) = arccos � 1 − ℓ |m|2q2 den(Q) � ≫Ω |m|−1 den(Q)−1/2, if ℓ ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This means that the number of possible second columns is OΩ(|m|2n−1 den(Q)(2n−1)/2) (this is rather ele- mentary, see also [BM16, Lemma 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let π, π′ ∈ Z[i] be distinct primes lying above distinct split rational primes, and let m ∈ Z[i] such that vπ(m) = vπ′(m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let further Q = diag(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , qn) such that qj ∈ Q and vπ(qj) = vπ′(qj) = 0 for all 1 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then for any 1 ⩽ ν ⩽ n, #Sm(Q, πν, π′ν, ∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' First observe that if 1 ⩽ ν ⩽ n − 1, then | det γ| 2 n = (ππ) ν(n−1) n � π′π′� ν n /∈ Q, hence | det γ|2/nγ∗Qγ ̸= Q, implying #Sm(Q, πν, π′ν, ∞) = 0 in this case, so from now on, we assume ν = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Any γ counted in #Sm(Q, πn, π′n, ∞) must have a column, say, the first one γ1 such that vπ(γ1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now we prove that any choice of γ2 leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let µ := vπ(γ2) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then we apply Lemma 1, in particular, (7) for γ1 and γ′ 2 := γ2/πµ, with the notation p = ππ, a, b ∈ Z coprime to p such that 1 ⩽ a, b, a′, b′ ⩽ pn−µ − 1, γ′ 2 ≡ aγ1 mod πn−µ, b ≡ ai mod πn−µ, a′, b′ are the multiplicative inverse of a, b modulo pn−µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We infer (10) 0 = π−µγ∗ 1Qγ2 = γ∗ 1Qγ′ 2 ≡ (a − bi)pn−1π′π′q1 + (a′ − b′i)pn−1−µπ′π′q2 mod pn−µ, which is immediately a contradiction for µ ⩾ 1, since the second term in the rightmost expression is not divisible by pn−µ, while the first one and the sum are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now assume µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By Lemma 1(c) we have π | a − bi and π | a′ − b′i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' So in (10) the first term on the right-hand side is not divisible by πn (since π ∤ a−bi as p ∤ a−bi), while the second one and the sum are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Exchanging matrices Recall that we want to prove the countings (5) and (6) for N(π), N(π′) ∈ [L, 2L) with an appropriate choice of L ⩾ L0 not exceeding a fixed power of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Our strategy is to switch Q for other matrices with better and better arithmetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Note that a priori, Q is a point on a real manifold with entries that might be highly transcendental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We first informally describe these switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' First we will write Q1 in place of Q in order to guarantee that S(Q, πν, π′ν, M) ⊆ S(Q1, πν, π′ν, ∞) for every admissible choice of π, π′, ν satisfying that N(π), N(π′) are between two fixed powers of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This new Q1 will have entries in a number field K ⊇ Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Using this Q1, we will be able to show that in an appropriate subinterval (again, the norms are between two fixed powers of L0), all the γ’s correspond to π = π′ or ν = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Secondly, in this subinterval, we will find a Q2, this time satisfying that S(Q, πν, π′ν, M) ⊆ S(Q2, πν, π′ν, ∞) for every admissible choice of π, π′, ν satisfying that N(π), N(π′) fall into an even shorter subinterval (again, between two powers of L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This new Q2 will have entries in Q(i), and we will have a control on their height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 6 Finally, we will diagonalize Q2 into Q3, again, with a control on the height of the entries, which necessarily will be rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This diagonalization process will affect the counted γ’s themselves, but not their number, and we will have a good control on the possible denominators of the newly counted γ’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' #S(Q, πν, π′ν, M) ⩽ #S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞) for π, π′, ν as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then to the rightmost count here, we will apply Lemmata 3–4, which in fact will be of the quality of (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now we carry out this plan in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' First we formulate a statement on effective computability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The field of algebraic numbers is denoted by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Given an algebraic number a, we define the complexity of a as comp(a) := inf a= b c b,c∈OQ(a) � max σ∈HomQ(Q(a),C) |σ(b)| + max σ′∈HomQ(Q(a),C) |σ′(c)| � + 1 with OQ(a) standing for the ring of integers of Q(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , am be algebraic numbers and put K := Q(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Assume that f : Q m → Q is a function of m variables that is computed altogether by t additions, subtractions, multiplications and divisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then we have comp(f(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , am)) ⩽ � max 1⩽j⩽m(comp(aj)) �Om,t,deg(K:Q)(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' See the first paragraph of the proof of [BM16, Lemma 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ Include Ω′ in some Ω1 which is still a compact subset of Pn in such a way that Ω′ ⊂ int(Ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We make this inclusion in a well-defined way, say, Ω1 is the set of self-adjoint, positive definite matrices of eigenvalues between a/2 and 2b, where a, b are respectively the smallest and the largest eigenvalues of matrices in Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then dist(Ω′, Pn \\ Ω1) ≫Ω 1, where by the distance of matrices, we mean the one induced by the maximum of the entrywise distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Given any matrix γ ∈ GLn(C), we define the following linear transformation Bγ : Sn → Sn: Bγ(A) := γ∗Aγ − | det(γ)| 2 n A, A ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any 0 < C1 < C2, we introduce the notation P(C1, C2) for the set of primes π ∈ Z[i] satisfying that ℜπ, ℑπ > 0 and C1 ⩽ N(π) < C2 (then in particular, elements of P(C1, C2) lie above distinct split rational primes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Another notation we introduce is T ⩾ 1, a fixed number depending only on n, which exceeds all the implicit constants in O(1)’s ever mentioned in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let D ⩾ 2 and E ⩾ 2 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then there exist some M ⩾ 2 (depending on n, D, E) and L1(Ω) ⩾ 2 (depending on n, D, E, M, Ω) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any Q ∈ Ω′, there exists some 1 ⩽ j ⩽ n2 + 1 satisfying that S(Q, πν, π′ν, M) = ∅ for all π, π′ ∈ P(2L(DE)j 0 , 2L(DE)j+1 0 ), unless π = π′ or ν = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Fix Q ∈ Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any L0 ⩾ 2, consider the subspaces (for the moment, with some unspecified M ⩾ 2) Hj := � γ∈S(Q,πν,π′ν,M) π,π′∈P(L0,2L(DE)j 0 ) 1⩽ν⩽n ker Bγ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , n2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then Sn ⊇ H1 ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' ⊇ Hn2+2 ⊇ {0} and dim(Sn) = n2 imply that Hj = Hj+1 for some 1 ⩽ j ⩽ n2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Fix the smallest such j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since dim Sn −dim Hj ⩽ n2, we in fact obtain Hj by intersecting only n2 many ker Bγ’s, say, Hj = ∩t ℓ=1 ker Bγℓ with some t ⩽ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since each entry of each such γℓ has complexity OΩ((L(DE)j 0 )O(1)), Hj is defined via a system of linear equations with entries from K = Q(i, π1/n 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , π1/n 2n2 ) of complexity OΩ((L(DE)j 0 )O(1)) by Lemma 5 (since the linear system defining Hj can be computed in O(1) steps from the used γ’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 7 By assumption, dist(Q, ker Bγ) ≪ L−M 0 , where by dist, we mean the distance in Sn as a real vector space of dimension n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We claim that this implies that dist(Q, Hj) = OΩ((L(DE)j 0 )O(1)L−M 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Indeed, take basis matrices V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , Vm of the linear span of the (ker Bγℓ)⊥’s such that each Vℓ′ is in one of the ker Bγℓ’s, its entries are in K of complexity OΩ((L(DE)j 0 )O(1)) (this can be done, because such a basis can be computed from the γℓ’s, and then Lemma 5 applies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Such a basis can be orthogonalized by Gram–Schmidt into V ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , V ′ m, and then dist(Q, Hj) = ∥ projH⊥ j (Q)∥ = ����� m � ℓ′=1 ⟨Q, V ′ ℓ′⟩ ⟨V ′ ℓ′, V ′ ℓ′⟩V ′ ℓ′ ����� = ����� m � ℓ′=1 �m ℓ′′=1 Uℓ′ℓ′′⟨Q, Vℓ′′⟩ ⟨V ′ ℓ′, V ′ ℓ′⟩ V ′ ℓ′ ����� , where U is the matrix standing for the Gram–Schmidt process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since ⟨Q, Vℓ′′⟩ ≪ L−M 0 (because for every ℓ′′, we have Vℓ′′ ∈ ker Bγℓ for some ℓ), and all the coefficients (including the entries of U, by Lemma 5) are of complexity OΩ((L(DE)j 0 )O(1)), we indeed see that dist(Q, Hj) = OΩ((L(DE)j 0 )O(1)L−M 0 ), applying Lemma 5 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Now if M is large enough, say, (11) M ⩾ T · (DE)n2+2 + 1, where recall that T ⩾ 1 is an upper bound on all the implicit constants in O(1)’s ever mentioned in the paper, then dist(Q, Hj) = OΩ(L−1 0 ) (since j ⩽ n2 + 1, the exponent n2 + 2 seems to be an overkill, but for a later reference, it is better to force M even a slightly larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then with the convenient choice of L1(Ω) ⩾ 2, if L0 ⩾ L1(Ω), then Hj ∩ int Ω1 ̸= ∅, in particular, Hj ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Further, since Hj is defined over K, we may find and fix some 0 ̸= Q1 ∈ Hj with entries in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By the definition of Hj and its choice Hj = Hj+1, we have that for any π, π′ ∈ P(L0, 2L(DE)j+1 0 ) and any 1 ⩽ ν ⩽ n, S(Q, πν, π′ν, M) ⊆ S(Q1, πν, π′ν, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Assume that γ ∈ S(Q, πν, π′ν, M) for some π, π′ ∈ P(2L(DE)j 0 , 2L(DE)j+1 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since Q1 has a nonzero entry, say, Qrs, this implies via the last display that γ∗ rQ1γs = | det(γ)| 2 n Qrs, where γr, γs stand for the rth and sth column of γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Here, the left-hand side is in K, so is the right-hand side, which implies that | det(γ)|2/n = (ππ)ν(n−1)/n(π′π′)ν/n ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By the independence of roots (a theorem of Besicovitch [Bes40]) and that K is defined by nth roots of primes of norm less than 2L(DE)j, while primes in the definition of Hj+1 have norms at least 2L(DE)j, we see that π = π′ or ν = n, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let D ⩾ 2 be arbitrary, and assume that E > Dn2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let M, L1(Ω) ⩾ 2 be given by Lemma 6 and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any Q ∈ Ω′, and for the corresponding j ⩽ n2 +1 given by Lemma 6, there exists a self-adjoint matrix Q2 with entries in Q(i) of complexity OΩ((LDk(DE)j 0 )O(1)) for some k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , n2} such that for any π, π′ ∈ P(2LDk(DE)j 0 , 2LDk+1(DE)j 0 ) and any 1 ⩽ ν ⩽ n, S(Q, πν, π′ν, M) ⊆ S(Q2, πν, π′ν, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Define the subspaces H′ k := � γ∈S(Q,πν,π′ν,M) π,π′∈P(2LDk(DE)j 0 ,2LDk+1(DE)j 0 ) 1⩽ν⩽n ker Bγ, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' , n2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then Sn ⊇ H′ 0 ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' ⊇ H′ n2+1 ⊇ {0} and dim(Sn) = n2 imply that H′ k = H′ k+1 for some 1 ⩽ k ⩽ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Fix the smallest such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' By our choice Dn2+1 < E we have 2L(DE)j 0 ⩽ 2LDk(DE)j 0 < 2LDk+1(DE)j 0 < 2L(DE)j+1 0 , so we may apply Lemma 6 to deduce S(Q, πν, π′ν, M) = ∅ unless π = π′ or ν = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In particular, we have | det(γ)|2/n = (ππ)ν(n−1)/n(π′π′)ν/n ∈ Z for all γ ∈ S(Q, πν, π′ν, M) in the definition of H′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Therefore all linear maps Bγ 8 in the definition of H′ k are defined over Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As in the proof of Lemma 6, we find a matrix Q2 ∈ int Ω1 ∩ H′ k with entries in Q(i) of complexity OΩ((LDk(DE)j 0 )O(1)) by Lemma 5 such that the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (Note that this is the point where we use the slightly stronger condition (11) put on M that it exceeds not only (DE)j · O(1), but Dk(DE)j · O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=') □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let D ⩾ 2 be arbitrary, and let E be as in Lemma 7, then M, L1(Ω) as given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any Q ∈ Ω1, let j, k be the numbers given also by Lemmata 6–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' There exists some m ∈ Z[i] of complexity OΩ((LDk(DE)j 0 )O(1)) and a matrix U ∈ SLn(Z[i, 1 m]) with the following properties: (a) All entries of U and U −1 have complexity OΩ((LDk(DE)j 0 )O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (b) Q3 := U ∗Q2U is diagonal with entries in Q of complexity OΩ((LDk(DE)j 0 )O(1)) and lies in a compact subset Ω2 ⊂ Pn depending only on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' (c) For any π, π′ ∈ P(2LDk(DE)j 0 , 2LDk+1(DE)j 0 ) not dividing m, and any 1 ⩽ ν ⩽ n, we have #S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We construct U in a way such that Q3 is a Gram–Schmidt orthogonalized form of Q2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Q3 = U ∗Q2U, where U is an �n 2 � long composition of elementary base changes idn +uEℓ1ℓ2 for ℓ1 ̸= ℓ2, where u is chosen to eliminate the entry at position ℓ1, ℓ2 (note that U is not canonically determined, since we can eliminate the elements in any order, fix one once for all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Noting that each entry of Q2 has complexity OΩ((LDk(DE)j 0 )O(1)), this proves (a) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As for (b), first we claim that in a Gram–Schmidt step applied to a positive definite matrix, one diagonal entry decreases in absolute value, and all the others remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' It suffices to check this for 2 × 2 blocks, where we see the calculation � 1 0 − b a 1 � � a b −b c � � 1 − b a 0 1 � = � a 0 0 c − |b|2 a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Here, 0 < c − |b|2/a < c, so the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' As a result, when we diagonalize any element of Ω1, the resulting diagonal matrix cannot have entry larger than the largest eigenvalue in Ω1, say, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' But then the smallest eigenvalue cannot be smaller than ∆/λn−1, where ∆ is the smallest determinant attained in Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Therefore, any Gram–Schmidt process applied to any element of Ω1 leads to positive definite matrices with eigevalues between ∆/λn−1 and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The set of such matrices is a good choice for Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The rationality of Q3 is obvious, since it is self-adjoint and with diagonal entries a priori in Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then observe that the matrix Q3 is computed in O(1) many steps from Q2, which verifies via Lemma 5 that the entries of Q3 are indeed of complexity OΩ((LDk(DE)j 0 )O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This proves (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Also, put m := den(U −1) den(U) (which is den(U)2, since det(U) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then m can be computed in O(1) steps, and referring to Lemma 5, we see that the complexity of m is OΩ((LDk(DE)j 0 )O(1)) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' In particular, we have mU −1γU ∈ Z[i]n×n for any γ ∈ SLn(Z[i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Finally, if π, π′ ∤ m, then the base change given by U does not alter the π- and π′-parts of the Smith normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Hence if γ ∈ S(Q2, πν, π′ν, ∞), then | det(γ)| 2 n Q3 = | det(γ)| 2 n U ∗Q2U = U ∗γ∗Q2γU = (U −1γU)∗U ∗Q2U(U −1γU) = (U −1γU)∗Q3(U −1γU) shows that U −1γU ∈ Sm(Q3, πν, π′ν, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Since the U-conjugation is a bijection, we obtain (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The endgame Let ε > 0 be given as in the input of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Choose then D ⩾ 2 such that Dε/2 > T + 1, where recall that T ⩾ 1 is an upper bound on all implicit constants in O(1)’s ever mentioned in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let then E ⩾ 2 as needed in Lemma 7, and then M ⩾ 2 as implied by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This M will be the M(ε) of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' 9 Now let Ω, hence Ω′, Ω1, Ω2 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let L(Ω) be large enough such that on the other hand L(Ω) ⩾ L1(Ω) of Lemmata 6–7, and on the other hand that for any L0 ⩾ L(Ω), and any m implied by Lemma 8 (for any possible 1 ⩽ j ⩽ n2 + 1 and 0 ⩽ k ⩽ n2), N(m) < LDk+1(DE)j 0 , which can be achieved, since N(m) = OΩ(LDk(DE)j 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This in particular implies π ∤ m for any π ∈ P(LDk+1(DE)j 0 , 2LDk+1(DE)j 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Note that at this point none of j, k, m is fixed, but we have the claimed bounds and the non-divisibility relations on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let then Q ∈ Ω′ and L0 ⩾ L(Ω) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Then there exist some 1 ⩽ j ⩽ n2 + 1 given by Lemma 6, 0 ⩽ k ⩽ n2 and Q2 given by Lemma 7, Q3 and m ∈ Z[i] given by Lemma 8 with the properties given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Let then L := LDk+1(DE)j 0 , which satisfies the magnitude requirement of Proposition 1 that L0 ⩽ L ⩽ Lα(ε) 0 , where α(ε) is a constant depending only on ε (we can take α(ε) = Dn2+1(DE)n2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' For any π, π′ ∈ P(L, 2L) and any 1 ⩽ ν ⩽ n, we have, combining Lemmata 7–8, that #S(Q, πν, π′ν, M) ⩽ #S(Q2, πν, π′ν, ∞) ⩽ #Sm(Q3, πν, π′ν, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Therefore, it suffices to estimate the rightmost expression from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' We apply Lemma 4 for π ̸= π′ to see the count is 0, which proves (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' When π = π′, then we apply Lemma 3 (with ε/2 written in place of ε there) to see it is OΩ((LDk(DE)j 0 )O(1)) · OΩ,ε((LDk+1(DE)j 0 )ν(n−1)+ ε 2 ) = OΩ,ε(Lν(n−1)+ε), where the last bound holds by the choice of D and by N(π) < 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' This proves (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' The proof of Proposition 1 and hence that of Theorem 1 are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' References [Bes40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Besicovitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' On the linear independence of fractional powers of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=', 15:3–6, 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' [BH10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Blomer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Harcos.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Available at http://publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='edu/sites/default/files/Sarnak_Letter_to_Morawetz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content=' Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest H-1364, Hungary Email address: magapeter@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='com, gergely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='zabradi@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='hu E¨otv¨os Lor´and University, Institute of Mathematics, P´azm´any P´eter s´et´any 1/C, Budapest H-1117, Hungary Email address: gergely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='zabradi@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE3T4oBgHgl3EQfPwkv/content/2301.04405v1.pdf'} +page_content='elte.' metadata={'source': 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